module sgexx ! A few LINPACK/LAPACK and supporting BLAS routines ! The routines have all been converted to use the shr_kind_r8 type which ! is set to double precision. The LINPACK naming convention isn't ! followed by the 's' routines (which should contain 'd's). use shr_kind_mod, only: r8 => shr_kind_r8 use cam_logfile, only: iulog use abortutils, only: endrun implicit none public :: & dgeco, & dgedi, & sgefa, & sdot, & isamax, & sasum, & saxpy, & sscal, & sswap, & sgeev, & sgebak, & sgebal, & sgehd2, & sgehrd, & shseqr, & slabad, & slacpy, & sladiv, & slahqr, & slahrd, & slaln2, & slange, & slanhs, & slanv2, & slapy2, & slarf, & slarfb, & slarfg, & slarft, & slarfx, & slartg, & slascl, & slaset, & slassq, & sorg2r, & sorghr, & sorgqr, & strevc, & ilaenv, & lsame, & slamch, & slamc1, & slamc2, & slamc3, & slamc4, & slamc5, & xerbla, & scopy, & sger, & snrm2, & srot, & sgemm, & sgemv, & strmm, & strmv !----------------------------------------------------------------------- contains !----------------------------------------------------------------------- SUBROUTINE DGECO (A,LDA,N,IPVT,RCOND,Z) ! INTEGER LDA,N,IPVT(N) REAL(r8) A(LDA,*),Z(*) REAL(r8) RCOND ! ! SGECO FACTORS A REAL MATRIX BY GAUSSIAN ELIMINATION ! AND ESTIMATES THE CONDITION OF THE MATRIX. ! ! IF RCOND IS NOT NEEDED, SGEFA IS SLIGHTLY FASTER. ! TO SOLVE A*X = B , FOLLOW SGECO BY SGESL. ! TO COMPUTE INVERSE(A)*C , FOLLOW SGECO BY SGESL. ! TO COMPUTE DETERMINANT(A) , FOLLOW SGECO BY SGEDI. ! TO COMPUTE INVERSE(A) , FOLLOW SGECO BY SGEDI. ! ! ON ENTRY ! ! A REAL(LDA, N) ! THE MATRIX TO BE FACTORED. ! ! LDA INTEGER ! THE LEADING DIMENSION OF THE ARRAY A . ! ! N INTEGER ! THE ORDER OF THE MATRIX A . ! ! ON RETURN ! ! A AN UPPER TRIANGULAR MATRIX AND THE MULTIPLIERS ! WHICH WERE USED TO OBTAIN IT. ! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE ! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER ! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR. ! ! IPVT INTEGER(N) ! AN INTEGER VECTOR OF PIVOT INDICES. ! ! RCOND REAL ! AN ESTIMATE OF THE RECIPROCAL CONDITION OF A . ! FOR THE SYSTEM A*X = B , RELATIVE PERTURBATIONS ! IN A AND B OF SIZE EPSILON MAY CAUSE ! RELATIVE PERTURBATIONS IN X OF SIZE EPSILON/RCOND . ! IF RCOND IS SO SMALL THAT THE LOGICAL EXPRESSION ! 1.0 + RCOND .EQ. 1.0 ! IS TRUE, THEN A MAY BE SINGULAR TO WORKING ! PRECISION. IN PARTICULAR, RCOND IS ZERO IF ! EXACT SINGULARITY IS DETECTED OR THE ESTIMATE ! UNDERFLOWS. ! ! Z REAL(N) ! A WORK VECTOR WHOSE CONTENTS ARE USUALLY UNIMPORTANT. ! IF A IS CLOSE TO A SINGULAR MATRIX, THEN Z IS ! AN APPROXIMATE NULL VECTOR IN THE SENSE THAT ! NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . ! ! LINPACK. THIS VERSION DATED 08/14/78 . ! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB. ! ! SUBROUTINES AND FUNCTIONS ! ! LINPACK SGEFA ! BLAS SAXPY,SDOT,SSCAL,SASUM ! FORTRAN ABS,MAX,SIGN ! ! INTERNAL VARIABLES ! REAL(r8) EK,T,WK,WKM REAL(r8) ANORM,S,SM,YNORM INTEGER INFO,J,K,KB,KP1,L ! ! ! ! COMPUTE 1-NORM OF A ! ANORM = 0.0E0_r8 DO 10 J = 1, N ANORM = MAX(ANORM,SASUM(N,A(1,J),1)) 10 CONTINUE ! ! FACTOR ! CALL SGEFA(A,LDA,N,IPVT,INFO) ! ! RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) . ! ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E . ! TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE ! CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE ! TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID ! OVERFLOW. ! ! SOLVE TRANS(U)*W = E ! EK = 1.0E0_r8 DO 20 J = 1, N Z(J) = 0.0E0_r8 20 CONTINUE DO 100 K = 1, N IF (Z(K) .NE. 0.0E0_r8) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. ABS(A(K,K))) GO TO 30 S = ABS(A(K,K))/ABS(EK-Z(K)) CALL SSCAL(N,S,Z,1) EK = S*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = ABS(WK) SM = ABS(WKM) IF (A(K,K) .EQ. 0.0E0_r8) GO TO 40 WK = WK/A(K,K) WKM = WKM/A(K,K) GO TO 50 40 CONTINUE WK = 1.0E0_r8 WKM = 1.0E0_r8 50 CONTINUE KP1 = K + 1 IF (KP1 .GT. N) GO TO 90 DO 60 J = KP1, N SM = SM + ABS(Z(J)+WKM*A(K,J)) Z(J) = Z(J) + WK*A(K,J) S = S + ABS(Z(J)) 60 CONTINUE IF (S .GE. SM) GO TO 80 T = WKM - WK WK = WKM DO 70 J = KP1, N Z(J) = Z(J) + T*A(K,J) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0E0_r8/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) ! ! SOLVE TRANS(L)*Y = W ! DO 120 KB = 1, N K = N + 1 - KB IF (K .LT. N) Z(K) = Z(K) + SDOT(N-K,A(K+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0_r8) GO TO 110 S = 1.0E0_r8/ABS(Z(K)) CALL SSCAL(N,S,Z,1) 110 CONTINUE L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T 120 CONTINUE S = 1.0E0_r8/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) ! YNORM = 1.0E0_r8 ! ! SOLVE L*V = Y ! DO 140 K = 1, N L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T IF (K .LT. N) CALL SAXPY(N-K,T,A(K+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0_r8) GO TO 130 S = 1.0E0_r8/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 130 CONTINUE 140 CONTINUE S = 1.0E0_r8/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM ! ! SOLVE U*Z = V ! DO 160 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 150 S = ABS(A(K,K))/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 150 CONTINUE IF (A(K,K) .NE. 0.0E0_r8) Z(K) = Z(K)/A(K,K) IF (A(K,K) .EQ. 0.0E0_r8) Z(K) = 1.0E0_r8 T = -Z(K) CALL SAXPY(K-1,T,A(1,K),1,Z(1),1) 160 CONTINUE ! MAKE ZNORM = 1.0 S = 1.0E0_r8/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM ! IF (ANORM .NE. 0.0E0_r8) RCOND = YNORM/ANORM IF (ANORM .EQ. 0.0E0_r8) RCOND = 0.0E0_r8 RETURN END SUBROUTINE DGECO !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE DGEDI (A,LDA,N,IPVT,DET,WORK,JOB) INTEGER LDA,N,IPVT(N),JOB REAL(r8) A(LDA,*),DET(2),WORK(*) ! ! SGEDI COMPUTES THE DETERMINANT AND INVERSE OF A MATRIX ! USING THE FACTORS COMPUTED BY SGECO OR SGEFA. ! ! ON ENTRY ! ! A REAL(LDA, N) ! THE OUTPUT FROM SGECO OR SGEFA. ! ! LDA INTEGER ! THE LEADING DIMENSION OF THE ARRAY A . ! ! N INTEGER ! THE ORDER OF THE MATRIX A . ! ! IPVT INTEGER(N) ! THE PIVOT VECTOR FROM SGECO OR SGEFA. ! ! WORK REAL(N) ! WORK VECTOR. CONTENTS DESTROYED. ! ! JOB INTEGER ! = 11 BOTH DETERMINANT AND INVERSE. ! = 01 INVERSE ONLY. ! = 10 DETERMINANT ONLY. ! ! ON RETURN ! ! A INVERSE OF ORIGINAL MATRIX IF REQUESTED. ! OTHERWISE UNCHANGED. ! ! DET REAL(2) ! DETERMINANT OF ORIGINAL MATRIX IF REQUESTED. ! OTHERWISE NOT REFERENCED. ! DETERMINANT = DET(1) * 10.0**DET(2) ! WITH 1.0 .LE. ABS(DET(1)) .LT. 10.0 ! OR DET(1) .EQ. 0.0 . ! ! ERROR CONDITION ! ! A DIVISION BY ZERO WILL OCCUR IF THE INPUT FACTOR CONTAINS ! A ZERO ON THE DIAGONAL AND THE INVERSE IS REQUESTED. ! IT WILL NOT OCCUR IF THE SUBROUTINES ARE CALLED CORRECTLY ! AND IF SGECO HAS SET RCOND .GT. 0.0 OR SGEFA HAS SET ! INFO .EQ. 0 . ! ! LINPACK. THIS VERSION DATED 08/14/78 . ! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB. ! ! SUBROUTINES AND FUNCTIONS ! ! BLAS SAXPY,SSCAL,SSWAP ! FORTRAN ABS,MOD ! ! INTERNAL VARIABLES ! REAL(r8) T REAL(r8) TEN INTEGER I,J,K,KB,KP1,L,NM1 ! ! ! ! COMPUTE DETERMINANT ! IF (JOB/10 .EQ. 0) GO TO 70 DET(1) = 1.0E0_r8 DET(2) = 0.0E0_r8 TEN = 10.0E0_r8 DO 50 I = 1, N IF (IPVT(I) .NE. I) DET(1) = -DET(1) DET(1) = A(I,I)*DET(1) ! ...EXIT IF (DET(1) .EQ. 0.0E0_r8) GO TO 60 10 IF (ABS(DET(1)) .GE. 1.0E0_r8) GO TO 20 DET(1) = TEN*DET(1) DET(2) = DET(2) - 1.0E0_r8 GO TO 10 20 CONTINUE 30 IF (ABS(DET(1)) .LT. TEN) GO TO 40 DET(1) = DET(1)/TEN DET(2) = DET(2) + 1.0E0_r8 GO TO 30 40 CONTINUE 50 CONTINUE 60 CONTINUE 70 CONTINUE ! ! COMPUTE INVERSE(U) ! IF (MOD(JOB,10) .EQ. 0) GO TO 150 DO 100 K = 1, N A(K,K) = 1.0E0_r8/A(K,K) T = -A(K,K) CALL SSCAL(K-1,T,A(1,K),1) KP1 = K + 1 IF (N .LT. KP1) GO TO 90 DO 80 J = KP1, N T = A(K,J) A(K,J) = 0.0E0_r8 CALL SAXPY(K,T,A(1,K),1,A(1,J),1) 80 CONTINUE 90 CONTINUE 100 CONTINUE ! ! FORM INVERSE(U)*INVERSE(L) ! NM1 = N - 1 IF (NM1 .LT. 1) GO TO 140 DO 130 KB = 1, NM1 K = N - KB KP1 = K + 1 DO 110 I = KP1, N WORK(I) = A(I,K) A(I,K) = 0.0E0_r8 110 CONTINUE DO 120 J = KP1, N T = WORK(J) CALL SAXPY(N,T,A(1,J),1,A(1,K),1) 120 CONTINUE L = IPVT(K) IF (L .NE. K) CALL SSWAP(N,A(1,K),1,A(1,L),1) 130 CONTINUE 140 CONTINUE 150 CONTINUE RETURN END SUBROUTINE DGEDI !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEFA (A,LDA,N,IPVT,INFO) INTEGER LDA,N,IPVT(N),INFO REAL(r8) A(LDA,*) ! ! SGEFA FACTORS A REAL MATRIX BY GAUSSIAN ELIMINATION. ! ! SGEFA IS USUALLY CALLED BY SGECO, BUT IT CAN BE CALLED ! DIRECTLY WITH A SAVING IN TIME IF RCOND IS NOT NEEDED. ! (TIME FOR SGECO) = (1 + 9/N)*(TIME FOR SGEFA) . ! ! ON ENTRY ! ! A REAL(LDA, N) ! THE MATRIX TO BE FACTORED. ! ! LDA INTEGER ! THE LEADING DIMENSION OF THE ARRAY A . ! ! N INTEGER ! THE ORDER OF THE MATRIX A . ! ! ON RETURN ! ! A AN UPPER TRIANGULAR MATRIX AND THE MULTIPLIERS ! WHICH WERE USED TO OBTAIN IT. ! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE ! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER ! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR. ! ! IPVT INTEGER(N) ! AN INTEGER VECTOR OF PIVOT INDICES. ! ! INFO INTEGER ! = 0 NORMAL VALUE. ! = K IF U(K,K) .EQ. 0.0 . THIS IS NOT AN ERROR ! CONDITION FOR THIS SUBROUTINE, BUT IT DOES ! INDICATE THAT SGESL OR SGEDI WILL DIVIDE BY ZERO ! IF CALLED. USE RCOND IN SGECO FOR A RELIABLE ! INDICATION OF SINGULARITY. ! ! LINPACK. THIS VERSION DATED 08/14/78 . ! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB. ! ! SUBROUTINES AND FUNCTIONS ! ! BLAS SAXPY,SSCAL,ISAMAX ! ! INTERNAL VARIABLES ! REAL(r8) T INTEGER J,K,KP1,L,NM1 ! ! ! ! GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING ! INFO = 0 NM1 = N - 1 IF (NM1 .LT. 1) GO TO 70 DO 60 K = 1, NM1 KP1 = K + 1 ! ! FIND L = PIVOT INDEX ! L = ISAMAX(N-K+1,A(K,K),1) + K - 1 IPVT(K) = L ! ! ZERO PIVOT IMPLIES THIS COLUMN ALREADY TRIANGULARIZED ! IF (A(L,K) .EQ. 0.0E0_r8) GO TO 40 ! ! INTERCHANGE IF NECESSARY ! IF (L .EQ. K) GO TO 10 T = A(L,K) A(L,K) = A(K,K) A(K,K) = T 10 CONTINUE ! ! COMPUTE MULTIPLIERS ! T = -1.0E0_r8/A(K,K) CALL SSCAL(N-K,T,A(K+1,K),1) ! ! ROW ELIMINATION WITH COLUMN INDEXING ! DO 30 J = KP1, N T = A(L,J) IF (L .EQ. K) GO TO 20 A(L,J) = A(K,J) A(K,J) = T 20 CONTINUE CALL SAXPY(N-K,T,A(K+1,K),1,A(K+1,J),1) 30 CONTINUE GO TO 50 40 CONTINUE INFO = K 50 CONTINUE 60 CONTINUE 70 CONTINUE IPVT(N) = N IF (A(N,N) .EQ. 0.0E0_r8) INFO = N RETURN END SUBROUTINE SGEFA !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SDOT(N,SX,INCX,SY,INCY) ! ! FORMS THE DOT PRODUCT OF TWO VECTORS. ! USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE. ! JACK DONGARRA, LINPACK, 3/11/78. ! ! real(r8) sdot REAL(r8) SX(*),SY(*),STEMP INTEGER I,INCX,INCY,IX,IY,M,MP1,N ! STEMP = 0.0E0_r8 SDOT = 0.0E0_r8 IF(N.LE.0)RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 ! ! CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS ! NOT EQUAL TO 1 ! IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N STEMP = STEMP + SX(IX)*SY(IY) IX = IX + INCX IY = IY + INCY 10 CONTINUE SDOT = STEMP RETURN ! ! CODE FOR BOTH INCREMENTS EQUAL TO 1 ! ! ! CLEAN-UP LOOP ! 20 M = MOD(N,5) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M STEMP = STEMP + SX(I)*SY(I) 30 CONTINUE IF( N .LT. 5 ) GO TO 60 40 MP1 = M + 1 DO 50 I = MP1,N,5 STEMP = STEMP + SX(I)*SY(I) + SX(I + 1)*SY(I + 1) + & SX(I + 2)*SY(I + 2) + SX(I + 3)*SY(I + 3) + SX(I + 4)*SY(I + 4) 50 CONTINUE 60 SDOT = STEMP RETURN END FUNCTION SDOT !--------1---------2---------3---------4---------5---------6---------7---------8 INTEGER FUNCTION ISAMAX(N,SX,INCX) ! ! FINDS THE INDEX OF ELEMENT HAVING MAX. ABSOLUTE VALUE. ! JACK DONGARRA, LINPACK, 3/11/78. ! REAL(r8) SX(*),SMAX INTEGER I,INCX,IX,N ! ISAMAX = 0 IF( N .LT. 1 ) RETURN ISAMAX = 1 IF(N.EQ.1)RETURN IF(INCX.EQ.1)GO TO 20 ! ! CODE FOR INCREMENT NOT EQUAL TO 1 ! IX = 1 SMAX = ABS(SX(1)) IX = IX + INCX DO 10 I = 2,N IF(ABS(SX(IX)).LE.SMAX) GO TO 5 ISAMAX = I SMAX = ABS(SX(IX)) 5 IX = IX + INCX 10 CONTINUE RETURN ! ! CODE FOR INCREMENT EQUAL TO 1 ! 20 SMAX = ABS(SX(1)) DO 30 I = 2,N IF(ABS(SX(I)).LE.SMAX) GO TO 30 ISAMAX = I SMAX = ABS(SX(I)) 30 CONTINUE RETURN END FUNCTION ISAMAX !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SASUM(N,SX,INCX) ! ! TAKES THE SUM OF THE ABSOLUTE VALUES. ! USES UNROLLED LOOPS FOR INCREMENT EQUAL TO ONE. ! JACK DONGARRA, LINPACK, 3/11/78. ! real(r8) sasum REAL(r8) SX(*),STEMP INTEGER I,INCX,M,MP1,N,NINCX ! SASUM = 0.0E0_r8 STEMP = 0.0E0_r8 IF(N.LE.0)RETURN IF(INCX.EQ.1)GO TO 20 ! ! CODE FOR INCREMENT NOT EQUAL TO 1 ! NINCX = N*INCX DO 10 I = 1,NINCX,INCX STEMP = STEMP + ABS(SX(I)) 10 CONTINUE SASUM = STEMP RETURN ! ! CODE FOR INCREMENT EQUAL TO 1 ! ! ! CLEAN-UP LOOP ! 20 M = MOD(N,6) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M STEMP = STEMP + ABS(SX(I)) 30 CONTINUE IF( N .LT. 6 ) GO TO 60 40 MP1 = M + 1 DO 50 I = MP1,N,6 STEMP = STEMP + ABS(SX(I)) + ABS(SX(I + 1)) + ABS(SX(I + 2)) & + ABS(SX(I + 3)) + ABS(SX(I + 4)) + ABS(SX(I + 5)) 50 CONTINUE 60 SASUM = STEMP RETURN END FUNCTION SASUM !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SAXPY(N,SA,SX,INCX,SY,INCY) ! ! CONSTANT TIMES A VECTOR PLUS A VECTOR. ! USES UNROLLED LOOP FOR INCREMENTS EQUAL TO ONE. ! JACK DONGARRA, LINPACK, 3/11/78. ! REAL(r8) SX(*),SY(*),SA INTEGER I,INCX,INCY,IX,IY,M,MP1,N ! IF(N.LE.0)RETURN IF (SA .EQ. 0.0_r8) RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 ! ! CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS ! NOT EQUAL TO 1 ! IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N SY(IY) = SY(IY) + SA*SX(IX) IX = IX + INCX IY = IY + INCY 10 CONTINUE RETURN ! ! CODE FOR BOTH INCREMENTS EQUAL TO 1 ! ! ! CLEAN-UP LOOP ! 20 M = MOD(N,4) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M SY(I) = SY(I) + SA*SX(I) 30 CONTINUE IF( N .LT. 4 ) RETURN 40 MP1 = M + 1 DO 50 I = MP1,N,4 SY(I) = SY(I) + SA*SX(I) SY(I + 1) = SY(I + 1) + SA*SX(I + 1) SY(I + 2) = SY(I + 2) + SA*SX(I + 2) SY(I + 3) = SY(I + 3) + SA*SX(I + 3) 50 CONTINUE RETURN END SUBROUTINE SAXPY !--------1---------2---------3---------4---------5---------6---------7---------8 subroutine sscal(n,sa,sx,incx) ! ! scales a vector by a constant. ! uses unrolled loops for increment equal to 1. ! jack dongarra, linpack, 3/11/78. ! modified 3/93 to return if incx .le. 0. ! modified 12/3/93, array(1) declarations changed to array(*) ! integer i,incx,m,mp1,n,nincx real(r8) sa,sx(*) ! if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 ! ! code for increment not equal to 1 ! nincx = n*incx do 10 i = 1,nincx,incx sx(i) = sa*sx(i) 10 continue return ! ! code for increment equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m sx(i) = sa*sx(i) 30 continue if( n .lt. 5 ) return 40 mp1 = m + 1 do 50 i = mp1,n,5 sx(i) = sa*sx(i) sx(i + 1) = sa*sx(i + 1) sx(i + 2) = sa*sx(i + 2) sx(i + 3) = sa*sx(i + 3) sx(i + 4) = sa*sx(i + 4) 50 continue return end subroutine sscal !--------1---------2---------3---------4---------5---------6---------7---------8 subroutine sswap (n,sx,incx,sy,incy) ! ! interchanges two vectors. ! uses unrolled loops for increments equal to 1. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! real(r8) sx(*),sy(*),stemp integer i,incx,incy,ix,iy,m,mp1,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments not equal ! to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n stemp = sx(ix) sx(ix) = sy(iy) sy(iy) = stemp ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,3) if( m .eq. 0 ) go to 40 do 30 i = 1,m stemp = sx(i) sx(i) = sy(i) sy(i) = stemp 30 continue if( n .lt. 3 ) return 40 mp1 = m + 1 do 50 i = mp1,n,3 stemp = sx(i) sx(i) = sy(i) sy(i) = stemp stemp = sx(i + 1) sx(i + 1) = sy(i + 1) sy(i + 1) = stemp stemp = sx(i + 2) sx(i + 2) = sy(i + 2) sy(i + 2) = stemp 50 continue return end subroutine sswap !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, & LDVR, WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDVL, LDVR, LWORK, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), & WI( * ), WORK( * ), WR( * ) ! .. ! ! Purpose ! ======= ! ! SGEEV computes for an N-by-N real nonsymmetric matrix A, the ! eigenvalues and, optionally, the left and/or right eigenvectors. ! ! The right eigenvector v(j) of A satisfies ! A * v(j) = lambda(j) * v(j) ! where lambda(j) is its eigenvalue. ! The left eigenvector u(j) of A satisfies ! u(j)**H * A = lambda(j) * u(j)**H ! where u(j)**H denotes the conjugate transpose of u(j). ! ! The computed eigenvectors are normalized to have Euclidean norm ! equal to 1 and largest component real. ! ! Arguments ! ========= ! ! JOBVL (input) CHARACTER*1 ! = 'N': left eigenvectors of A are not computed; ! = 'V': left eigenvectors of A are computed. ! ! JOBVR (input) CHARACTER*1 ! = 'N': right eigenvectors of A are not computed; ! = 'V': right eigenvectors of A are computed. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the N-by-N matrix A. ! On exit, A has been overwritten. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! WR (output) REAL array, dimension (N) ! WI (output) REAL array, dimension (N) ! WR and WI contain the real and imaginary parts, ! respectively, of the computed eigenvalues. Complex ! conjugate pairs of eigenvalues appear consecutively ! with the eigenvalue having the positive imaginary part ! first. ! ! VL (output) REAL array, dimension (LDVL,N) ! If JOBVL = 'V', the left eigenvectors u(j) are stored one ! after another in the columns of VL, in the same order ! as their eigenvalues. ! If JOBVL = 'N', VL is not referenced. ! If the j-th eigenvalue is real, then u(j) = VL(:,j), ! the j-th column of VL. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and ! u(j+1) = VL(:,j) - i*VL(:,j+1). ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= 1; if ! JOBVL = 'V', LDVL >= N. ! ! VR (output) REAL array, dimension (LDVR,N) ! If JOBVR = 'V', the right eigenvectors v(j) are stored one ! after another in the columns of VR, in the same order ! as their eigenvalues. ! If JOBVR = 'N', VR is not referenced. ! If the j-th eigenvalue is real, then v(j) = VR(:,j), ! the j-th column of VR. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and ! v(j+1) = VR(:,j) - i*VR(:,j+1). ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= 1; if ! JOBVR = 'V', LDVR >= N. ! ! WORK (workspace/output) REAL array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,3*N), and ! if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good ! performance, LWORK must generally be larger. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the QR algorithm failed to compute all the ! eigenvalues, and no eigenvectors have been computed; ! elements i+1:N of WR and WI contain eigenvalues which ! have converged. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E0_r8, ONE = 1.0E0_r8 ) ! .. ! .. Local Scalars .. LOGICAL SCALEA, WANTVL, WANTVR CHARACTER SIDE INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K, & MAXB, MAXWRK, MINWRK, NOUT REAL(r8) ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, & SN ! .. ! .. Local Arrays .. LOGICAL SELECT( 1 ) REAL(r8) DUM( 1 ) ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 WANTVL = LSAME( JOBVL, 'V' ) WANTVR = LSAME( JOBVR, 'V' ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN INFO = -9 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN INFO = -11 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. ! HSWORK refers to the workspace preferred by SHSEQR, as ! calculated below. HSWORK is computed assuming ILO=1 and IHI=N, ! the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN MAXWRK = 2*N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN MINWRK = MAX( 1, 3*N ) MAXB = MAX( ILAENV( 8, 'SHSEQR', 'EN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'SHSEQR', 'EN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+1, N+HSWORK ) ELSE MINWRK = MAX( 1, 4*N ) MAXWRK = MAX( MAXWRK, 2*N+( N-1 )* & ILAENV( 1, 'SORGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'SHSEQR', 'SV', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'SHSEQR', 'SV', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+1, N+HSWORK ) MAXWRK = MAX( MAXWRK, 4*N ) END IF WORK( 1 ) = MAXWRK END IF IF( LWORK.LT.MINWRK ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEEV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = SLAMCH( 'P' ) SMLNUM = SLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = SLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) & CALL SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) ! ! Balance the matrix ! (Workspace: need N) ! IBAL = 1 CALL SGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) ! ! Reduce to upper Hessenberg form ! (Workspace: need 3*N, prefer 2*N+N*NB) ! ITAU = IBAL + N IWRK = ITAU + N CALL SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! IF( WANTVL ) THEN ! ! Want left eigenvectors ! Copy Householder vectors to VL ! SIDE = 'L' CALL SLACPY( 'L', N, N, A, LDA, VL, LDVL ) ! ! Generate orthogonal matrix in VL ! (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL SORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VL ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! IF( WANTVR ) THEN ! ! Want left and right eigenvectors ! Copy Schur vectors to VR ! SIDE = 'B' CALL SLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF ! ELSE IF( WANTVR ) THEN ! ! Want right eigenvectors ! Copy Householder vectors to VR ! SIDE = 'R' CALL SLACPY( 'L', N, N, A, LDA, VR, LDVR ) ! ! Generate orthogonal matrix in VR ! (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL SORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VR ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! ELSE ! ! Compute eigenvalues only ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL SHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF ! ! If INFO > 0 from SHSEQR, then quit ! IF( INFO.GT.0 ) & GO TO 50 ! IF( WANTVL .OR. WANTVR ) THEN ! ! Compute left and/or right eigenvectors ! (Workspace: need 4*N) ! CALL STREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, & N, NOUT, WORK( IWRK ), IERR ) END IF ! IF( WANTVL ) THEN ! ! Undo balancing of left eigenvectors ! (Workspace: need N) ! CALL SGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL, & IERR ) ! ! Normalize left eigenvectors and make largest component real ! DO 20 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / SNRM2( N, VL( 1, I ), 1 ) CALL SSCAL( N, SCL, VL( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / SLAPY2( SNRM2( N, VL( 1, I ), 1 ), & SNRM2( N, VL( 1, I+1 ), 1 ) ) CALL SSCAL( N, SCL, VL( 1, I ), 1 ) CALL SSCAL( N, SCL, VL( 1, I+1 ), 1 ) DO 10 K = 1, N WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2 10 CONTINUE K = ISAMAX( N, WORK( IWRK ), 1 ) CALL SLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) CALL SROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN ) VL( K, I+1 ) = ZERO END IF 20 CONTINUE END IF ! IF( WANTVR ) THEN ! ! Undo balancing of right eigenvectors ! (Workspace: need N) ! CALL SGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR, & IERR ) ! ! Normalize right eigenvectors and make largest component real ! DO 40 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / SNRM2( N, VR( 1, I ), 1 ) CALL SSCAL( N, SCL, VR( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / SLAPY2( SNRM2( N, VR( 1, I ), 1 ), & SNRM2( N, VR( 1, I+1 ), 1 ) ) CALL SSCAL( N, SCL, VR( 1, I ), 1 ) CALL SSCAL( N, SCL, VR( 1, I+1 ), 1 ) DO 30 K = 1, N WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2 30 CONTINUE K = ISAMAX( N, WORK( IWRK ), 1 ) CALL SLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) CALL SROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN ) VR( K, I+1 ) = ZERO END IF 40 CONTINUE END IF ! ! Undo scaling if necessary ! 50 CONTINUE IF( SCALEA ) THEN CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) IF( INFO.GT.0 ) THEN CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, & IERR ) CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, & IERR ) END IF END IF ! WORK( 1 ) = MAXWRK RETURN ! ! End of SGEEV ! END SUBROUTINE SGEEV !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, & INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB, SIDE INTEGER IHI, ILO, INFO, LDV, M, N ! .. ! .. Array Arguments .. REAL(r8) V( LDV, * ), SCALE( * ) ! .. ! ! Purpose ! ======= ! ! SGEBAK forms the right or left eigenvectors of a real general matrix ! by backward transformation on the computed eigenvectors of the ! balanced matrix output by SGEBAL. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the type of backward transformation required: ! = 'N', do nothing, return immediately; ! = 'P', do backward transformation for permutation only; ! = 'S', do backward transformation for scaling only; ! = 'B', do backward transformations for both permutation and ! scaling. ! JOB must be the same as the argument JOB supplied to SGEBAL. ! ! SIDE (input) CHARACTER*1 ! = 'R': V contains right eigenvectors; ! = 'L': V contains left eigenvectors. ! ! N (input) INTEGER ! The number of rows of the matrix V. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! The integers ILO and IHI determined by SGEBAL. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! SCALE (input) REAL array, dimension (N) ! Details of the permutation and scaling factors, as returned ! by SGEBAL. ! ! M (input) INTEGER ! The number of columns of the matrix V. M >= 0. ! ! V (input/output) REAL array, dimension (LDV,M) ! On entry, the matrix of right or left eigenvectors to be ! transformed, as returned by SHSEIN or STREVC. ! On exit, V is overwritten by the transformed eigenvectors. ! ! LDV (input) INTEGER ! The leading dimension of the array V. LDV >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE PARAMETER ( ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. LOGICAL LEFTV, RIGHTV INTEGER I, II, K REAL(r8) S ! .. ! .. Executable Statements .. ! ! Decode and Test the input parameters ! RIGHTV = LSAME( SIDE, 'R' ) LEFTV = LSAME( SIDE, 'L' ) ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( M.LT.0 ) THEN INFO = -7 ELSE IF( LDV.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEBAK', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( M.EQ.0 ) & RETURN IF( LSAME( JOB, 'N' ) ) & RETURN ! IF( ILO.EQ.IHI ) & GO TO 30 ! ! Backward balance ! IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN ! IF( RIGHTV ) THEN DO 10 I = ILO, IHI S = SCALE( I ) CALL SSCAL( M, S, V( I, 1 ), LDV ) 10 CONTINUE END IF ! IF( LEFTV ) THEN DO 20 I = ILO, IHI S = ONE / SCALE( I ) CALL SSCAL( M, S, V( I, 1 ), LDV ) 20 CONTINUE END IF ! END IF ! ! Backward permutation ! ! For I = ILO-1 step -1 until 1, ! IHI+1 step 1 until N do -- ! 30 CONTINUE IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN IF( RIGHTV ) THEN DO 40 II = 1, N I = II IF( I.GE.ILO .AND. I.LE.IHI ) & GO TO 40 IF( I.LT.ILO ) & I = ILO - II K = SCALE( I ) IF( K.EQ.I ) & GO TO 40 CALL SSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 40 CONTINUE END IF ! IF( LEFTV ) THEN DO 50 II = 1, N I = II IF( I.GE.ILO .AND. I.LE.IHI ) & GO TO 50 IF( I.LT.ILO ) & I = ILO - II K = SCALE( I ) IF( K.EQ.I ) & GO TO 50 CALL SSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 50 CONTINUE END IF END IF ! RETURN ! ! End of SGEBAK ! END SUBROUTINE SGEBAK !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB INTEGER IHI, ILO, INFO, LDA, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), SCALE( * ) ! .. ! ! Purpose ! ======= ! ! SGEBAL balances a general real matrix A. This involves, first, ! permuting A by a similarity transformation to isolate eigenvalues ! in the first 1 to ILO-1 and last IHI+1 to N elements on the ! diagonal; and second, applying a diagonal similarity transformation ! to rows and columns ILO to IHI to make the rows and columns as ! close in norm as possible. Both steps are optional. ! ! Balancing may reduce the 1-norm of the matrix, and improve the ! accuracy of the computed eigenvalues and/or eigenvectors. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the operations to be performed on A: ! = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 ! for i = 1,...,N; ! = 'P': permute only; ! = 'S': scale only; ! = 'B': both permute and scale. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the input matrix A. ! On exit, A is overwritten by the balanced matrix. ! If JOB = 'N', A is not referenced. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! ILO (output) INTEGER ! IHI (output) INTEGER ! ILO and IHI are set to integers such that on exit ! A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. ! If JOB = 'N' or 'S', ILO = 1 and IHI = N. ! ! SCALE (output) REAL array, dimension (N) ! Details of the permutations and scaling factors applied to ! A. If P(j) is the index of the row and column interchanged ! with row and column j and D(j) is the scaling factor ! applied to row and column j, then ! SCALE(j) = P(j) for j = 1,...,ILO-1 ! = D(j) for j = ILO,...,IHI ! = P(j) for j = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The permutations consist of row and column interchanges which put ! the matrix in the form ! ! ( T1 X Y ) ! P A P = ( 0 B Z ) ! ( 0 0 T2 ) ! ! where T1 and T2 are upper triangular matrices whose eigenvalues lie ! along the diagonal. The column indices ILO and IHI mark the starting ! and ending columns of the submatrix B. Balancing consists of applying ! a diagonal similarity transformation inv(D) * B * D to make the ! 1-norms of each row of B and its corresponding column nearly equal. ! The output matrix is ! ! ( T1 X*D Y ) ! ( 0 inv(D)*B*D inv(D)*Z ). ! ( 0 0 T2 ) ! ! Information about the permutations P and the diagonal matrix D is ! returned in the vector SCALE. ! ! This subroutine is based on the EISPACK routine BALANC. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) REAL(r8) SCLFAC PARAMETER ( SCLFAC = 1.0E+1_r8 ) REAL(r8) FACTOR PARAMETER ( FACTOR = 0.95E+0_r8 ) ! .. ! .. Local Scalars .. LOGICAL NOCONV INTEGER I, ICA, IEXC, IRA, J, K, L, M REAL(r8) C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1, & SFMIN2 ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEBAL', -INFO ) RETURN END IF ! K = 1 L = N ! IF( N.EQ.0 ) & GO TO 210 ! IF( LSAME( JOB, 'N' ) ) THEN DO 10 I = 1, N SCALE( I ) = ONE 10 CONTINUE GO TO 210 END IF ! IF( LSAME( JOB, 'S' ) ) & GO TO 120 ! ! Permutation to isolate eigenvalues if possible ! GO TO 50 ! ! Row and column exchange. ! 20 CONTINUE SCALE( M ) = J IF( J.EQ.M ) & GO TO 30 ! CALL SSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) CALL SSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA ) ! 30 CONTINUE GO TO ( 40, 80 )IEXC ! ! Search for rows isolating an eigenvalue and push them down. ! 40 CONTINUE IF( L.EQ.1 ) & GO TO 210 L = L - 1 ! 50 CONTINUE DO 70 J = L, 1, -1 ! DO 60 I = 1, L IF( I.EQ.J ) & GO TO 60 IF( A( J, I ).NE.ZERO ) & GO TO 70 60 CONTINUE ! M = L IEXC = 1 GO TO 20 70 CONTINUE ! GO TO 90 ! ! Search for columns isolating an eigenvalue and push them left. ! 80 CONTINUE K = K + 1 ! 90 CONTINUE DO 110 J = K, L ! DO 100 I = K, L IF( I.EQ.J ) & GO TO 100 IF( A( I, J ).NE.ZERO ) & GO TO 110 100 CONTINUE ! M = K IEXC = 2 GO TO 20 110 CONTINUE ! 120 CONTINUE DO 130 I = K, L SCALE( I ) = ONE 130 CONTINUE ! IF( LSAME( JOB, 'P' ) ) & GO TO 210 ! ! Balance the submatrix in rows K to L. ! ! Iterative loop for norm reduction ! SFMIN1 = SLAMCH( 'S' ) / SLAMCH( 'P' ) SFMAX1 = ONE / SFMIN1 SFMIN2 = SFMIN1*SCLFAC SFMAX2 = ONE / SFMIN2 140 CONTINUE NOCONV = .FALSE. ! DO 200 I = K, L C = ZERO R = ZERO ! DO 150 J = K, L IF( J.EQ.I ) & GO TO 150 C = C + ABS( A( J, I ) ) R = R + ABS( A( I, J ) ) 150 CONTINUE ICA = ISAMAX( L, A( 1, I ), 1 ) CA = ABS( A( ICA, I ) ) IRA = ISAMAX( N-K+1, A( I, K ), LDA ) RA = ABS( A( I, IRA+K-1 ) ) ! ! Guard against zero C or R due to underflow. ! IF( C.EQ.ZERO .OR. R.EQ.ZERO ) & GO TO 200 G = R / SCLFAC F = ONE S = C + R 160 CONTINUE IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR. & MIN( R, G, RA ).LE.SFMIN2 )GO TO 170 F = F*SCLFAC C = C*SCLFAC CA = CA*SCLFAC R = R / SCLFAC G = G / SCLFAC RA = RA / SCLFAC GO TO 160 ! 170 CONTINUE G = C / SCLFAC 180 CONTINUE IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR. & MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190 F = F / SCLFAC C = C / SCLFAC G = G / SCLFAC CA = CA / SCLFAC R = R*SCLFAC RA = RA*SCLFAC GO TO 180 ! ! Now balance. ! 190 CONTINUE IF( ( C+R ).GE.FACTOR*S ) & GO TO 200 IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN IF( F*SCALE( I ).LE.SFMIN1 ) & GO TO 200 END IF IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN IF( SCALE( I ).GE.SFMAX1 / F ) & GO TO 200 END IF G = ONE / F SCALE( I ) = SCALE( I )*F NOCONV = .TRUE. ! CALL SSCAL( N-K+1, G, A( I, K ), LDA ) CALL SSCAL( L, F, A( 1, I ), 1 ) ! 200 CONTINUE ! IF( NOCONV ) & GO TO 140 ! 210 CONTINUE ILO = K IHI = L ! RETURN ! ! End of SGEBAL ! END SUBROUTINE SGEBAL !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SGEHD2 reduces a real general matrix A to upper Hessenberg form H by ! an orthogonal similarity transformation: Q' * A * Q = H . ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to SGEBAL; otherwise they should be ! set to 1 and N respectively. See Further Details. ! 1 <= ILO <= IHI <= max(1,N). ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the n by n general matrix to be reduced. ! On exit, the upper triangle and the first subdiagonal of A ! are overwritten with the upper Hessenberg matrix H, and the ! elements below the first subdiagonal, with the array TAU, ! represent the orthogonal matrix Q as a product of elementary ! reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) REAL array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) REAL array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of (ihi-ilo) elementary ! reflectors ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on ! exit in A(i+2:ihi,i), and tau in TAU(i). ! ! The contents of A are illustrated by the following example, with ! n = 7, ilo = 2 and ihi = 6: ! ! on entry, on exit, ! ! ( a a a a a a a ) ( a a h h h h a ) ! ( a a a a a a ) ( a h h h h a ) ! ( a a a a a a ) ( h h h h h h ) ! ( a a a a a a ) ( v2 h h h h h ) ! ( a a a a a a ) ( v2 v3 h h h h ) ! ( a a a a a a ) ( v2 v3 v4 h h h ) ! ( a ) ( a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE PARAMETER ( ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I REAL(r8) AII ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEHD2', -INFO ) RETURN END IF ! DO 10 I = ILO, IHI - 1 ! ! Compute elementary reflector H(i) to annihilate A(i+2:ihi,i) ! CALL SLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAU( I ) ) AII = A( I+1, I ) A( I+1, I ) = ONE ! ! Apply H(i) to A(1:ihi,i+1:ihi) from the right ! CALL SLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), & A( 1, I+1 ), LDA, WORK ) ! ! Apply H(i) to A(i+1:ihi,i+1:n) from the left ! CALL SLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ), & A( I+1, I+1 ), LDA, WORK ) ! A( I+1, I ) = AII 10 CONTINUE ! RETURN ! ! End of SGEHD2 ! END SUBROUTINE SGEHD2 !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), TAU( * ), WORK( LWORK ) ! .. ! ! Purpose ! ======= ! ! SGEHRD reduces a real general matrix A to upper Hessenberg form H by ! an orthogonal similarity transformation: Q' * A * Q = H . ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to SGEBAL; otherwise they should be ! set to 1 and N respectively. See Further Details. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the N-by-N general matrix to be reduced. ! On exit, the upper triangle and the first subdiagonal of A ! are overwritten with the upper Hessenberg matrix H, and the ! elements below the first subdiagonal, with the array TAU, ! represent the orthogonal matrix Q as a product of elementary ! reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) REAL array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to ! zero. ! ! WORK (workspace/output) REAL array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of (ihi-ilo) elementary ! reflectors ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on ! exit in A(i+2:ihi,i), and tau in TAU(i). ! ! The contents of A are illustrated by the following example, with ! n = 7, ilo = 2 and ihi = 6: ! ! on entry, on exit, ! ! ( a a a a a a a ) ( a a h h h h a ) ! ( a a a a a a ) ( a h h h h a ) ! ( a a a a a a ) ( h h h h h h ) ! ( a a a a a a ) ( v2 h h h h h ) ! ( a a a a a a ) ( v2 v3 h h h h ) ! ( a a a a a a ) ( v2 v3 v4 h h h ) ! ( a ) ( a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, IB, IINFO, IWS, LDWORK, NB, NBMIN, NH, NX REAL(r8) EI ! .. ! .. Local Arrays .. REAL(r8) T( LDT, NBMAX ) ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEHRD', -INFO ) RETURN END IF ! ! Set elements 1:ILO-1 and IHI:N-1 of TAU to zero ! DO 10 I = 1, ILO - 1 TAU( I ) = ZERO 10 CONTINUE DO 20 I = MAX( 1, IHI ), N - 1 TAU( I ) = ZERO 20 CONTINUE ! ! Quick return if possible ! NH = IHI - ILO + 1 IF( NH.LE.1 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Determine the block size. ! NB = MIN( NBMAX, ILAENV( 1, 'SGEHRD', ' ', N, ILO, IHI, -1 ) ) NBMIN = 2 IWS = 1 IF( NB.GT.1 .AND. NB.LT.NH ) THEN ! ! Determine when to cross over from blocked to unblocked code ! (last block is always handled by unblocked code). ! NX = MAX( NB, ILAENV( 3, 'SGEHRD', ' ', N, ILO, IHI, -1 ) ) IF( NX.LT.NH ) THEN ! ! Determine if workspace is large enough for blocked code. ! IWS = N*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: determine the ! minimum value of NB, and reduce NB or force use of ! unblocked code. ! NBMIN = MAX( 2, ILAENV( 2, 'SGEHRD', ' ', N, ILO, IHI, & -1 ) ) IF( LWORK.GE.N*NBMIN ) THEN NB = LWORK / N ELSE NB = 1 END IF END IF END IF END IF LDWORK = N ! IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN ! ! Use unblocked code below ! I = ILO ! ELSE ! ! Use blocked code ! DO 30 I = ILO, IHI - 1 - NX, NB IB = MIN( NB, IHI-I ) ! ! Reduce columns i:i+ib-1 to Hessenberg form, returning the ! matrices V and T of the block reflector H = I - V*T*V' ! which performs the reduction, and also the matrix Y = A*V*T ! CALL SLAHRD( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT, & WORK, LDWORK ) ! ! Apply the block reflector H to A(1:ihi,i+ib:ihi) from the ! right, computing A := A - Y * V'. V(i+ib,ib-1) must be set ! to 1. ! EI = A( I+IB, I+IB-1 ) A( I+IB, I+IB-1 ) = ONE CALL SGEMM( 'No transpose', 'Transpose', IHI, IHI-I-IB+1, & IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE, & A( 1, I+IB ), LDA ) A( I+IB, I+IB-1 ) = EI ! ! Apply the block reflector H to A(i+1:ihi,i+ib:n) from the ! left ! CALL SLARFB( 'Left', 'Transpose', 'Forward', 'Columnwise', & IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT, & A( I+1, I+IB ), LDA, WORK, LDWORK ) 30 CONTINUE END IF ! ! Use unblocked code to reduce the rest of the matrix ! CALL SGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) WORK( 1 ) = IWS ! RETURN ! ! End of SGEHRD ! END SUBROUTINE SGEHRD !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, & LDZ, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER COMPZ, JOB INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N ! .. ! .. Array Arguments .. REAL(r8) H( LDH, * ), WI( * ), WORK( * ), WR( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! SHSEQR computes the eigenvalues of a real upper Hessenberg matrix H ! and, optionally, the matrices T and Z from the Schur decomposition ! H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur ! form), and Z is the orthogonal matrix of Schur vectors. ! ! Optionally Z may be postmultiplied into an input orthogonal matrix Q, ! so that this routine can give the Schur factorization of a matrix A ! which has been reduced to the Hessenberg form H by the orthogonal ! matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! = 'E': compute eigenvalues only; ! = 'S': compute eigenvalues and the Schur form T. ! ! COMPZ (input) CHARACTER*1 ! = 'N': no Schur vectors are computed; ! = 'I': Z is initialized to the unit matrix and the matrix Z ! of Schur vectors of H is returned; ! = 'V': Z must contain an orthogonal matrix Q on entry, and ! the product Q*Z is returned. ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that H is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to SGEBAL, and then passed to SGEHRD ! when the matrix output by SGEBAL is reduced to Hessenberg ! form. Otherwise ILO and IHI should be set to 1 and N ! respectively. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! H (input/output) REAL array, dimension (LDH,N) ! On entry, the upper Hessenberg matrix H. ! On exit, if JOB = 'S', H contains the upper quasi-triangular ! matrix T from the Schur decomposition (the Schur form); ! 2-by-2 diagonal blocks (corresponding to complex conjugate ! pairs of eigenvalues) are returned in standard form, with ! H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If JOB = 'E', ! the contents of H are unspecified on exit. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (output) REAL array, dimension (N) ! WI (output) REAL array, dimension (N) ! The real and imaginary parts, respectively, of the computed ! eigenvalues. If two eigenvalues are computed as a complex ! conjugate pair, they are stored in consecutive elements of ! WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and ! WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in the ! same order as on the diagonal of the Schur form returned in ! H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 ! diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and ! WI(i+1) = -WI(i). ! ! Z (input/output) REAL array, dimension (LDZ,N) ! If COMPZ = 'N': Z is not referenced. ! If COMPZ = 'I': on entry, Z need not be set, and on exit, Z ! contains the orthogonal matrix Z of the Schur vectors of H. ! If COMPZ = 'V': on entry Z must contain an N-by-N matrix Q, ! which is assumed to be equal to the unit matrix except for ! the submatrix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z. ! Normally Q is the orthogonal matrix generated by SORGHR after ! the call to SGEHRD which formed the Hessenberg matrix H. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. ! LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise. ! ! WORK (workspace) REAL array, dimension (N) ! ! LWORK (input) INTEGER ! This argument is currently redundant. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, SHSEQR failed to compute all of the ! eigenvalues in a total of 30*(IHI-ILO+1) iterations; ! elements 1:ilo-1 and i+1:n of WR and WI contain those ! eigenvalues which have been successfully computed. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8, TWO = 2.0E+0_r8 ) REAL(r8) CONST PARAMETER ( CONST = 1.5E+0_r8 ) INTEGER NSMAX, LDS PARAMETER ( NSMAX = 15, LDS = NSMAX ) ! .. ! .. Local Scalars .. LOGICAL INITZ, WANTT, WANTZ INTEGER I, I1, I2, IERR, II, ITEMP, ITN, ITS, J, K, L, & MAXB, NH, NR, NS, NV REAL(r8) ABSW, OVFL, SMLNUM, TAU, TEMP, TST1, ULP, UNFL ! .. ! .. Local Arrays .. REAL(r8) S( LDS, NSMAX ), V( NSMAX+1 ), VV( NSMAX+1 ) ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! WANTT = LSAME( JOB, 'S' ) INITZ = LSAME( COMPZ, 'I' ) WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) ! INFO = 0 IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN INFO = -1 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. LDZ.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SHSEQR', -INFO ) RETURN END IF ! ! Initialize Z, if necessary ! IF( INITZ ) & CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! ! Store the eigenvalues isolated by SGEBAL. ! DO 10 I = 1, ILO - 1 WR( I ) = H( I, I ) WI( I ) = ZERO 10 CONTINUE DO 20 I = IHI + 1, N WR( I ) = H( I, I ) WI( I ) = ZERO 20 CONTINUE ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF ! ! Set rows and columns ILO to IHI to zero below the first ! subdiagonal. ! DO 40 J = ILO, IHI - 2 DO 30 I = J + 2, N H( I, J ) = ZERO 30 CONTINUE 40 CONTINUE NH = IHI - ILO + 1 ! ! Determine the order of the multi-shift QR algorithm to be used. ! NS = ILAENV( 4, 'SHSEQR', JOB // COMPZ, N, ILO, IHI, -1 ) MAXB = ILAENV( 8, 'SHSEQR', JOB // COMPZ, N, ILO, IHI, -1 ) IF( NS.LE.2 .OR. NS.GT.NH .OR. MAXB.GE.NH ) THEN ! ! Use the standard double-shift algorithm ! CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, & IHI, Z, LDZ, INFO ) RETURN END IF MAXB = MAX( 3, MAXB ) NS = MIN( NS, MAXB, NSMAX ) ! ! Now 2 < NS <= MAXB < NH. ! ! Set machine-dependent constants for the stopping criterion. ! If norm(H) <= sqrt(OVFL), overflow should not occur. ! UNFL = SLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL SLABAD( UNFL, OVFL ) ULP = SLAMCH( 'Precision' ) SMLNUM = UNFL*( NH / ULP ) ! ! I1 and I2 are the indices of the first row and last column of H ! to which transformations must be applied. If eigenvalues only are ! being computed, I1 and I2 are set inside the main loop. ! IF( WANTT ) THEN I1 = 1 I2 = N END IF ! ! ITN is the total number of multiple-shift QR iterations allowed. ! ITN = 30*NH ! ! The main loop begins here. I is the loop index and decreases from ! IHI to ILO in steps of at most MAXB. Each iteration of the loop ! works with the active submatrix in rows and columns L to I. ! Eigenvalues I+1 to IHI have already converged. Either L = ILO or ! H(L,L-1) is negligible so that the matrix splits. ! I = IHI 50 CONTINUE L = ILO IF( I.LT.ILO ) & GO TO 170 ! ! Perform multiple-shift QR iterations on rows and columns ILO to I ! until a submatrix of order at most MAXB splits off at the bottom ! because a subdiagonal element has become negligible. ! DO 150 ITS = 0, ITN ! ! Look for a single small subdiagonal element. ! DO 60 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) & TST1 = SLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) & GO TO 70 60 CONTINUE 70 CONTINUE L = K IF( L.GT.ILO ) THEN ! ! H(L,L-1) is negligible. ! H( L, L-1 ) = ZERO END IF ! ! Exit from loop if a submatrix of order <= MAXB has split off. ! IF( L.GE.I-MAXB+1 ) & GO TO 160 ! ! Now the active submatrix is in rows and columns L to I. If ! eigenvalues only are being computed, only the active submatrix ! need be transformed. ! IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF ! IF( ITS.EQ.20 .OR. ITS.EQ.30 ) THEN ! ! Exceptional shifts. ! DO 80 II = I - NS + 1, I WR( II ) = CONST*( ABS( H( II, II-1 ) )+ & ABS( H( II, II ) ) ) WI( II ) = ZERO 80 CONTINUE ELSE ! ! Use eigenvalues of trailing submatrix of order NS as shifts. ! CALL SLACPY( 'Full', NS, NS, H( I-NS+1, I-NS+1 ), LDH, S, & LDS ) CALL SLAHQR( .FALSE., .FALSE., NS, 1, NS, S, LDS, & WR( I-NS+1 ), WI( I-NS+1 ), 1, NS, Z, LDZ, & IERR ) IF( IERR.GT.0 ) THEN ! ! If SLAHQR failed to compute all NS eigenvalues, use the ! unconverged diagonal elements as the remaining shifts. ! DO 90 II = 1, IERR WR( I-NS+II ) = S( II, II ) WI( I-NS+II ) = ZERO 90 CONTINUE END IF END IF ! ! Form the first column of (G-w(1)) (G-w(2)) . . . (G-w(ns)) ! where G is the Hessenberg submatrix H(L:I,L:I) and w is ! the vector of shifts (stored in WR and WI). The result is ! stored in the local array V. ! V( 1 ) = ONE DO 100 II = 2, NS + 1 V( II ) = ZERO 100 CONTINUE NV = 1 DO 120 J = I - NS + 1, I IF( WI( J ).GE.ZERO ) THEN IF( WI( J ).EQ.ZERO ) THEN ! ! real shift ! CALL SCOPY( NV+1, V, 1, VV, 1 ) CALL SGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ), & LDH, VV, 1, -WR( J ), V, 1 ) NV = NV + 1 ELSE IF( WI( J ).GT.ZERO ) THEN ! ! complex conjugate pair of shifts ! CALL SCOPY( NV+1, V, 1, VV, 1 ) CALL SGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ), & LDH, V, 1, -TWO*WR( J ), VV, 1 ) ITEMP = ISAMAX( NV+1, VV, 1 ) TEMP = ONE / MAX( ABS( VV( ITEMP ) ), SMLNUM ) CALL SSCAL( NV+1, TEMP, VV, 1 ) ABSW = SLAPY2( WR( J ), WI( J ) ) TEMP = ( TEMP*ABSW )*ABSW CALL SGEMV( 'No transpose', NV+2, NV+1, ONE, & H( L, L ), LDH, VV, 1, TEMP, V, 1 ) NV = NV + 2 END IF ! ! Scale V(1:NV) so that max(abs(V(i))) = 1. If V is zero, ! reset it to the unit vector. ! ITEMP = ISAMAX( NV, V, 1 ) TEMP = ABS( V( ITEMP ) ) IF( TEMP.EQ.ZERO ) THEN V( 1 ) = ONE DO 110 II = 2, NV V( II ) = ZERO 110 CONTINUE ELSE TEMP = MAX( TEMP, SMLNUM ) CALL SSCAL( NV, ONE / TEMP, V, 1 ) END IF END IF 120 CONTINUE ! ! Multiple-shift QR step ! DO 140 K = L, I - 1 ! ! The first iteration of this loop determines a reflection G ! from the vector V and applies it from left and right to H, ! thus creating a nonzero bulge below the subdiagonal. ! ! Each subsequent iteration determines a reflection G to ! restore the Hessenberg form in the (K-1)th column, and thus ! chases the bulge one step toward the bottom of the active ! submatrix. NR is the order of G. ! NR = MIN( NS+1, I-K+1 ) IF( K.GT.L ) & CALL SCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL SLARFG( NR, V( 1 ), V( 2 ), 1, TAU ) IF( K.GT.L ) THEN H( K, K-1 ) = V( 1 ) DO 130 II = K + 1, I H( II, K-1 ) = ZERO 130 CONTINUE END IF V( 1 ) = ONE ! ! Apply G from the left to transform the rows of the matrix in ! columns K to I2. ! CALL SLARFX( 'Left', NR, I2-K+1, V, TAU, H( K, K ), LDH, & WORK ) ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+NR,I). ! CALL SLARFX( 'Right', MIN( K+NR, I )-I1+1, NR, V, TAU, & H( I1, K ), LDH, WORK ) ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! CALL SLARFX( 'Right', NH, NR, V, TAU, Z( ILO, K ), LDZ, & WORK ) END IF 140 CONTINUE ! 150 CONTINUE ! ! Failure to converge in remaining number of iterations ! INFO = I RETURN ! 160 CONTINUE ! ! A submatrix of order <= MAXB in rows and columns L to I has split ! off. Use the double-shift QR algorithm to handle it. ! CALL SLAHQR( WANTT, WANTZ, N, L, I, H, LDH, WR, WI, ILO, IHI, Z, & LDZ, INFO ) IF( INFO.GT.0 ) & RETURN ! ! Decrement number of remaining iterations, and return to start of ! the main loop with a new value of I. ! ITN = ITN - ITS I = L - 1 GO TO 50 ! 170 CONTINUE RETURN ! ! End of SHSEQR ! END SUBROUTINE SHSEQR !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLABAD( SMALL, LARGE ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. REAL(r8) LARGE, SMALL ! .. ! ! Purpose ! ======= ! ! SLABAD takes as input the values computed by SLAMCH for underflow and ! overflow, and returns the square root of each of these values if the ! log of LARGE is sufficiently large. This subroutine is intended to ! identify machines with a large exponent range, such as the Crays, and ! redefine the underflow and overflow limits to be the square roots of ! the values computed by SLAMCH. This subroutine is needed because ! SLAMCH does not compensate for poor arithmetic in the upper half of ! the exponent range, as is found on a Cray. ! ! Arguments ! ========= ! ! SMALL (input/output) REAL ! On entry, the underflow threshold as computed by SLAMCH. ! On exit, if LOG10(LARGE) is sufficiently large, the square ! root of SMALL, otherwise unchanged. ! ! LARGE (input/output) REAL ! On entry, the overflow threshold as computed by SLAMCH. ! On exit, if LOG10(LARGE) is sufficiently large, the square ! root of LARGE, otherwise unchanged. ! ! ===================================================================== ! ! .. ! .. Executable Statements .. ! ! If it looks like we're on a Cray, take the square root of ! SMALL and LARGE to avoid overflow and underflow problems. ! IF( LOG10( LARGE ).GT.2000._r8 ) THEN SMALL = SQRT( SMALL ) LARGE = SQRT( LARGE ) END IF ! RETURN ! ! End of SLABAD ! END SUBROUTINE SLABAD !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLACPY( UPLO, M, N, A, LDA, B, LDB ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDB, M, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! SLACPY copies all or part of a two-dimensional matrix A to another ! matrix B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies the part of the matrix A to be copied to B. ! = 'U': Upper triangular part ! = 'L': Lower triangular part ! Otherwise: All of the matrix A ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input) REAL array, dimension (LDA,N) ! The m by n matrix A. If UPLO = 'U', only the upper triangle ! or trapezoid is accessed; if UPLO = 'L', only the lower ! triangle or trapezoid is accessed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (output) REAL array, dimension (LDB,N) ! On exit, B = A in the locations specified by UPLO. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,M). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J ! .. ! .. Executable Statements .. ! IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = 1, MIN( J, M ) B( I, J ) = A( I, J ) 10 CONTINUE 20 CONTINUE ELSE IF( LSAME( UPLO, 'L' ) ) THEN DO 40 J = 1, N DO 30 I = J, M B( I, J ) = A( I, J ) 30 CONTINUE 40 CONTINUE ELSE DO 60 J = 1, N DO 50 I = 1, M B( I, J ) = A( I, J ) 50 CONTINUE 60 CONTINUE END IF RETURN ! ! End of SLACPY ! END SUBROUTINE SLACPY !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLADIV( A, B, C, D, P, Q ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. REAL(r8) A, B, C, D, P, Q ! .. ! ! Purpose ! ======= ! ! SLADIV performs complex division in real arithmetic ! ! a + i*b ! p + i*q = --------- ! c + i*d ! ! The algorithm is due to Robert L. Smith and can be found ! in D. Knuth, The art of Computer Programming, Vol.2, p.195 ! ! Arguments ! ========= ! ! A (input) REAL ! B (input) REAL ! C (input) REAL ! D (input) REAL ! The scalars a, b, c, and d in the above expression. ! ! P (output) REAL ! Q (output) REAL ! The scalars p and q in the above expression. ! ! ===================================================================== ! ! .. Local Scalars .. REAL(r8) E, F ! .. ! .. Executable Statements .. ! IF( ABS( D ).LT.ABS( C ) ) THEN E = D / C F = C + D*E P = ( A+B*E ) / F Q = ( B-A*E ) / F ELSE E = C / D F = D + C*E P = ( B+A*E ) / F Q = ( -A+B*E ) / F END IF ! RETURN ! ! End of SLADIV ! END SUBROUTINE SLADIV !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, & ILOZ, IHIZ, Z, LDZ, INFO ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL WANTT, WANTZ INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N ! .. ! .. Array Arguments .. REAL(r8) H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! SLAHQR is an auxiliary routine called by SHSEQR to update the ! eigenvalues and Schur decomposition already computed by SHSEQR, by ! dealing with the Hessenberg submatrix in rows and columns ILO to IHI. ! ! Arguments ! ========= ! ! WANTT (input) LOGICAL ! = .TRUE. : the full Schur form T is required; ! = .FALSE.: only eigenvalues are required. ! ! WANTZ (input) LOGICAL ! = .TRUE. : the matrix of Schur vectors Z is required; ! = .FALSE.: Schur vectors are not required. ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that H is already upper quasi-triangular in ! rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ! ILO = 1). SLAHQR works primarily with the Hessenberg ! submatrix in rows and columns ILO to IHI, but applies ! transformations to all of H if WANTT is .TRUE.. ! 1 <= ILO <= max(1,IHI); IHI <= N. ! ! H (input/output) REAL array, dimension (LDH,N) ! On entry, the upper Hessenberg matrix H. ! On exit, if WANTT is .TRUE., H is upper quasi-triangular in ! rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in ! standard form. If WANTT is .FALSE., the contents of H are ! unspecified on exit. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (output) REAL array, dimension (N) ! WI (output) REAL array, dimension (N) ! The real and imaginary parts, respectively, of the computed ! eigenvalues ILO to IHI are stored in the corresponding ! elements of WR and WI. If two eigenvalues are computed as a ! complex conjugate pair, they are stored in consecutive ! elements of WR and WI, say the i-th and (i+1)th, with ! WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the ! eigenvalues are stored in the same order as on the diagonal ! of the Schur form returned in H, with WR(i) = H(i,i), and, if ! H(i:i+1,i:i+1) is a 2-by-2 diagonal block, ! WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). ! ! ILOZ (input) INTEGER ! IHIZ (input) INTEGER ! Specify the rows of Z to which transformations must be ! applied if WANTZ is .TRUE.. ! 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. ! ! Z (input/output) REAL array, dimension (LDZ,N) ! If WANTZ is .TRUE., on entry Z must contain the current ! matrix Z of transformations accumulated by SHSEQR, and on ! exit Z has been updated; transformations are applied only to ! the submatrix Z(ILOZ:IHIZ,ILO:IHI). ! If WANTZ is .FALSE., Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: SLAHQR failed to compute all the eigenvalues ILO to IHI ! in a total of 30*(IHI-ILO+1) iterations; if INFO = i, ! elements i+1:ihi of WR and WI contain those eigenvalues ! which have been successfully computed. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) REAL(r8) DAT1, DAT2 PARAMETER ( DAT1 = 0.75E+0_r8, DAT2 = -0.4375E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, I1, I2, ITN, ITS, J, K, L, M, NH, NR, NZ REAL(r8) CS, H00, H10, H11, H12, H21, H22, H33, H33S, & H43H34, H44, H44S, OVFL, S, SMLNUM, SN, SUM, & T1, T2, T3, TST1, ULP, UNFL, V1, V2, V3 ! .. ! .. Local Arrays .. REAL(r8) V( 3 ), WORK( 1 ) ! .. ! .. Executable Statements .. ! INFO = 0 ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF ! NH = IHI - ILO + 1 NZ = IHIZ - ILOZ + 1 ! ! Set machine-dependent constants for the stopping criterion. ! If norm(H) <= sqrt(OVFL), overflow should not occur. ! UNFL = SLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL SLABAD( UNFL, OVFL ) ULP = SLAMCH( 'Precision' ) SMLNUM = UNFL*( NH / ULP ) ! ! I1 and I2 are the indices of the first row and last column of H ! to which transformations must be applied. If eigenvalues only are ! being computed, I1 and I2 are set inside the main loop. ! IF( WANTT ) THEN I1 = 1 I2 = N END IF ! ! ITN is the total number of QR iterations allowed. ! ITN = 30*NH ! ! The main loop begins here. I is the loop index and decreases from ! IHI to ILO in steps of 1 or 2. Each iteration of the loop works ! with the active submatrix in rows and columns L to I. ! Eigenvalues I+1 to IHI have already converged. Either L = ILO or ! H(L,L-1) is negligible so that the matrix splits. ! I = IHI 10 CONTINUE L = ILO IF( I.LT.ILO ) & GO TO 150 ! ! Perform QR iterations on rows and columns ILO to I until a ! submatrix of order 1 or 2 splits off at the bottom because a ! subdiagonal element has become negligible. ! DO 130 ITS = 0, ITN ! ! Look for a single small subdiagonal element. ! DO 20 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) & TST1 = SLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) & GO TO 30 20 CONTINUE 30 CONTINUE L = K IF( L.GT.ILO ) THEN ! ! H(L,L-1) is negligible ! H( L, L-1 ) = ZERO END IF ! ! Exit from loop if a submatrix of order 1 or 2 has split off. ! IF( L.GE.I-1 ) & GO TO 140 ! ! Now the active submatrix is in rows and columns L to I. If ! eigenvalues only are being computed, only the active submatrix ! need be transformed. ! IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF ! IF( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN ! ! Exceptional shift. ! S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) H44 = DAT1*S H33 = H44 H43H34 = DAT2*S*S ELSE ! ! Prepare to use Wilkinson's double shift ! H44 = H( I, I ) H33 = H( I-1, I-1 ) H43H34 = H( I, I-1 )*H( I-1, I ) END IF ! ! Look for two consecutive small subdiagonal elements. ! DO 40 M = I - 2, L, -1 ! ! Determine the effect of starting the double-shift QR ! iteration at row M, and see if this would make H(M,M-1) ! negligible. ! H11 = H( M, M ) H22 = H( M+1, M+1 ) H21 = H( M+1, M ) H12 = H( M, M+1 ) H44S = H44 - H11 H33S = H33 - H11 V1 = ( H33S*H44S-H43H34 ) / H21 + H12 V2 = H22 - H11 - H33S - H44S V3 = H( M+2, M+1 ) S = ABS( V1 ) + ABS( V2 ) + ABS( V3 ) V1 = V1 / S V2 = V2 / S V3 = V3 / S V( 1 ) = V1 V( 2 ) = V2 V( 3 ) = V3 IF( M.EQ.L ) & GO TO 50 H00 = H( M-1, M-1 ) H10 = H( M, M-1 ) TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) ) IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).LE.ULP*TST1 ) & GO TO 50 40 CONTINUE 50 CONTINUE ! ! Double-shift QR step ! DO 120 K = M, I - 1 ! ! The first iteration of this loop determines a reflection G ! from the vector V and applies it from left and right to H, ! thus creating a nonzero bulge below the subdiagonal. ! ! Each subsequent iteration determines a reflection G to ! restore the Hessenberg form in the (K-1)th column, and thus ! chases the bulge one step toward the bottom of the active ! submatrix. NR is the order of G. ! NR = MIN( 3, I-K+1 ) IF( K.GT.M ) & CALL SCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL SLARFG( NR, V( 1 ), V( 2 ), 1, T1 ) IF( K.GT.M ) THEN H( K, K-1 ) = V( 1 ) H( K+1, K-1 ) = ZERO IF( K.LT.I-1 ) & H( K+2, K-1 ) = ZERO ELSE IF( M.GT.L ) THEN H( K, K-1 ) = -H( K, K-1 ) END IF V2 = V( 2 ) T2 = T1*V2 IF( NR.EQ.3 ) THEN V3 = V( 3 ) T3 = T1*V3 ! ! Apply G from the left to transform the rows of the matrix ! in columns K to I2. ! DO 60 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 H( K+2, J ) = H( K+2, J ) - SUM*T3 60 CONTINUE ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+3,I). ! DO 70 J = I1, MIN( K+3, I ) SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 H( J, K+2 ) = H( J, K+2 ) - SUM*T3 70 CONTINUE ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! DO 80 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3 80 CONTINUE END IF ELSE IF( NR.EQ.2 ) THEN ! ! Apply G from the left to transform the rows of the matrix ! in columns K to I2. ! DO 90 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 90 CONTINUE ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+3,I). ! DO 100 J = I1, I SUM = H( J, K ) + V2*H( J, K+1 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 100 CONTINUE ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! DO 110 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 110 CONTINUE END IF END IF 120 CONTINUE ! 130 CONTINUE ! ! Failure to converge in remaining number of iterations ! INFO = I RETURN ! 140 CONTINUE ! IF( L.EQ.I ) THEN ! ! H(I,I-1) is negligible: one eigenvalue has converged. ! WR( I ) = H( I, I ) WI( I ) = ZERO ELSE IF( L.EQ.I-1 ) THEN ! ! H(I-1,I-2) is negligible: a pair of eigenvalues have converged. ! ! Transform the 2-by-2 submatrix to standard Schur form, ! and compute and store the eigenvalues. ! CALL SLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ), & H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ), & CS, SN ) ! IF( WANTT ) THEN ! ! Apply the transformation to the rest of H. ! IF( I2.GT.I ) & CALL SROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH, & CS, SN ) CALL SROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN ) END IF IF( WANTZ ) THEN ! ! Apply the transformation to Z. ! CALL SROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN ) END IF END IF ! ! Decrement number of remaining iterations, and return to start of ! the main loop with new value of I. ! ITN = ITN - ITS I = L - 1 GO TO 10 ! 150 CONTINUE RETURN ! ! End of SLAHQR ! END SUBROUTINE SLAHQR !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER K, LDA, LDT, LDY, N, NB ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), T( LDT, NB ), TAU( NB ), & Y( LDY, NB ) ! .. ! ! Purpose ! ======= ! ! SLAHRD reduces the first NB columns of a real general n-by-(n-k+1) ! matrix A so that elements below the k-th subdiagonal are zero. The ! reduction is performed by an orthogonal similarity transformation ! Q' * A * Q. The routine returns the matrices V and T which determine ! Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. ! ! This is an auxiliary routine called by SGEHRD. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. ! ! K (input) INTEGER ! The offset for the reduction. Elements below the k-th ! subdiagonal in the first NB columns are reduced to zero. ! ! NB (input) INTEGER ! The number of columns to be reduced. ! ! A (input/output) REAL array, dimension (LDA,N-K+1) ! On entry, the n-by-(n-k+1) general matrix A. ! On exit, the elements on and above the k-th subdiagonal in ! the first NB columns are overwritten with the corresponding ! elements of the reduced matrix; the elements below the k-th ! subdiagonal, with the array TAU, represent the matrix Q as a ! product of elementary reflectors. The other columns of A are ! unchanged. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) REAL array, dimension (NB) ! The scalar factors of the elementary reflectors. See Further ! Details. ! ! T (output) REAL array, dimension (NB,NB) ! The upper triangular matrix T. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= NB. ! ! Y (output) REAL array, dimension (LDY,NB) ! The n-by-nb matrix Y. ! ! LDY (input) INTEGER ! The leading dimension of the array Y. LDY >= N. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of nb elementary reflectors ! ! Q = H(1) H(2) . . . H(nb). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in ! A(i+k+1:n,i), and tau in TAU(i). ! ! The elements of the vectors v together form the (n-k+1)-by-nb matrix ! V which is needed, with T and Y, to apply the transformation to the ! unreduced part of the matrix, using an update of the form: ! A := (I - V*T*V') * (A - Y*V'). ! ! The contents of A on exit are illustrated by the following example ! with n = 7, k = 3 and nb = 2: ! ! ( a h a a a ) ! ( a h a a a ) ! ( a h a a a ) ! ( h h a a a ) ! ( v1 h a a a ) ! ( v1 v2 a a a ) ! ( v1 v2 a a a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I REAL(r8) EI ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! DO 10 I = 1, NB IF( I.GT.1 ) THEN ! ! Update A(1:n,i) ! ! Compute i-th column of A - Y * V' ! CALL SGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, & A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) ! ! Apply I - V * T' * V' to this column (call it b) from the ! left, using the last column of T as workspace ! ! Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) ! ( V2 ) ( b2 ) ! ! where V1 is unit lower triangular ! ! w := V1' * b1 ! CALL SCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL STRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), & LDA, T( 1, NB ), 1 ) ! ! w := w + V2'*b2 ! CALL SGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), & LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) ! ! w := T'*w ! CALL STRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, & T( 1, NB ), 1 ) ! ! b2 := b2 - V2*w ! CALL SGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), & LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) ! ! b1 := b1 - V1*w ! CALL STRMV( 'Lower', 'No transpose', 'Unit', I-1, & A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL SAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) ! A( K+I-1, I-1 ) = EI END IF ! ! Generate the elementary reflector H(i) to annihilate ! A(k+i+1:n,i) ! CALL SLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, & TAU( I ) ) EI = A( K+I, I ) A( K+I, I ) = ONE ! ! Compute Y(1:n,i) ! CALL SGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, & A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL SGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA, & A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) CALL SGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, & ONE, Y( 1, I ), 1 ) CALL SSCAL( N, TAU( I ), Y( 1, I ), 1 ) ! ! Compute T(1:i,i) ! CALL SSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL STRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, & T( 1, I ), 1 ) T( I, I ) = TAU( I ) ! 10 CONTINUE A( K+NB, NB ) = EI ! RETURN ! ! End of SLAHRD ! END SUBROUTINE SLAHRD !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, & LDB, WR, WI, X, LDX, SCALE, XNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL LTRANS INTEGER INFO, LDA, LDB, LDX, NA, NW REAL(r8) CA, D1, D2, SCALE, SMIN, WI, WR, XNORM ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), B( LDB, * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! SLALN2 solves a system of the form (ca A - w D ) X = s B ! or (ca A' - w D) X = s B with possible scaling ("s") and ! perturbation of A. (A' means A-transpose.) ! ! A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA ! real diagonal matrix, w is a real or complex value, and X and B are ! NA x 1 matrices -- real if w is real, complex if w is complex. NA ! may be 1 or 2. ! ! If w is complex, X and B are represented as NA x 2 matrices, ! the first column of each being the real part and the second ! being the imaginary part. ! ! "s" is a scaling factor (.LE. 1), computed by SLALN2, which is ! so chosen that X can be computed without overflow. X is further ! scaled if necessary to assure that norm(ca A - w D)*norm(X) is less ! than overflow. ! ! If both singular values of (ca A - w D) are less than SMIN, ! SMIN*identity will be used instead of (ca A - w D). If only one ! singular value is less than SMIN, one element of (ca A - w D) will be ! perturbed enough to make the smallest singular value roughly SMIN. ! If both singular values are at least SMIN, (ca A - w D) will not be ! perturbed. In any case, the perturbation will be at most some small ! multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values ! are computed by infinity-norm approximations, and thus will only be ! correct to a factor of 2 or so. ! ! Note: all input quantities are assumed to be smaller than overflow ! by a reasonable factor. (See BIGNUM.) ! ! Arguments ! ========== ! ! LTRANS (input) LOGICAL ! =.TRUE.: A-transpose will be used. ! =.FALSE.: A will be used (not transposed.) ! ! NA (input) INTEGER ! The size of the matrix A. It may (only) be 1 or 2. ! ! NW (input) INTEGER ! 1 if "w" is real, 2 if "w" is complex. It may only be 1 ! or 2. ! ! SMIN (input) REAL ! The desired lower bound on the singular values of A. This ! should be a safe distance away from underflow or overflow, ! say, between (underflow/machine precision) and (machine ! precision * overflow ). (See BIGNUM and ULP.) ! ! CA (input) REAL ! The coefficient c, which A is multiplied by. ! ! A (input) REAL array, dimension (LDA,NA) ! The NA x NA matrix A. ! ! LDA (input) INTEGER ! The leading dimension of A. It must be at least NA. ! ! D1 (input) REAL ! The 1,1 element in the diagonal matrix D. ! ! D2 (input) REAL ! The 2,2 element in the diagonal matrix D. Not used if NW=1. ! ! B (input) REAL array, dimension (LDB,NW) ! The NA x NW matrix B (right-hand side). If NW=2 ("w" is ! complex), column 1 contains the real part of B and column 2 ! contains the imaginary part. ! ! LDB (input) INTEGER ! The leading dimension of B. It must be at least NA. ! ! WR (input) REAL ! The real part of the scalar "w". ! ! WI (input) REAL ! The imaginary part of the scalar "w". Not used if NW=1. ! ! X (output) REAL array, dimension (LDX,NW) ! The NA x NW matrix X (unknowns), as computed by SLALN2. ! If NW=2 ("w" is complex), on exit, column 1 will contain ! the real part of X and column 2 will contain the imaginary ! part. ! ! LDX (input) INTEGER ! The leading dimension of X. It must be at least NA. ! ! SCALE (output) REAL ! The scale factor that B must be multiplied by to insure ! that overflow does not occur when computing X. Thus, ! (ca A - w D) X will be SCALE*B, not B (ignoring ! perturbations of A.) It will be at most 1. ! ! XNORM (output) REAL ! The infinity-norm of X, when X is regarded as an NA x NW ! real matrix. ! ! INFO (output) INTEGER ! An error flag. It will be set to zero if no error occurs, ! a negative number if an argument is in error, or a positive ! number if ca A - w D had to be perturbed. ! The possible values are: ! = 0: No error occurred, and (ca A - w D) did not have to be ! perturbed. ! = 1: (ca A - w D) had to be perturbed to make its smallest ! (or only) singular value greater than SMIN. ! NOTE: In the interests of speed, this routine does not ! check the inputs for errors. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E0_r8, ONE = 1.0E0_r8 ) REAL(r8) TWO PARAMETER ( TWO = 2.0E0_r8 ) ! .. ! .. Local Scalars .. INTEGER ICMAX, J REAL(r8) BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21, & CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21, & LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R, & UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S, & UR22, XI1, XI2, XR1, XR2 ! .. ! .. Local Arrays .. LOGICAL CSWAP( 4 ), RSWAP( 4 ) INTEGER IPIVOT( 4, 4 ) REAL(r8) CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 ) ! .. ! .. Equivalences .. EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ), & ( CR( 1, 1 ), CRV( 1 ) ) ! .. ! .. Data statements .. DATA CSWAP / .FALSE., .FALSE., .TRUE., .TRUE. / DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. / DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4, & 3, 2, 1 / ! .. ! .. Executable Statements .. ! ! Compute BIGNUM ! SMLNUM = TWO*SLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM SMINI = MAX( SMIN, SMLNUM ) ! ! Don't check for input errors ! INFO = 0 ! ! Standard Initializations ! SCALE = ONE ! IF( NA.EQ.1 ) THEN ! ! 1 x 1 (i.e., scalar) system C X = B ! IF( NW.EQ.1 ) THEN ! ! Real 1x1 system. ! ! C = ca A - w D ! CSR = CA*A( 1, 1 ) - WR*D1 CNORM = ABS( CSR ) ! ! If | C | < SMINI, use C = SMINI ! IF( CNORM.LT.SMINI ) THEN CSR = SMINI CNORM = SMINI INFO = 1 END IF ! ! Check scaling for X = B / C ! BNORM = ABS( B( 1, 1 ) ) IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*CNORM ) & SCALE = ONE / BNORM END IF ! ! Compute X ! X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR XNORM = ABS( X( 1, 1 ) ) ELSE ! ! Complex 1x1 system (w is complex) ! ! C = ca A - w D ! CSR = CA*A( 1, 1 ) - WR*D1 CSI = -WI*D1 CNORM = ABS( CSR ) + ABS( CSI ) ! ! If | C | < SMINI, use C = SMINI ! IF( CNORM.LT.SMINI ) THEN CSR = SMINI CSI = ZERO CNORM = SMINI INFO = 1 END IF ! ! Check scaling for X = B / C ! BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) ) IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*CNORM ) & SCALE = ONE / BNORM END IF ! ! Compute X ! CALL SLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI, & X( 1, 1 ), X( 1, 2 ) ) XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) END IF ! ELSE ! ! 2x2 System ! ! Compute the real part of C = ca A - w D (or ca A' - w D ) ! CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1 CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2 IF( LTRANS ) THEN CR( 1, 2 ) = CA*A( 2, 1 ) CR( 2, 1 ) = CA*A( 1, 2 ) ELSE CR( 2, 1 ) = CA*A( 2, 1 ) CR( 1, 2 ) = CA*A( 1, 2 ) END IF ! IF( NW.EQ.1 ) THEN ! ! Real 2x2 system (w is real) ! ! Find the largest element in C ! CMAX = ZERO ICMAX = 0 ! DO 10 J = 1, 4 IF( ABS( CRV( J ) ).GT.CMAX ) THEN CMAX = ABS( CRV( J ) ) ICMAX = J END IF 10 CONTINUE ! ! If norm(C) < SMINI, use SMINI*identity. ! IF( CMAX.LT.SMINI ) THEN BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) ) IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*SMINI ) & SCALE = ONE / BNORM END IF TEMP = SCALE / SMINI X( 1, 1 ) = TEMP*B( 1, 1 ) X( 2, 1 ) = TEMP*B( 2, 1 ) XNORM = TEMP*BNORM INFO = 1 RETURN END IF ! ! Gaussian elimination with complete pivoting. ! UR11 = CRV( ICMAX ) CR21 = CRV( IPIVOT( 2, ICMAX ) ) UR12 = CRV( IPIVOT( 3, ICMAX ) ) CR22 = CRV( IPIVOT( 4, ICMAX ) ) UR11R = ONE / UR11 LR21 = UR11R*CR21 UR22 = CR22 - UR12*LR21 ! ! If smaller pivot < SMINI, use SMINI ! IF( ABS( UR22 ).LT.SMINI ) THEN UR22 = SMINI INFO = 1 END IF IF( RSWAP( ICMAX ) ) THEN BR1 = B( 2, 1 ) BR2 = B( 1, 1 ) ELSE BR1 = B( 1, 1 ) BR2 = B( 2, 1 ) END IF BR2 = BR2 - LR21*BR1 BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) ) IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN IF( BBND.GE.BIGNUM*ABS( UR22 ) ) & SCALE = ONE / BBND END IF ! XR2 = ( BR2*SCALE ) / UR22 XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 ) IF( CSWAP( ICMAX ) ) THEN X( 1, 1 ) = XR2 X( 2, 1 ) = XR1 ELSE X( 1, 1 ) = XR1 X( 2, 1 ) = XR2 END IF XNORM = MAX( ABS( XR1 ), ABS( XR2 ) ) ! ! Further scaling if norm(A) norm(X) > overflow ! IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN IF( XNORM.GT.BIGNUM / CMAX ) THEN TEMP = CMAX / BIGNUM X( 1, 1 ) = TEMP*X( 1, 1 ) X( 2, 1 ) = TEMP*X( 2, 1 ) XNORM = TEMP*XNORM SCALE = TEMP*SCALE END IF END IF ELSE ! ! Complex 2x2 system (w is complex) ! ! Find the largest element in C ! CI( 1, 1 ) = -WI*D1 CI( 2, 1 ) = ZERO CI( 1, 2 ) = ZERO CI( 2, 2 ) = -WI*D2 CMAX = ZERO ICMAX = 0 ! DO 20 J = 1, 4 IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) ) ICMAX = J END IF 20 CONTINUE ! ! If norm(C) < SMINI, use SMINI*identity. ! IF( CMAX.LT.SMINI ) THEN BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ), & ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) ) IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*SMINI ) & SCALE = ONE / BNORM END IF TEMP = SCALE / SMINI X( 1, 1 ) = TEMP*B( 1, 1 ) X( 2, 1 ) = TEMP*B( 2, 1 ) X( 1, 2 ) = TEMP*B( 1, 2 ) X( 2, 2 ) = TEMP*B( 2, 2 ) XNORM = TEMP*BNORM INFO = 1 RETURN END IF ! ! Gaussian elimination with complete pivoting. ! UR11 = CRV( ICMAX ) UI11 = CIV( ICMAX ) CR21 = CRV( IPIVOT( 2, ICMAX ) ) CI21 = CIV( IPIVOT( 2, ICMAX ) ) UR12 = CRV( IPIVOT( 3, ICMAX ) ) UI12 = CIV( IPIVOT( 3, ICMAX ) ) CR22 = CRV( IPIVOT( 4, ICMAX ) ) CI22 = CIV( IPIVOT( 4, ICMAX ) ) IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN ! ! Code when off-diagonals of pivoted C are real ! IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN TEMP = UI11 / UR11 UR11R = ONE / ( UR11*( ONE+TEMP**2 ) ) UI11R = -TEMP*UR11R ELSE TEMP = UR11 / UI11 UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) ) UR11R = -TEMP*UI11R END IF LR21 = CR21*UR11R LI21 = CR21*UI11R UR12S = UR12*UR11R UI12S = UR12*UI11R UR22 = CR22 - UR12*LR21 UI22 = CI22 - UR12*LI21 ELSE ! ! Code when diagonals of pivoted C are real ! UR11R = ONE / UR11 UI11R = ZERO LR21 = CR21*UR11R LI21 = CI21*UR11R UR12S = UR12*UR11R UI12S = UI12*UR11R UR22 = CR22 - UR12*LR21 + UI12*LI21 UI22 = -UR12*LI21 - UI12*LR21 END IF U22ABS = ABS( UR22 ) + ABS( UI22 ) ! ! If smaller pivot < SMINI, use SMINI ! IF( U22ABS.LT.SMINI ) THEN UR22 = SMINI UI22 = ZERO INFO = 1 END IF IF( RSWAP( ICMAX ) ) THEN BR2 = B( 1, 1 ) BR1 = B( 2, 1 ) BI2 = B( 1, 2 ) BI1 = B( 2, 2 ) ELSE BR1 = B( 1, 1 ) BR2 = B( 2, 1 ) BI1 = B( 1, 2 ) BI2 = B( 2, 2 ) END IF BR2 = BR2 - LR21*BR1 + LI21*BI1 BI2 = BI2 - LI21*BR1 - LR21*BI1 BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )* & ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ), & ABS( BR2 )+ABS( BI2 ) ) IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN IF( BBND.GE.BIGNUM*U22ABS ) THEN SCALE = ONE / BBND BR1 = SCALE*BR1 BI1 = SCALE*BI1 BR2 = SCALE*BR2 BI2 = SCALE*BI2 END IF END IF ! CALL SLADIV( BR2, BI2, UR22, UI22, XR2, XI2 ) XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2 XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2 IF( CSWAP( ICMAX ) ) THEN X( 1, 1 ) = XR2 X( 2, 1 ) = XR1 X( 1, 2 ) = XI2 X( 2, 2 ) = XI1 ELSE X( 1, 1 ) = XR1 X( 2, 1 ) = XR2 X( 1, 2 ) = XI1 X( 2, 2 ) = XI2 END IF XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) ) ! ! Further scaling if norm(A) norm(X) > overflow ! IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN IF( XNORM.GT.BIGNUM / CMAX ) THEN TEMP = CMAX / BIGNUM X( 1, 1 ) = TEMP*X( 1, 1 ) X( 2, 1 ) = TEMP*X( 2, 1 ) X( 1, 2 ) = TEMP*X( 1, 2 ) X( 2, 2 ) = TEMP*X( 2, 2 ) XNORM = TEMP*XNORM SCALE = TEMP*SCALE END IF END IF END IF END IF ! RETURN ! ! End of SLALN2 ! END SUBROUTINE SLALN2 !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SLANGE( NORM, M, N, A, LDA, WORK ) real(r8) slange ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER LDA, M, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SLANGE returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real matrix A. ! ! Description ! =========== ! ! SLANGE returns the value ! ! SLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in SLANGE as described ! above. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. When M = 0, ! SLANGE is set to zero. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. When N = 0, ! SLANGE is set to zero. ! ! A (input) REAL array, dimension (LDA,N) ! The m by n matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(M,1). ! ! WORK (workspace) REAL array, dimension (LWORK), ! where LWORK >= M when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, J REAL(r8) SCALE, SUM, VALUE ! .. ! .. Executable Statements .. ! IF( MIN( M, N ).EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO DO 20 J = 1, N DO 10 I = 1, M VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO DO 40 J = 1, N SUM = ZERO DO 30 I = 1, M SUM = SUM + ABS( A( I, J ) ) 30 CONTINUE VALUE = MAX( VALUE, SUM ) 40 CONTINUE ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! DO 50 I = 1, M WORK( I ) = ZERO 50 CONTINUE DO 70 J = 1, N DO 60 I = 1, M WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 60 CONTINUE 70 CONTINUE VALUE = ZERO DO 80 I = 1, M VALUE = MAX( VALUE, WORK( I ) ) 80 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE DO 90 J = 1, N CALL SLASSQ( M, A( 1, J ), 1, SCALE, SUM ) 90 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! SLANGE = VALUE RETURN ! ! End of SLANGE ! END FUNCTION SLANGE !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SLANHS( NORM, N, A, LDA, WORK ) real(r8) slanhs ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER LDA, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SLANHS returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! Hessenberg matrix A. ! ! Description ! =========== ! ! SLANHS returns the value ! ! SLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in SLANHS as described ! above. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, SLANHS is ! set to zero. ! ! A (input) REAL array, dimension (LDA,N) ! The n by n upper Hessenberg matrix A; the part of A below the ! first sub-diagonal is not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(N,1). ! ! WORK (workspace) REAL array, dimension (LWORK), ! where LWORK >= N when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, J REAL(r8) SCALE, SUM, VALUE ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO DO 20 J = 1, N DO 10 I = 1, MIN( N, J+1 ) VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO DO 40 J = 1, N SUM = ZERO DO 30 I = 1, MIN( N, J+1 ) SUM = SUM + ABS( A( I, J ) ) 30 CONTINUE VALUE = MAX( VALUE, SUM ) 40 CONTINUE ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! DO 50 I = 1, N WORK( I ) = ZERO 50 CONTINUE DO 70 J = 1, N DO 60 I = 1, MIN( N, J+1 ) WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 60 CONTINUE 70 CONTINUE VALUE = ZERO DO 80 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 80 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE DO 90 J = 1, N CALL SLASSQ( MIN( N, J+1 ), A( 1, J ), 1, SCALE, SUM ) 90 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! SLANHS = VALUE RETURN ! ! End of SLANHS ! END FUNCTION SLANHS !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. REAL(r8) A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN ! .. ! ! Purpose ! ======= ! ! SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric ! matrix in standard form: ! ! [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] ! [ C D ] [ SN CS ] [ CC DD ] [-SN CS ] ! ! where either ! 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or ! 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex ! conjugate eigenvalues. ! ! Arguments ! ========= ! ! A (input/output) REAL ! B (input/output) REAL ! C (input/output) REAL ! D (input/output) REAL ! On entry, the elements of the input matrix. ! On exit, they are overwritten by the elements of the ! standardised Schur form. ! ! RT1R (output) REAL ! RT1I (output) REAL ! RT2R (output) REAL ! RT2I (output) REAL ! The real and imaginary parts of the eigenvalues. If the ! eigenvalues are both real, abs(RT1R) >= abs(RT2R); if the ! eigenvalues are a complex conjugate pair, RT1I > 0. ! ! CS (output) REAL ! SN (output) REAL ! Parameters of the rotation matrix. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, HALF, ONE PARAMETER ( ZERO = 0.0E+0_r8, HALF = 0.5E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. REAL(r8) AA, BB, CC, CS1, DD, P, SAB, SAC, SIGMA, SN1, & TAU, TEMP ! .. ! .. Executable Statements .. ! ! Initialize CS and SN ! CS = ONE SN = ZERO ! IF( C.EQ.ZERO ) THEN GO TO 10 ! ELSE IF( B.EQ.ZERO ) THEN ! ! Swap rows and columns ! CS = ZERO SN = ONE TEMP = D D = A A = TEMP B = -C C = ZERO GO TO 10 ELSE IF( (A-D).EQ.ZERO .AND. SIGN( ONE, B ).NE. & SIGN( ONE, C ) ) THEN GO TO 10 ELSE ! ! Make diagonal elements equal ! TEMP = A - D P = HALF*TEMP SIGMA = B + C TAU = SLAPY2( SIGMA, TEMP ) CS1 = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) ) SN1 = -( P / ( TAU*CS1 ) )*SIGN( ONE, SIGMA ) ! ! Compute [ AA BB ] = [ A B ] [ CS1 -SN1 ] ! [ CC DD ] [ C D ] [ SN1 CS1 ] ! AA = A*CS1 + B*SN1 BB = -A*SN1 + B*CS1 CC = C*CS1 + D*SN1 DD = -C*SN1 + D*CS1 ! ! Compute [ A B ] = [ CS1 SN1 ] [ AA BB ] ! [ C D ] [-SN1 CS1 ] [ CC DD ] ! A = AA*CS1 + CC*SN1 B = BB*CS1 + DD*SN1 C = -AA*SN1 + CC*CS1 D = -BB*SN1 + DD*CS1 ! ! Accumulate transformation ! TEMP = CS*CS1 - SN*SN1 SN = CS*SN1 + SN*CS1 CS = TEMP ! TEMP = HALF*( A+D ) A = TEMP D = TEMP ! IF( C.NE.ZERO ) THEN IF ( B.NE.ZERO ) THEN IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN ! ! Real eigenvalues: reduce to upper triangular form ! SAB = SQRT( ABS( B ) ) SAC = SQRT( ABS( C ) ) P = SIGN( SAB*SAC, C ) TAU = ONE / SQRT( ABS( B+C ) ) A = TEMP + P D = TEMP - P B = B - C C = ZERO CS1 = SAB*TAU SN1 = SAC*TAU TEMP = CS*CS1 - SN*SN1 SN = CS*SN1 + SN*CS1 CS = TEMP END IF ELSE B = -C C = ZERO TEMP = CS CS = -SN SN = TEMP ENDIF ENDIF END IF ! 10 CONTINUE ! ! Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I). ! RT1R = A RT2R = D IF( C.EQ.ZERO ) THEN RT1I = ZERO RT2I = ZERO ELSE RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) ) RT2I = -RT1I END IF RETURN ! ! End of SLANV2 ! END SUBROUTINE SLANV2 !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SLAPY2( X, Y ) real(r8) slapy2 ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. REAL(r8) X, Y ! .. ! ! Purpose ! ======= ! ! SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary ! overflow. ! ! Arguments ! ========= ! ! X (input) REAL ! Y (input) REAL ! X and Y specify the values x and y. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E0_r8 ) REAL(r8) ONE PARAMETER ( ONE = 1.0E0_r8 ) ! .. ! .. Local Scalars .. REAL(r8) W, XABS, YABS, Z ! .. ! .. Executable Statements .. ! XABS = ABS( X ) YABS = ABS( Y ) W = MAX( XABS, YABS ) Z = MIN( XABS, YABS ) IF( Z.EQ.ZERO ) THEN SLAPY2 = W ELSE SLAPY2 = W*SQRT( ONE+( Z / W )**2 ) END IF RETURN ! ! End of SLAPY2 ! END FUNCTION SLAPY2 !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER INCV, LDC, M, N REAL(r8) TAU ! .. ! .. Array Arguments .. REAL(r8) C( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SLARF applies a real elementary reflector H to a real m by n matrix ! C, from either the left or the right. H is represented in the form ! ! H = I - tau * v * v' ! ! where tau is a real scalar and v is a real vector. ! ! If tau = 0, then H is taken to be the unit matrix. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form H * C ! = 'R': form C * H ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! V (input) REAL array, dimension ! (1 + (M-1)*abs(INCV)) if SIDE = 'L' ! or (1 + (N-1)*abs(INCV)) if SIDE = 'R' ! The vector v in the representation of H. V is not used if ! TAU = 0. ! ! INCV (input) INTEGER ! The increment between elements of v. INCV <> 0. ! ! TAU (input) REAL ! The value tau in the representation of H. ! ! C (input/output) REAL array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by the matrix H * C if SIDE = 'L', ! or C * H if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) REAL array, dimension ! (N) if SIDE = 'L' ! or (M) if SIDE = 'R' ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Executable Statements .. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C ! IF( TAU.NE.ZERO ) THEN ! ! w := C' * v ! CALL SGEMV( 'Transpose', M, N, ONE, C, LDC, V, INCV, ZERO, & WORK, 1 ) ! ! C := C - v * w' ! CALL SGER( M, N, -TAU, V, INCV, WORK, 1, C, LDC ) END IF ELSE ! ! Form C * H ! IF( TAU.NE.ZERO ) THEN ! ! w := C * v ! CALL SGEMV( 'No transpose', M, N, ONE, C, LDC, V, INCV, & ZERO, WORK, 1 ) ! ! C := C - w * v' ! CALL SGER( M, N, -TAU, WORK, 1, V, INCV, C, LDC ) END IF END IF RETURN ! ! End of SLARF ! END SUBROUTINE SLARF !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, & T, LDT, C, LDC, WORK, LDWORK ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER DIRECT, SIDE, STOREV, TRANS INTEGER K, LDC, LDT, LDV, LDWORK, M, N ! .. ! .. Array Arguments .. REAL(r8) C( LDC, * ), T( LDT, * ), V( LDV, * ), & WORK( LDWORK, * ) ! .. ! ! Purpose ! ======= ! ! SLARFB applies a real block reflector H or its transpose H' to a ! real m by n matrix C, from either the left or the right. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply H or H' from the Left ! = 'R': apply H or H' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply H (No transpose) ! = 'T': apply H' (Transpose) ! ! DIRECT (input) CHARACTER*1 ! Indicates how H is formed from a product of elementary ! reflectors ! = 'F': H = H(1) H(2) . . . H(k) (Forward) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Indicates how the vectors which define the elementary ! reflectors are stored: ! = 'C': Columnwise ! = 'R': Rowwise ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! K (input) INTEGER ! The order of the matrix T (= the number of elementary ! reflectors whose product defines the block reflector). ! ! V (input) REAL array, dimension ! (LDV,K) if STOREV = 'C' ! (LDV,M) if STOREV = 'R' and SIDE = 'L' ! (LDV,N) if STOREV = 'R' and SIDE = 'R' ! The matrix V. See further details. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); ! if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); ! if STOREV = 'R', LDV >= K. ! ! T (input) REAL array, dimension (LDT,K) ! The triangular k by k matrix T in the representation of the ! block reflector. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! C (input/output) REAL array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by H*C or H'*C or C*H or C*H'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDA >= max(1,M). ! ! WORK (workspace) REAL array, dimension (LDWORK,K) ! ! LDWORK (input) INTEGER ! The leading dimension of the array WORK. ! If SIDE = 'L', LDWORK >= max(1,N); ! if SIDE = 'R', LDWORK >= max(1,M). ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE PARAMETER ( ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. CHARACTER TRANST INTEGER I, J ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) & RETURN ! IF( LSAME( TRANS, 'N' ) ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! IF( LSAME( STOREV, 'C' ) ) THEN ! IF( LSAME( DIRECT, 'F' ) ) THEN ! ! Let V = ( V1 ) (first K rows) ! ( V2 ) ! where V1 is unit lower triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK) ! ! W := C1' ! DO 10 J = 1, K CALL SCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 ) 10 CONTINUE ! ! W := W * V1 ! CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', N, & K, ONE, V, LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C2'*V2 ! CALL SGEMM( 'Transpose', 'No transpose', N, K, M-K, & ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL STRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V * W' ! IF( M.GT.K ) THEN ! ! C2 := C2 - V2 * W' ! CALL SGEMM( 'No transpose', 'Transpose', M-K, N, K, & -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE, & C( K+1, 1 ), LDC ) END IF ! ! W := W * V1' ! CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K, & ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W' ! DO 30 J = 1, K DO 20 I = 1, N C( J, I ) = C( J, I ) - WORK( I, J ) 20 CONTINUE 30 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V = (C1*V1 + C2*V2) (stored in WORK) ! ! W := C1 ! DO 40 J = 1, K CALL SCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 ) 40 CONTINUE ! ! W := W * V1 ! CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', M, & K, ONE, V, LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C2 * V2 ! CALL SGEMM( 'No transpose', 'No transpose', M, K, N-K, & ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL STRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V' ! IF( N.GT.K ) THEN ! ! C2 := C2 - W * V2' ! CALL SGEMM( 'No transpose', 'Transpose', M, N-K, K, & -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE, & C( 1, K+1 ), LDC ) END IF ! ! W := W * V1' ! CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K, & ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 60 J = 1, K DO 50 I = 1, M C( I, J ) = C( I, J ) - WORK( I, J ) 50 CONTINUE 60 CONTINUE END IF ! ELSE ! ! Let V = ( V1 ) ! ( V2 ) (last K rows) ! where V2 is unit upper triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK) ! ! W := C2' ! DO 70 J = 1, K CALL SCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 ) 70 CONTINUE ! ! W := W * V2 ! CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', N, & K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C1'*V1 ! CALL SGEMM( 'Transpose', 'No transpose', N, K, M-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL STRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V * W' ! IF( M.GT.K ) THEN ! ! C1 := C1 - V1 * W' ! CALL SGEMM( 'No transpose', 'Transpose', M-K, N, K, & -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC ) END IF ! ! W := W * V2' ! CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K, & ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W' ! DO 90 J = 1, K DO 80 I = 1, N C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J ) 80 CONTINUE 90 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V = (C1*V1 + C2*V2) (stored in WORK) ! ! W := C2 ! DO 100 J = 1, K CALL SCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 ) 100 CONTINUE ! ! W := W * V2 ! CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', M, & K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C1 * V1 ! CALL SGEMM( 'No transpose', 'No transpose', M, K, N-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL STRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V' ! IF( N.GT.K ) THEN ! ! C1 := C1 - W * V1' ! CALL SGEMM( 'No transpose', 'Transpose', M, N-K, K, & -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC ) END IF ! ! W := W * V2' ! CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K, & ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W ! DO 120 J = 1, K DO 110 I = 1, M C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J ) 110 CONTINUE 120 CONTINUE END IF END IF ! ELSE IF( LSAME( STOREV, 'R' ) ) THEN ! IF( LSAME( DIRECT, 'F' ) ) THEN ! ! Let V = ( V1 V2 ) (V1: first K columns) ! where V1 is unit upper triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK) ! ! W := C1' ! DO 130 J = 1, K CALL SCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 ) 130 CONTINUE ! ! W := W * V1' ! CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K, & ONE, V, LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C2'*V2' ! CALL SGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE, & C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE, & WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL STRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V' * W' ! IF( M.GT.K ) THEN ! ! C2 := C2 - V2' * W' ! CALL SGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE, & V( 1, K+1 ), LDV, WORK, LDWORK, ONE, & C( K+1, 1 ), LDC ) END IF ! ! W := W * V1 ! CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', N, & K, ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W' ! DO 150 J = 1, K DO 140 I = 1, N C( J, I ) = C( J, I ) - WORK( I, J ) 140 CONTINUE 150 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V' = (C1*V1' + C2*V2') (stored in WORK) ! ! W := C1 ! DO 160 J = 1, K CALL SCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 ) 160 CONTINUE ! ! W := W * V1' ! CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K, & ONE, V, LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C2 * V2' ! CALL SGEMM( 'No transpose', 'Transpose', M, K, N-K, & ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL STRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V ! IF( N.GT.K ) THEN ! ! C2 := C2 - W * V2 ! CALL SGEMM( 'No transpose', 'No transpose', M, N-K, K, & -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE, & C( 1, K+1 ), LDC ) END IF ! ! W := W * V1 ! CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', M, & K, ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 180 J = 1, K DO 170 I = 1, M C( I, J ) = C( I, J ) - WORK( I, J ) 170 CONTINUE 180 CONTINUE ! END IF ! ELSE ! ! Let V = ( V1 V2 ) (V2: last K columns) ! where V2 is unit lower triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK) ! ! W := C2' ! DO 190 J = 1, K CALL SCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 ) 190 CONTINUE ! ! W := W * V2' ! CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K, & ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C1'*V1' ! CALL SGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE, & C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL STRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V' * W' ! IF( M.GT.K ) THEN ! ! C1 := C1 - V1' * W' ! CALL SGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE, & V, LDV, WORK, LDWORK, ONE, C, LDC ) END IF ! ! W := W * V2 ! CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', N, & K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W' ! DO 210 J = 1, K DO 200 I = 1, N C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J ) 200 CONTINUE 210 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V' = (C1*V1' + C2*V2') (stored in WORK) ! ! W := C2 ! DO 220 J = 1, K CALL SCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 ) 220 CONTINUE ! ! W := W * V2' ! CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K, & ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C1 * V1' ! CALL SGEMM( 'No transpose', 'Transpose', M, K, N-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL STRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V ! IF( N.GT.K ) THEN ! ! C1 := C1 - W * V1 ! CALL SGEMM( 'No transpose', 'No transpose', M, N-K, K, & -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC ) END IF ! ! W := W * V2 ! CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', M, & K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 240 J = 1, K DO 230 I = 1, M C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J ) 230 CONTINUE 240 CONTINUE ! END IF ! END IF END IF ! RETURN ! ! End of SLARFB ! END SUBROUTINE SLARFB !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARFG( N, ALPHA, X, INCX, TAU ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INCX, N REAL(r8) ALPHA, TAU ! .. ! .. Array Arguments .. REAL(r8) X( * ) ! .. ! ! Purpose ! ======= ! ! SLARFG generates a real elementary reflector H of order n, such ! that ! ! H * ( alpha ) = ( beta ), H' * H = I. ! ( x ) ( 0 ) ! ! where alpha and beta are scalars, and x is an (n-1)-element real ! vector. H is represented in the form ! ! H = I - tau * ( 1 ) * ( 1 v' ) , ! ( v ) ! ! where tau is a real scalar and v is a real (n-1)-element ! vector. ! ! If the elements of x are all zero, then tau = 0 and H is taken to be ! the unit matrix. ! ! Otherwise 1 <= tau <= 2. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the elementary reflector. ! ! ALPHA (input/output) REAL ! On entry, the value alpha. ! On exit, it is overwritten with the value beta. ! ! X (input/output) REAL array, dimension ! (1+(N-2)*abs(INCX)) ! On entry, the vector x. ! On exit, it is overwritten with the vector v. ! ! INCX (input) INTEGER ! The increment between elements of X. INCX > 0. ! ! TAU (output) REAL ! The value tau. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER J, KNT REAL(r8) BETA, RSAFMN, SAFMIN, XNORM ! .. ! .. Executable Statements .. ! IF( N.LE.1 ) THEN TAU = ZERO RETURN END IF ! XNORM = SNRM2( N-1, X, INCX ) ! IF( XNORM.EQ.ZERO ) THEN ! ! H = I ! TAU = ZERO ELSE ! ! general case ! BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA ) SAFMIN = SLAMCH( 'S' ) / SLAMCH( 'E' ) IF( ABS( BETA ).LT.SAFMIN ) THEN ! ! XNORM, BETA may be inaccurate; scale X and recompute them ! RSAFMN = ONE / SAFMIN KNT = 0 10 CONTINUE KNT = KNT + 1 CALL SSCAL( N-1, RSAFMN, X, INCX ) BETA = BETA*RSAFMN ALPHA = ALPHA*RSAFMN IF( ABS( BETA ).LT.SAFMIN ) & GO TO 10 ! ! New BETA is at most 1, at least SAFMIN ! XNORM = SNRM2( N-1, X, INCX ) BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA ) TAU = ( BETA-ALPHA ) / BETA CALL SSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX ) ! ! If ALPHA is subnormal, it may lose relative accuracy ! ALPHA = BETA DO 20 J = 1, KNT ALPHA = ALPHA*SAFMIN 20 CONTINUE ELSE TAU = ( BETA-ALPHA ) / BETA CALL SSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX ) ALPHA = BETA END IF END IF ! RETURN ! ! End of SLARFG ! END SUBROUTINE SLARFG !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER DIRECT, STOREV INTEGER K, LDT, LDV, N ! .. ! .. Array Arguments .. REAL(r8) T( LDT, * ), TAU( * ), V( LDV, * ) ! .. ! ! Purpose ! ======= ! ! SLARFT forms the triangular factor T of a real block reflector H ! of order n, which is defined as a product of k elementary reflectors. ! ! If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; ! ! If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. ! ! If STOREV = 'C', the vector which defines the elementary reflector ! H(i) is stored in the i-th column of the array V, and ! ! H = I - V * T * V' ! ! If STOREV = 'R', the vector which defines the elementary reflector ! H(i) is stored in the i-th row of the array V, and ! ! H = I - V' * T * V ! ! Arguments ! ========= ! ! DIRECT (input) CHARACTER*1 ! Specifies the order in which the elementary reflectors are ! multiplied to form the block reflector: ! = 'F': H = H(1) H(2) . . . H(k) (Forward) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Specifies how the vectors which define the elementary ! reflectors are stored (see also Further Details): ! = 'C': columnwise ! = 'R': rowwise ! ! N (input) INTEGER ! The order of the block reflector H. N >= 0. ! ! K (input) INTEGER ! The order of the triangular factor T (= the number of ! elementary reflectors). K >= 1. ! ! V (input/output) REAL array, dimension ! (LDV,K) if STOREV = 'C' ! (LDV,N) if STOREV = 'R' ! The matrix V. See further details. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. ! ! TAU (input) REAL array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i). ! ! T (output) REAL array, dimension (LDT,K) ! The k by k triangular factor T of the block reflector. ! If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is ! lower triangular. The rest of the array is not used. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! Further Details ! =============== ! ! The shape of the matrix V and the storage of the vectors which define ! the H(i) is best illustrated by the following example with n = 5 and ! k = 3. The elements equal to 1 are not stored; the corresponding ! array elements are modified but restored on exit. The rest of the ! array is not used. ! ! DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ! ! V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ! ( v1 1 ) ( 1 v2 v2 v2 ) ! ( v1 v2 1 ) ( 1 v3 v3 ) ! ( v1 v2 v3 ) ! ( v1 v2 v3 ) ! ! DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ! ! V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ! ( v1 v2 v3 ) ( v2 v2 v2 1 ) ! ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ! ( 1 v3 ) ! ( 1 ) ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, J REAL(r8) VII ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( LSAME( DIRECT, 'F' ) ) THEN DO 20 I = 1, K IF( TAU( I ).EQ.ZERO ) THEN ! ! H(i) = I ! DO 10 J = 1, I T( J, I ) = ZERO 10 CONTINUE ELSE ! ! general case ! VII = V( I, I ) V( I, I ) = ONE IF( LSAME( STOREV, 'C' ) ) THEN ! ! T(1:i-1,i) := - tau(i) * V(i:n,1:i-1)' * V(i:n,i) ! CALL SGEMV( 'Transpose', N-I+1, I-1, -TAU( I ), & V( I, 1 ), LDV, V( I, I ), 1, ZERO, & T( 1, I ), 1 ) ELSE ! ! T(1:i-1,i) := - tau(i) * V(1:i-1,i:n) * V(i,i:n)' ! CALL SGEMV( 'No transpose', I-1, N-I+1, -TAU( I ), & V( 1, I ), LDV, V( I, I ), LDV, ZERO, & T( 1, I ), 1 ) END IF V( I, I ) = VII ! ! T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i) ! CALL STRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, & LDT, T( 1, I ), 1 ) T( I, I ) = TAU( I ) END IF 20 CONTINUE ELSE DO 40 I = K, 1, -1 IF( TAU( I ).EQ.ZERO ) THEN ! ! H(i) = I ! DO 30 J = I, K T( J, I ) = ZERO 30 CONTINUE ELSE ! ! general case ! IF( I.LT.K ) THEN IF( LSAME( STOREV, 'C' ) ) THEN VII = V( N-K+I, I ) V( N-K+I, I ) = ONE ! ! T(i+1:k,i) := ! - tau(i) * V(1:n-k+i,i+1:k)' * V(1:n-k+i,i) ! CALL SGEMV( 'Transpose', N-K+I, K-I, -TAU( I ), & V( 1, I+1 ), LDV, V( 1, I ), 1, ZERO, & T( I+1, I ), 1 ) V( N-K+I, I ) = VII ELSE VII = V( I, N-K+I ) V( I, N-K+I ) = ONE ! ! T(i+1:k,i) := ! - tau(i) * V(i+1:k,1:n-k+i) * V(i,1:n-k+i)' ! CALL SGEMV( 'No transpose', K-I, N-K+I, -TAU( I ), & V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO, & T( I+1, I ), 1 ) V( I, N-K+I ) = VII END IF ! ! T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i) ! CALL STRMV( 'Lower', 'No transpose', 'Non-unit', K-I, & T( I+1, I+1 ), LDT, T( I+1, I ), 1 ) END IF T( I, I ) = TAU( I ) END IF 40 CONTINUE END IF RETURN ! ! End of SLARFT ! END SUBROUTINE SLARFT !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARFX( SIDE, M, N, V, TAU, C, LDC, WORK ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER LDC, M, N REAL(r8) TAU ! .. ! .. Array Arguments .. REAL(r8) C( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SLARFX applies a real elementary reflector H to a real m by n ! matrix C, from either the left or the right. H is represented in the ! form ! ! H = I - tau * v * v' ! ! where tau is a real scalar and v is a real vector. ! ! If tau = 0, then H is taken to be the unit matrix ! ! This version uses inline code if H has order < 11. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form H * C ! = 'R': form C * H ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! V (input) REAL array, dimension (M) if SIDE = 'L' ! or (N) if SIDE = 'R' ! The vector v in the representation of H. ! ! TAU (input) REAL ! The value tau in the representation of H. ! ! C (input/output) REAL array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by the matrix H * C if SIDE = 'L', ! or C * H if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDA >= (1,M). ! ! WORK (workspace) REAL array, dimension ! (N) if SIDE = 'L' ! or (M) if SIDE = 'R' ! WORK is not referenced if H has order < 11. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER J REAL(r8) SUM, T1, T10, T2, T3, T4, T5, T6, T7, T8, T9, & V1, V10, V2, V3, V4, V5, V6, V7, V8, V9 ! .. ! .. Executable Statements .. ! IF( TAU.EQ.ZERO ) & RETURN IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C, where H has order m. ! GO TO ( 10, 30, 50, 70, 90, 110, 130, 150, & 170, 190 )M ! ! Code for general M ! ! w := C'*v ! CALL SGEMV( 'Transpose', M, N, ONE, C, LDC, V, 1, ZERO, WORK, & 1 ) ! ! C := C - tau * v * w' ! CALL SGER( M, N, -TAU, V, 1, WORK, 1, C, LDC ) GO TO 410 10 CONTINUE ! ! Special code for 1 x 1 Householder ! T1 = ONE - TAU*V( 1 )*V( 1 ) DO 20 J = 1, N C( 1, J ) = T1*C( 1, J ) 20 CONTINUE GO TO 410 30 CONTINUE ! ! Special code for 2 x 2 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 DO 40 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 40 CONTINUE GO TO 410 50 CONTINUE ! ! Special code for 3 x 3 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 DO 60 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 60 CONTINUE GO TO 410 70 CONTINUE ! ! Special code for 4 x 4 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 DO 80 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 80 CONTINUE GO TO 410 90 CONTINUE ! ! Special code for 5 x 5 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 DO 100 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 100 CONTINUE GO TO 410 110 CONTINUE ! ! Special code for 6 x 6 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 DO 120 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 120 CONTINUE GO TO 410 130 CONTINUE ! ! Special code for 7 x 7 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 DO 140 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 140 CONTINUE GO TO 410 150 CONTINUE ! ! Special code for 8 x 8 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 DO 160 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 160 CONTINUE GO TO 410 170 CONTINUE ! ! Special code for 9 x 9 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 DO 180 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 C( 9, J ) = C( 9, J ) - SUM*T9 180 CONTINUE GO TO 410 190 CONTINUE ! ! Special code for 10 x 10 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 V10 = V( 10 ) T10 = TAU*V10 DO 200 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) + & V10*C( 10, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 C( 9, J ) = C( 9, J ) - SUM*T9 C( 10, J ) = C( 10, J ) - SUM*T10 200 CONTINUE GO TO 410 ELSE ! ! Form C * H, where H has order n. ! GO TO ( 210, 230, 250, 270, 290, 310, 330, 350, & 370, 390 )N ! ! Code for general N ! ! w := C * v ! CALL SGEMV( 'No transpose', M, N, ONE, C, LDC, V, 1, ZERO, & WORK, 1 ) ! ! C := C - tau * w * v' ! CALL SGER( M, N, -TAU, WORK, 1, V, 1, C, LDC ) GO TO 410 210 CONTINUE ! ! Special code for 1 x 1 Householder ! T1 = ONE - TAU*V( 1 )*V( 1 ) DO 220 J = 1, M C( J, 1 ) = T1*C( J, 1 ) 220 CONTINUE GO TO 410 230 CONTINUE ! ! Special code for 2 x 2 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 DO 240 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 240 CONTINUE GO TO 410 250 CONTINUE ! ! Special code for 3 x 3 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 DO 260 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 260 CONTINUE GO TO 410 270 CONTINUE ! ! Special code for 4 x 4 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 DO 280 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 280 CONTINUE GO TO 410 290 CONTINUE ! ! Special code for 5 x 5 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 DO 300 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 300 CONTINUE GO TO 410 310 CONTINUE ! ! Special code for 6 x 6 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 DO 320 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 320 CONTINUE GO TO 410 330 CONTINUE ! ! Special code for 7 x 7 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 DO 340 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 340 CONTINUE GO TO 410 350 CONTINUE ! ! Special code for 8 x 8 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 DO 360 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 360 CONTINUE GO TO 410 370 CONTINUE ! ! Special code for 9 x 9 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 DO 380 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 C( J, 9 ) = C( J, 9 ) - SUM*T9 380 CONTINUE GO TO 410 390 CONTINUE ! ! Special code for 10 x 10 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 V10 = V( 10 ) T10 = TAU*V10 DO 400 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) + & V10*C( J, 10 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 C( J, 9 ) = C( J, 9 ) - SUM*T9 C( J, 10 ) = C( J, 10 ) - SUM*T10 400 CONTINUE GO TO 410 END IF 410 RETURN ! ! End of SLARFX ! END SUBROUTINE SLARFX !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLARTG( F, G, CS, SN, R ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. REAL(r8) CS, F, G, R, SN ! .. ! ! Purpose ! ======= ! ! SLARTG generate a plane rotation so that ! ! [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. ! [ -SN CS ] [ G ] [ 0 ] ! ! This is a slower, more accurate version of the BLAS1 routine SROTG, ! with the following other differences: ! F and G are unchanged on return. ! If G=0, then CS=1 and SN=0. ! If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any ! floating point operations (saves work in SBDSQR when ! there are zeros on the diagonal). ! ! If F exceeds G in magnitude, CS will be positive. ! ! Arguments ! ========= ! ! F (input) REAL ! The first component of vector to be rotated. ! ! G (input) REAL ! The second component of vector to be rotated. ! ! CS (output) REAL ! The cosine of the rotation. ! ! SN (output) REAL ! The sine of the rotation. ! ! R (output) REAL ! The nonzero component of the rotated vector. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E0_r8 ) REAL(r8) ONE PARAMETER ( ONE = 1.0E0_r8 ) REAL(r8) TWO PARAMETER ( TWO = 2.0E0_r8 ) ! .. ! .. Local Scalars .. LOGICAL FIRST INTEGER COUNT, I REAL(r8) EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE ! .. ! .. Save statement .. SAVE FIRST, SAFMX2, SAFMIN, SAFMN2 ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. SAFMIN = SLAMCH( 'S' ) EPS = SLAMCH( 'E' ) SAFMN2 = SLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) / & LOG( SLAMCH( 'B' ) ) / TWO ) SAFMX2 = ONE / SAFMN2 END IF IF( G.EQ.ZERO ) THEN CS = ONE SN = ZERO R = F ELSE IF( F.EQ.ZERO ) THEN CS = ZERO SN = ONE R = G ELSE F1 = F G1 = G SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.GE.SAFMX2 ) THEN COUNT = 0 10 CONTINUE COUNT = COUNT + 1 F1 = F1*SAFMN2 G1 = G1*SAFMN2 SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.GE.SAFMX2 ) & GO TO 10 R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R DO 20 I = 1, COUNT R = R*SAFMX2 20 CONTINUE ELSE IF( SCALE.LE.SAFMN2 ) THEN COUNT = 0 30 CONTINUE COUNT = COUNT + 1 F1 = F1*SAFMX2 G1 = G1*SAFMX2 SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.LE.SAFMN2 ) & GO TO 30 R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R DO 40 I = 1, COUNT R = R*SAFMN2 40 CONTINUE ELSE R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R END IF IF( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN CS = -CS SN = -SN R = -R END IF END IF RETURN ! ! End of SLARTG ! END SUBROUTINE SLARTG !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER TYPE INTEGER INFO, KL, KU, LDA, M, N REAL(r8) CFROM, CTO ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! SLASCL multiplies the M by N real matrix A by the real scalar ! CTO/CFROM. This is done without over/underflow as long as the final ! result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that ! A may be full, upper triangular, lower triangular, upper Hessenberg, ! or banded. ! ! Arguments ! ========= ! ! TYPE (input) CHARACTER*1 ! TYPE indices the storage type of the input matrix. ! = 'G': A is a full matrix. ! = 'L': A is a lower triangular matrix. ! = 'U': A is an upper triangular matrix. ! = 'H': A is an upper Hessenberg matrix. ! = 'B': A is a symmetric band matrix with lower bandwidth KL ! and upper bandwidth KU and with the only the lower ! half stored. ! = 'Q': A is a symmetric band matrix with lower bandwidth KL ! and upper bandwidth KU and with the only the upper ! half stored. ! = 'Z': A is a band matrix with lower bandwidth KL and upper ! bandwidth KU. ! ! KL (input) INTEGER ! The lower bandwidth of A. Referenced only if TYPE = 'B', ! 'Q' or 'Z'. ! ! KU (input) INTEGER ! The upper bandwidth of A. Referenced only if TYPE = 'B', ! 'Q' or 'Z'. ! ! CFROM (input) REAL ! CTO (input) REAL ! The matrix A is multiplied by CTO/CFROM. A(I,J) is computed ! without over/underflow if the final result CTO*A(I,J)/CFROM ! can be represented without over/underflow. CFROM must be ! nonzero. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) REAL array, dimension (LDA,M) ! The matrix to be multiplied by CTO/CFROM. See TYPE for the ! storage type. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! INFO (output) INTEGER ! 0 - successful exit ! <0 - if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E0_r8, ONE = 1.0E0_r8 ) ! .. ! .. Local Scalars .. LOGICAL DONE INTEGER I, ITYPE, J, K1, K2, K3, K4 REAL(r8) BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 ! IF( LSAME( TYPE, 'G' ) ) THEN ITYPE = 0 ELSE IF( LSAME( TYPE, 'L' ) ) THEN ITYPE = 1 ELSE IF( LSAME( TYPE, 'U' ) ) THEN ITYPE = 2 ELSE IF( LSAME( TYPE, 'H' ) ) THEN ITYPE = 3 ELSE IF( LSAME( TYPE, 'B' ) ) THEN ITYPE = 4 ELSE IF( LSAME( TYPE, 'Q' ) ) THEN ITYPE = 5 ELSE IF( LSAME( TYPE, 'Z' ) ) THEN ITYPE = 6 ELSE ITYPE = -1 END IF ! IF( ITYPE.EQ.-1 ) THEN INFO = -1 ELSE IF( CFROM.EQ.ZERO ) THEN INFO = -4 ELSE IF( M.LT.0 ) THEN INFO = -6 ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR. & ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN INFO = -7 ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN INFO = -9 ELSE IF( ITYPE.GE.4 ) THEN IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN INFO = -2 ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR. & ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) ) & THEN INFO = -3 ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR. & ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR. & ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN INFO = -9 END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLASCL', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. M.EQ.0 ) & RETURN ! ! Get machine parameters ! SMLNUM = SLAMCH( 'S' ) BIGNUM = ONE / SMLNUM ! CFROMC = CFROM CTOC = CTO ! 10 CONTINUE CFROM1 = CFROMC*SMLNUM CTO1 = CTOC / BIGNUM IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN MUL = SMLNUM DONE = .FALSE. CFROMC = CFROM1 ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN MUL = BIGNUM DONE = .FALSE. CTOC = CTO1 ELSE MUL = CTOC / CFROMC DONE = .TRUE. END IF ! IF( ITYPE.EQ.0 ) THEN ! ! Full matrix ! DO 30 J = 1, N DO 20 I = 1, M A( I, J ) = A( I, J )*MUL 20 CONTINUE 30 CONTINUE ! ELSE IF( ITYPE.EQ.1 ) THEN ! ! Lower triangular matrix ! DO 50 J = 1, N DO 40 I = J, M A( I, J ) = A( I, J )*MUL 40 CONTINUE 50 CONTINUE ! ELSE IF( ITYPE.EQ.2 ) THEN ! ! Upper triangular matrix ! DO 70 J = 1, N DO 60 I = 1, MIN( J, M ) A( I, J ) = A( I, J )*MUL 60 CONTINUE 70 CONTINUE ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! Upper Hessenberg matrix ! DO 90 J = 1, N DO 80 I = 1, MIN( J+1, M ) A( I, J ) = A( I, J )*MUL 80 CONTINUE 90 CONTINUE ! ELSE IF( ITYPE.EQ.4 ) THEN ! ! Lower half of a symmetric band matrix ! K3 = KL + 1 K4 = N + 1 DO 110 J = 1, N DO 100 I = 1, MIN( K3, K4-J ) A( I, J ) = A( I, J )*MUL 100 CONTINUE 110 CONTINUE ! ELSE IF( ITYPE.EQ.5 ) THEN ! ! Upper half of a symmetric band matrix ! K1 = KU + 2 K3 = KU + 1 DO 130 J = 1, N DO 120 I = MAX( K1-J, 1 ), K3 A( I, J ) = A( I, J )*MUL 120 CONTINUE 130 CONTINUE ! ELSE IF( ITYPE.EQ.6 ) THEN ! ! Band matrix ! K1 = KL + KU + 2 K2 = KL + 1 K3 = 2*KL + KU + 1 K4 = KL + KU + 1 + M DO 150 J = 1, N DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J ) A( I, J ) = A( I, J )*MUL 140 CONTINUE 150 CONTINUE ! END IF ! IF( .NOT.DONE ) & GO TO 10 ! RETURN ! ! End of SLASCL ! END SUBROUTINE SLASCL !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLASET( UPLO, M, N, ALPHA, BETA, A, LDA ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, M, N REAL(r8) ALPHA, BETA ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! SLASET initializes an m-by-n matrix A to BETA on the diagonal and ! ALPHA on the offdiagonals. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies the part of the matrix A to be set. ! = 'U': Upper triangular part is set; the strictly lower ! triangular part of A is not changed. ! = 'L': Lower triangular part is set; the strictly upper ! triangular part of A is not changed. ! Otherwise: All of the matrix A is set. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! ALPHA (input) REAL ! The constant to which the offdiagonal elements are to be set. ! ! BETA (input) REAL ! The constant to which the diagonal elements are to be set. ! ! A (input/output) REAL array, dimension (LDA,N) ! On exit, the leading m-by-n submatrix of A is set as follows: ! ! if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n, ! if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, ! otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, ! ! and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J ! .. ! .. Executable Statements .. ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Set the strictly upper triangular or trapezoidal part of the ! array to ALPHA. ! DO 20 J = 2, N DO 10 I = 1, MIN( J-1, M ) A( I, J ) = ALPHA 10 CONTINUE 20 CONTINUE ! ELSE IF( LSAME( UPLO, 'L' ) ) THEN ! ! Set the strictly lower triangular or trapezoidal part of the ! array to ALPHA. ! DO 40 J = 1, MIN( M, N ) DO 30 I = J + 1, M A( I, J ) = ALPHA 30 CONTINUE 40 CONTINUE ! ELSE ! ! Set the leading m-by-n submatrix to ALPHA. ! DO 60 J = 1, N DO 50 I = 1, M A( I, J ) = ALPHA 50 CONTINUE 60 CONTINUE END IF ! ! Set the first min(M,N) diagonal elements to BETA. ! DO 70 I = 1, MIN( M, N ) A( I, I ) = BETA 70 CONTINUE ! RETURN ! ! End of SLASET ! END SUBROUTINE SLASET !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SLASSQ( N, X, INCX, SCALE, SUMSQ ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER INCX, N REAL(r8) SCALE, SUMSQ ! .. ! .. Array Arguments .. REAL(r8) X( * ) ! .. ! ! Purpose ! ======= ! ! SLASSQ returns the values scl and smsq such that ! ! ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, ! ! where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is ! assumed to be non-negative and scl returns the value ! ! scl = max( scale, abs( x( i ) ) ). ! ! scale and sumsq must be supplied in SCALE and SUMSQ and ! scl and smsq are overwritten on SCALE and SUMSQ respectively. ! ! The routine makes only one pass through the vector x. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of elements to be used from the vector X. ! ! X (input) REAL ! The vector for which a scaled sum of squares is computed. ! x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n. ! ! INCX (input) INTEGER ! The increment between successive values of the vector X. ! INCX > 0. ! ! SCALE (input/output) REAL ! On entry, the value scale in the equation above. ! On exit, SCALE is overwritten with scl , the scaling factor ! for the sum of squares. ! ! SUMSQ (input/output) REAL ! On entry, the value sumsq in the equation above. ! On exit, SUMSQ is overwritten with smsq , the basic sum of ! squares from which scl has been factored out. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER IX REAL(r8) ABSXI ! .. ! .. Executable Statements .. ! IF( N.GT.0 ) THEN DO 10 IX = 1, 1 + ( N-1 )*INCX, INCX IF( X( IX ).NE.ZERO ) THEN ABSXI = ABS( X( IX ) ) IF( SCALE.LT.ABSXI ) THEN SUMSQ = 1 + SUMSQ*( SCALE / ABSXI )**2 SCALE = ABSXI ELSE SUMSQ = SUMSQ + ( ABSXI / SCALE )**2 END IF END IF 10 CONTINUE END IF RETURN ! ! End of SLASSQ ! END SUBROUTINE SLASSQ !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SORG2R( M, N, K, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, M, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! SORG2R generates an m by n real matrix Q with orthonormal columns, ! which is defined as the first n columns of a product of k elementary ! reflectors of order m ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by SGEQRF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the i-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by SGEQRF in the first k columns of its array ! argument A. ! On exit, the m-by-n matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) REAL array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by SGEQRF. ! ! WORK (workspace) REAL array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, J, L ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SORG2R', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! ! Initialise columns k+1:n to columns of the unit matrix ! DO 20 J = K + 1, N DO 10 L = 1, M A( L, J ) = ZERO 10 CONTINUE A( J, J ) = ONE 20 CONTINUE ! DO 40 I = K, 1, -1 ! ! Apply H(i) to A(i:m,i:n) from the left ! IF( I.LT.N ) THEN A( I, I ) = ONE CALL SLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ), & A( I, I+1 ), LDA, WORK ) END IF IF( I.LT.M ) & CALL SSCAL( M-I, -TAU( I ), A( I+1, I ), 1 ) A( I, I ) = ONE - TAU( I ) ! ! Set A(1:i-1,i) to zero ! DO 30 L = 1, I - 1 A( L, I ) = ZERO 30 CONTINUE 40 CONTINUE RETURN ! ! End of SORG2R ! END SUBROUTINE SORG2R !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), TAU( * ), WORK( LWORK ) ! .. ! ! Purpose ! ======= ! ! SORGHR generates a real orthogonal matrix Q which is defined as the ! product of IHI-ILO elementary reflectors of order N, as returned by ! SGEHRD: ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix Q. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! ILO and IHI must have the same values as in the previous call ! of SGEHRD. Q is equal to the unit matrix except in the ! submatrix Q(ilo+1:ihi,ilo+1:ihi). ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the vectors which define the elementary reflectors, ! as returned by SGEHRD. ! On exit, the N-by-N orthogonal matrix Q. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (input) REAL array, dimension (N-1) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by SGEHRD. ! ! WORK (workspace/output) REAL array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= IHI-ILO. ! For optimum performance LWORK >= (IHI-ILO)*NB, where NB is ! the optimal blocksize. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, IINFO, J, NH ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, IHI-ILO ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SORGHR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Shift the vectors which define the elementary reflectors one ! column to the right, and set the first ilo and the last n-ihi ! rows and columns to those of the unit matrix ! DO 40 J = IHI, ILO + 1, -1 DO 10 I = 1, J - 1 A( I, J ) = ZERO 10 CONTINUE DO 20 I = J + 1, IHI A( I, J ) = A( I, J-1 ) 20 CONTINUE DO 30 I = IHI + 1, N A( I, J ) = ZERO 30 CONTINUE 40 CONTINUE DO 60 J = 1, ILO DO 50 I = 1, N A( I, J ) = ZERO 50 CONTINUE A( J, J ) = ONE 60 CONTINUE DO 80 J = IHI + 1, N DO 70 I = 1, N A( I, J ) = ZERO 70 CONTINUE A( J, J ) = ONE 80 CONTINUE ! NH = IHI - ILO IF( NH.GT.0 ) THEN ! ! Generate Q(ilo+1:ihi,ilo+1:ihi) ! CALL SORGQR( NH, NH, NH, A( ILO+1, ILO+1 ), LDA, TAU( ILO ), & WORK, LWORK, IINFO ) END IF RETURN ! ! End of SORGHR ! END SUBROUTINE SORGHR !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. REAL(r8) A( LDA, * ), TAU( * ), WORK( LWORK ) ! .. ! ! Purpose ! ======= ! ! SORGQR generates an M-by-N real matrix Q with orthonormal columns, ! which is defined as the first N columns of a product of K elementary ! reflectors of order M ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by SGEQRF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) REAL array, dimension (LDA,N) ! On entry, the i-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by SGEQRF in the first k columns of its array ! argument A. ! On exit, the M-by-N matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) REAL array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by SGEQRF. ! ! WORK (workspace/output) REAL array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK, NB, & NBMIN, NX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SORGQR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Determine the block size. ! NB = ILAENV( 1, 'SORGQR', ' ', M, N, K, -1 ) NBMIN = 2 NX = 0 IWS = N IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'SORGQR', ' ', M, N, K, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'SORGQR', ' ', M, N, K, -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code after the last block. ! The first kk columns are handled by the block method. ! KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) ! ! Set A(1:kk,kk+1:n) to zero. ! DO 20 J = KK + 1, N DO 10 I = 1, KK A( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE KK = 0 END IF ! ! Use unblocked code for the last or only block. ! IF( KK.LT.N ) & CALL SORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA, & TAU( KK+1 ), WORK, IINFO ) ! IF( KK.GT.0 ) THEN ! ! Use blocked code ! DO 50 I = KI + 1, 1, -NB IB = MIN( NB, K-I+1 ) IF( I+IB.LE.N ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL SLARFT( 'Forward', 'Columnwise', M-I+1, IB, & A( I, I ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(i:m,i+ib:n) from the left ! CALL SLARFB( 'Left', 'No transpose', 'Forward', & 'Columnwise', M-I+1, N-I-IB+1, IB, & A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ), & LDA, WORK( IB+1 ), LDWORK ) END IF ! ! Apply H to rows i:m of current block ! CALL SORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) ! ! Set rows 1:i-1 of current block to zero ! DO 40 J = I, I + IB - 1 DO 30 L = 1, I - 1 A( L, J ) = ZERO 30 CONTINUE 40 CONTINUE 50 CONTINUE END IF ! WORK( 1 ) = IWS RETURN ! ! End of SORGQR ! END SUBROUTINE SORGQR !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE STREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, & LDVR, MM, M, WORK, INFO ) ! ! -- LAPACK routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER HOWMNY, SIDE INTEGER INFO, LDT, LDVL, LDVR, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) REAL(r8) T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! STREVC computes some or all of the right and/or left eigenvectors of ! a real upper quasi-triangular matrix T. ! ! The right eigenvector x and the left eigenvector y of T corresponding ! to an eigenvalue w are defined by: ! ! T*x = w*x, y'*T = w*y' ! ! where y' denotes the conjugate transpose of the vector y. ! ! If all eigenvectors are requested, the routine may either return the ! matrices X and/or Y of right or left eigenvectors of T, or the ! products Q*X and/or Q*Y, where Q is an input orthogonal ! matrix. If T was obtained from the real-Schur factorization of an ! original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of ! right or left eigenvectors of A. ! ! T must be in Schur canonical form (as returned by SHSEQR), that is, ! block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each ! 2-by-2 diagonal block has its diagonal elements equal and its ! off-diagonal elements of opposite sign. Corresponding to each 2-by-2 ! diagonal block is a complex conjugate pair of eigenvalues and ! eigenvectors; only one eigenvector of the pair is computed, namely ! the one corresponding to the eigenvalue with positive imaginary part. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'R': compute right eigenvectors only; ! = 'L': compute left eigenvectors only; ! = 'B': compute both right and left eigenvectors. ! ! HOWMNY (input) CHARACTER*1 ! = 'A': compute all right and/or left eigenvectors; ! = 'B': compute all right and/or left eigenvectors, ! and backtransform them using the input matrices ! supplied in VR and/or VL; ! = 'S': compute selected right and/or left eigenvectors, ! specified by the logical array SELECT. ! ! SELECT (input/output) LOGICAL array, dimension (N) ! If HOWMNY = 'S', SELECT specifies the eigenvectors to be ! computed. ! If HOWMNY = 'A' or 'B', SELECT is not referenced. ! To select the real eigenvector corresponding to a real ! eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select ! the complex eigenvector corresponding to a complex conjugate ! pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be ! set to .TRUE.; then on exit SELECT(j) is .TRUE. and ! SELECT(j+1) is .FALSE.. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input) REAL array, dimension (LDT,N) ! The upper quasi-triangular matrix T in Schur canonical form. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! VL (input/output) REAL array, dimension (LDVL,MM) ! On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must ! contain an N-by-N matrix Q (usually the orthogonal matrix Q ! of Schur vectors returned by SHSEQR). ! On exit, if SIDE = 'L' or 'B', VL contains: ! if HOWMNY = 'A', the matrix Y of left eigenvectors of T; ! if HOWMNY = 'B', the matrix Q*Y; ! if HOWMNY = 'S', the left eigenvectors of T specified by ! SELECT, stored consecutively in the columns ! of VL, in the same order as their ! eigenvalues. ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part, and the second the imaginary part. ! If SIDE = 'R', VL is not referenced. ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= max(1,N) if ! SIDE = 'L' or 'B'; LDVL >= 1 otherwise. ! ! VR (input/output) REAL array, dimension (LDVR,MM) ! On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must ! contain an N-by-N matrix Q (usually the orthogonal matrix Q ! of Schur vectors returned by SHSEQR). ! On exit, if SIDE = 'R' or 'B', VR contains: ! if HOWMNY = 'A', the matrix X of right eigenvectors of T; ! if HOWMNY = 'B', the matrix Q*X; ! if HOWMNY = 'S', the right eigenvectors of T specified by ! SELECT, stored consecutively in the columns ! of VR, in the same order as their ! eigenvalues. ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part and the second the imaginary part. ! If SIDE = 'L', VR is not referenced. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= max(1,N) if ! SIDE = 'R' or 'B'; LDVR >= 1 otherwise. ! ! MM (input) INTEGER ! The number of columns in the arrays VL and/or VR. MM >= M. ! ! M (output) INTEGER ! The number of columns in the arrays VL and/or VR actually ! used to store the eigenvectors. ! If HOWMNY = 'A' or 'B', M is set to N. ! Each selected real eigenvector occupies one column and each ! selected complex eigenvector occupies two columns. ! ! WORK (workspace) REAL array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The algorithm used in this program is basically backward (forward) ! substitution, with scaling to make the the code robust against ! possible overflow. ! ! Each eigenvector is normalized so that the element of largest ! magnitude has magnitude 1; here the magnitude of a complex number ! (x,y) is taken to be |x| + |y|. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E+0_r8, ONE = 1.0E+0_r8 ) ! .. ! .. Local Scalars .. LOGICAL ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2 REAL(r8) BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE, & SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR, & XNORM ! .. ! .. Local Arrays .. REAL(r8) X( 2, 2 ) ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! BOTHV = LSAME( SIDE, 'B' ) RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV ! ALLV = LSAME( HOWMNY, 'A' ) OVER = LSAME( HOWMNY, 'B' ) .OR. LSAME( HOWMNY, 'O' ) SOMEV = LSAME( HOWMNY, 'S' ) ! INFO = 0 IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -1 ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN INFO = -8 ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN INFO = -10 ELSE ! ! Set M to the number of columns required to store the selected ! eigenvectors, standardize the array SELECT if necessary, and ! test MM. ! IF( SOMEV ) THEN M = 0 PAIR = .FALSE. DO 10 J = 1, N IF( PAIR ) THEN PAIR = .FALSE. SELECT( J ) = .FALSE. ELSE IF( J.LT.N ) THEN IF( T( J+1, J ).EQ.ZERO ) THEN IF( SELECT( J ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN SELECT( J ) = .TRUE. M = M + 2 END IF END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF ! IF( MM.LT.M ) THEN INFO = -11 END IF END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'STREVC', -INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! Set the constants to control overflow. ! UNFL = SLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL SLABAD( UNFL, OVFL ) ULP = SLAMCH( 'Precision' ) SMLNUM = UNFL*( N / ULP ) BIGNUM = ( ONE-ULP ) / SMLNUM ! ! Compute 1-norm of each column of strictly upper triangular ! part of T to control overflow in triangular solver. ! WORK( 1 ) = ZERO DO 30 J = 2, N WORK( J ) = ZERO DO 20 I = 1, J - 1 WORK( J ) = WORK( J ) + ABS( T( I, J ) ) 20 CONTINUE 30 CONTINUE ! ! Index IP is used to specify the real or complex eigenvalue: ! IP = 0, real eigenvalue, ! 1, first of conjugate complex pair: (wr,wi) ! -1, second of conjugate complex pair: (wr,wi) ! N2 = 2*N ! IF( RIGHTV ) THEN ! ! Compute right eigenvectors. ! IP = 0 IS = M DO 140 KI = N, 1, -1 ! IF( IP.EQ.1 ) & GO TO 130 IF( KI.EQ.1 ) & GO TO 40 IF( T( KI, KI-1 ).EQ.ZERO ) & GO TO 40 IP = -1 ! 40 CONTINUE IF( SOMEV ) THEN IF( IP.EQ.0 ) THEN IF( .NOT.SELECT( KI ) ) & GO TO 130 ELSE IF( .NOT.SELECT( KI-1 ) ) & GO TO 130 END IF END IF ! ! Compute the KI-th eigenvalue (WR,WI). ! WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) & WI = SQRT( ABS( T( KI, KI-1 ) ) )* & SQRT( ABS( T( KI-1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) ! IF( IP.EQ.0 ) THEN ! ! Real right eigenvector ! WORK( KI+N ) = ONE ! ! Form right-hand side ! DO 50 K = 1, KI - 1 WORK( K+N ) = -T( K, KI ) 50 CONTINUE ! ! Solve the upper quasi-triangular system: ! (T(1:KI-1,1:KI-1) - WR)*X = SCALE*WORK. ! JNXT = KI - 1 DO 60 J = KI - 1, 1, -1 IF( J.GT.JNXT ) & GO TO 60 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! CALL SLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale X(1,1) to avoid overflow when updating ! the right-hand side. ! IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL SSCAL( KI, SCALE, WORK( 1+N ), 1 ) WORK( J+N ) = X( 1, 1 ) ! ! Update right-hand side ! CALL SAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) ! ELSE ! ! 2-by-2 diagonal block ! CALL SLALN2( .FALSE., 2, 1, SMIN, ONE, & T( J-1, J-1 ), LDT, ONE, ONE, & WORK( J-1+N ), N, WR, ZERO, X, 2, & SCALE, XNORM, IERR ) ! ! Scale X(1,1) and X(2,1) to avoid overflow when ! updating the right-hand side. ! IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 2, 1 ) = X( 2, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL SSCAL( KI, SCALE, WORK( 1+N ), 1 ) WORK( J-1+N ) = X( 1, 1 ) WORK( J+N ) = X( 2, 1 ) ! ! Update right-hand side ! CALL SAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, & WORK( 1+N ), 1 ) CALL SAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) END IF 60 CONTINUE ! ! Copy the vector x or Q*x to VR and normalize. ! IF( .NOT.OVER ) THEN CALL SCOPY( KI, WORK( 1+N ), 1, VR( 1, IS ), 1 ) ! II = ISAMAX( KI, VR( 1, IS ), 1 ) REMAX = ONE / ABS( VR( II, IS ) ) CALL SSCAL( KI, REMAX, VR( 1, IS ), 1 ) ! DO 70 K = KI + 1, N VR( K, IS ) = ZERO 70 CONTINUE ELSE IF( KI.GT.1 ) & CALL SGEMV( 'N', N, KI-1, ONE, VR, LDVR, & WORK( 1+N ), 1, WORK( KI+N ), & VR( 1, KI ), 1 ) ! II = ISAMAX( N, VR( 1, KI ), 1 ) REMAX = ONE / ABS( VR( II, KI ) ) CALL SSCAL( N, REMAX, VR( 1, KI ), 1 ) END IF ! ELSE ! ! Complex right eigenvector. ! ! Initial solve ! [ (T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I* WI)]*X = 0. ! [ (T(KI,KI-1) T(KI,KI) ) ] ! IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN WORK( KI-1+N ) = ONE WORK( KI+N2 ) = WI / T( KI-1, KI ) ELSE WORK( KI-1+N ) = -WI / T( KI, KI-1 ) WORK( KI+N2 ) = ONE END IF WORK( KI+N ) = ZERO WORK( KI-1+N2 ) = ZERO ! ! Form right-hand side ! DO 80 K = 1, KI - 2 WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 ) WORK( K+N2 ) = -WORK( KI+N2 )*T( K, KI ) 80 CONTINUE ! ! Solve upper quasi-triangular system: ! (T(1:KI-2,1:KI-2) - (WR+i*WI))*X = SCALE*(WORK+i*WORK2) ! JNXT = KI - 2 DO 90 J = KI - 2, 1, -1 IF( J.GT.JNXT ) & GO TO 90 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! CALL SLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, WI, & X, 2, SCALE, XNORM, IERR ) ! ! Scale X(1,1) and X(1,2) to avoid overflow when ! updating the right-hand side. ! IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 1, 2 ) = X( 1, 2 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL SSCAL( KI, SCALE, WORK( 1+N ), 1 ) CALL SSCAL( KI, SCALE, WORK( 1+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) ! ! Update the right-hand side ! CALL SAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) CALL SAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1, & WORK( 1+N2 ), 1 ) ! ELSE ! ! 2-by-2 diagonal block ! CALL SLALN2( .FALSE., 2, 2, SMIN, ONE, & T( J-1, J-1 ), LDT, ONE, ONE, & WORK( J-1+N ), N, WR, WI, X, 2, SCALE, & XNORM, IERR ) ! ! Scale X to avoid overflow when updating ! the right-hand side. ! IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN REC = ONE / XNORM X( 1, 1 ) = X( 1, 1 )*REC X( 1, 2 ) = X( 1, 2 )*REC X( 2, 1 ) = X( 2, 1 )*REC X( 2, 2 ) = X( 2, 2 )*REC SCALE = SCALE*REC END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL SSCAL( KI, SCALE, WORK( 1+N ), 1 ) CALL SSCAL( KI, SCALE, WORK( 1+N2 ), 1 ) END IF WORK( J-1+N ) = X( 1, 1 ) WORK( J+N ) = X( 2, 1 ) WORK( J-1+N2 ) = X( 1, 2 ) WORK( J+N2 ) = X( 2, 2 ) ! ! Update the right-hand side ! CALL SAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, & WORK( 1+N ), 1 ) CALL SAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) CALL SAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1, & WORK( 1+N2 ), 1 ) CALL SAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1, & WORK( 1+N2 ), 1 ) END IF 90 CONTINUE ! ! Copy the vector x or Q*x to VR and normalize. ! IF( .NOT.OVER ) THEN CALL SCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 ) CALL SCOPY( KI, WORK( 1+N2 ), 1, VR( 1, IS ), 1 ) ! EMAX = ZERO DO 100 K = 1, KI EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+ & ABS( VR( K, IS ) ) ) 100 CONTINUE ! REMAX = ONE / EMAX CALL SSCAL( KI, REMAX, VR( 1, IS-1 ), 1 ) CALL SSCAL( KI, REMAX, VR( 1, IS ), 1 ) ! DO 110 K = KI + 1, N VR( K, IS-1 ) = ZERO VR( K, IS ) = ZERO 110 CONTINUE ! ELSE ! IF( KI.GT.2 ) THEN CALL SGEMV( 'N', N, KI-2, ONE, VR, LDVR, & WORK( 1+N ), 1, WORK( KI-1+N ), & VR( 1, KI-1 ), 1 ) CALL SGEMV( 'N', N, KI-2, ONE, VR, LDVR, & WORK( 1+N2 ), 1, WORK( KI+N2 ), & VR( 1, KI ), 1 ) ELSE CALL SSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ), 1 ) CALL SSCAL( N, WORK( KI+N2 ), VR( 1, KI ), 1 ) END IF ! EMAX = ZERO DO 120 K = 1, N EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+ & ABS( VR( K, KI ) ) ) 120 CONTINUE REMAX = ONE / EMAX CALL SSCAL( N, REMAX, VR( 1, KI-1 ), 1 ) CALL SSCAL( N, REMAX, VR( 1, KI ), 1 ) END IF END IF ! IS = IS - 1 IF( IP.NE.0 ) & IS = IS - 1 130 CONTINUE IF( IP.EQ.1 ) & IP = 0 IF( IP.EQ.-1 ) & IP = 1 140 CONTINUE END IF ! IF( LEFTV ) THEN ! ! Compute left eigenvectors. ! IP = 0 IS = 1 DO 260 KI = 1, N ! IF( IP.EQ.-1 ) & GO TO 250 IF( KI.EQ.N ) & GO TO 150 IF( T( KI+1, KI ).EQ.ZERO ) & GO TO 150 IP = 1 ! 150 CONTINUE IF( SOMEV ) THEN IF( .NOT.SELECT( KI ) ) & GO TO 250 END IF ! ! Compute the KI-th eigenvalue (WR,WI). ! WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) & WI = SQRT( ABS( T( KI, KI+1 ) ) )* & SQRT( ABS( T( KI+1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) ! IF( IP.EQ.0 ) THEN ! ! Real left eigenvector. ! WORK( KI+N ) = ONE ! ! Form right-hand side ! DO 160 K = KI + 1, N WORK( K+N ) = -T( KI, K ) 160 CONTINUE ! ! Solve the quasi-triangular system: ! (T(KI+1:N,KI+1:N) - WR)'*X = SCALE*WORK ! VMAX = ONE VCRIT = BIGNUM ! JNXT = KI + 1 DO 170 J = KI + 1, N IF( J.LT.JNXT ) & GO TO 170 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side. ! IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL SSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & SDOT( J-KI-1, T( KI+1, J ), 1, & WORK( KI+1+N ), 1 ) ! ! Solve (T(J,J)-WR)'*X = WORK ! CALL SLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL SSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) WORK( J+N ) = X( 1, 1 ) VMAX = MAX( ABS( WORK( J+N ) ), VMAX ) VCRIT = BIGNUM / VMAX ! ELSE ! ! 2-by-2 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side. ! BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL SSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & SDOT( J-KI-1, T( KI+1, J ), 1, & WORK( KI+1+N ), 1 ) ! WORK( J+1+N ) = WORK( J+1+N ) - & SDOT( J-KI-1, T( KI+1, J+1 ), 1, & WORK( KI+1+N ), 1 ) ! ! Solve ! [T(J,J)-WR T(J,J+1) ]'* X = SCALE*( WORK1 ) ! [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) ! CALL SLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL SSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) WORK( J+N ) = X( 1, 1 ) WORK( J+1+N ) = X( 2, 1 ) ! VMAX = MAX( ABS( WORK( J+N ) ), & ABS( WORK( J+1+N ) ), VMAX ) VCRIT = BIGNUM / VMAX ! END IF 170 CONTINUE ! ! Copy the vector x or Q*x to VL and normalize. ! IF( .NOT.OVER ) THEN CALL SCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 ) ! II = ISAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1 REMAX = ONE / ABS( VL( II, IS ) ) CALL SSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) ! DO 180 K = 1, KI - 1 VL( K, IS ) = ZERO 180 CONTINUE ! ELSE ! IF( KI.LT.N ) & CALL SGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ), LDVL, & WORK( KI+1+N ), 1, WORK( KI+N ), & VL( 1, KI ), 1 ) ! II = ISAMAX( N, VL( 1, KI ), 1 ) REMAX = ONE / ABS( VL( II, KI ) ) CALL SSCAL( N, REMAX, VL( 1, KI ), 1 ) ! END IF ! ELSE ! ! Complex left eigenvector. ! ! Initial solve: ! ((T(KI,KI) T(KI,KI+1) )' - (WR - I* WI))*X = 0. ! ((T(KI+1,KI) T(KI+1,KI+1)) ) ! IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN WORK( KI+N ) = WI / T( KI, KI+1 ) WORK( KI+1+N2 ) = ONE ELSE WORK( KI+N ) = ONE WORK( KI+1+N2 ) = -WI / T( KI+1, KI ) END IF WORK( KI+1+N ) = ZERO WORK( KI+N2 ) = ZERO ! ! Form right-hand side ! DO 190 K = KI + 2, N WORK( K+N ) = -WORK( KI+N )*T( KI, K ) WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K ) 190 CONTINUE ! ! Solve complex quasi-triangular system: ! ( T(KI+2,N:KI+2,N) - (WR-i*WI) )*X = WORK1+i*WORK2 ! VMAX = ONE VCRIT = BIGNUM ! JNXT = KI + 2 DO 200 J = KI + 2, N IF( J.LT.JNXT ) & GO TO 200 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! ! Scale if necessary to avoid overflow when ! forming the right-hand side elements. ! IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL SSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) CALL SSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & SDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N ), 1 ) WORK( J+N2 ) = WORK( J+N2 ) - & SDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N2 ), 1 ) ! ! Solve (T(J,J)-(WR-i*WI))*(X11+i*X12)= WK+I*WK2 ! CALL SLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & -WI, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL SSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) CALL SSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) VMAX = MAX( ABS( WORK( J+N ) ), & ABS( WORK( J+N2 ) ), VMAX ) VCRIT = BIGNUM / VMAX ! ELSE ! ! 2-by-2 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side elements. ! BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL SSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) CALL SSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & SDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N ), 1 ) ! WORK( J+N2 ) = WORK( J+N2 ) - & SDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N2 ), 1 ) ! WORK( J+1+N ) = WORK( J+1+N ) - & SDOT( J-KI-2, T( KI+2, J+1 ), 1, & WORK( KI+2+N ), 1 ) ! WORK( J+1+N2 ) = WORK( J+1+N2 ) - & SDOT( J-KI-2, T( KI+2, J+1 ), 1, & WORK( KI+2+N2 ), 1 ) ! ! Solve 2-by-2 complex linear equation ! ([T(j,j) T(j,j+1) ]'-(wr-i*wi)*I)*X = SCALE*B ! ([T(j+1,j) T(j+1,j+1)] ) ! CALL SLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & -WI, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL SSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) CALL SSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) WORK( J+1+N ) = X( 2, 1 ) WORK( J+1+N2 ) = X( 2, 2 ) VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ), & ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), VMAX ) VCRIT = BIGNUM / VMAX ! END IF 200 CONTINUE ! ! Copy the vector x or Q*x to VL and normalize. ! 210 CONTINUE IF( .NOT.OVER ) THEN CALL SCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 ) CALL SCOPY( N-KI+1, WORK( KI+N2 ), 1, VL( KI, IS+1 ), & 1 ) ! EMAX = ZERO DO 220 K = KI, N EMAX = MAX( EMAX, ABS( VL( K, IS ) )+ & ABS( VL( K, IS+1 ) ) ) 220 CONTINUE REMAX = ONE / EMAX CALL SSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) CALL SSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 ) ! DO 230 K = 1, KI - 1 VL( K, IS ) = ZERO VL( K, IS+1 ) = ZERO 230 CONTINUE ELSE IF( KI.LT.N-1 ) THEN CALL SGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ), & LDVL, WORK( KI+2+N ), 1, WORK( KI+N ), & VL( 1, KI ), 1 ) CALL SGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ), & LDVL, WORK( KI+2+N2 ), 1, & WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 ) ELSE CALL SSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 ) CALL SSCAL( N, WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 ) END IF ! EMAX = ZERO DO 240 K = 1, N EMAX = MAX( EMAX, ABS( VL( K, KI ) )+ & ABS( VL( K, KI+1 ) ) ) 240 CONTINUE REMAX = ONE / EMAX CALL SSCAL( N, REMAX, VL( 1, KI ), 1 ) CALL SSCAL( N, REMAX, VL( 1, KI+1 ), 1 ) ! END IF ! END IF ! IS = IS + 1 IF( IP.NE.0 ) & IS = IS + 1 250 CONTINUE IF( IP.EQ.-1 ) & IP = 0 IF( IP.EQ.1 ) & IP = -1 ! 260 CONTINUE ! END IF ! RETURN ! ! End of STREVC ! END SUBROUTINE STREVC !--------1---------2---------3---------4---------5---------6---------7---------8 INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, & N4 ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER*( * ) NAME, OPTS INTEGER ISPEC, N1, N2, N3, N4 ! .. ! ! Purpose ! ======= ! ! ILAENV is called from the LAPACK routines to choose problem-dependent ! parameters for the local environment. See ISPEC for a description of ! the parameters. ! ! This version provides a set of parameters which should give good, ! but not optimal, performance on many of the currently available ! computers. Users are encouraged to modify this subroutine to set ! the tuning parameters for their particular machine using the option ! and problem size information in the arguments. ! ! This routine will not function correctly if it is converted to all ! lower case. Converting it to all upper case is allowed. ! ! Arguments ! ========= ! ! ISPEC (input) INTEGER ! Specifies the parameter to be returned as the value of ! ILAENV. ! = 1: the optimal blocksize; if this value is 1, an unblocked ! algorithm will give the best performance. ! = 2: the minimum block size for which the block routine ! should be used; if the usable block size is less than ! this value, an unblocked routine should be used. ! = 3: the crossover point (in a block routine, for N less ! than this value, an unblocked routine should be used) ! = 4: the number of shifts, used in the nonsymmetric ! eigenvalue routines ! = 5: the minimum column dimension for blocking to be used; ! rectangular blocks must have dimension at least k by m, ! where k is given by ILAENV(2,...) and m by ILAENV(5,...) ! = 6: the crossover point for the SVD (when reducing an m by n ! matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds ! this value, a QR factorization is used first to reduce ! the matrix to a triangular form.) ! = 7: the number of processors ! = 8: the crossover point for the multishift QR and QZ methods ! for nonsymmetric eigenvalue problems. ! ! NAME (input) CHARACTER*(*) ! The name of the calling subroutine, in either upper case or ! lower case. ! ! OPTS (input) CHARACTER*(*) ! The character options to the subroutine NAME, concatenated ! into a single character string. For example, UPLO = 'U', ! TRANS = 'T', and DIAG = 'N' for a triangular routine would ! be specified as OPTS = 'UTN'. ! ! N1 (input) INTEGER ! N2 (input) INTEGER ! N3 (input) INTEGER ! N4 (input) INTEGER ! Problem dimensions for the subroutine NAME; these may not all ! be required. ! ! (ILAENV) (output) INTEGER ! >= 0: the value of the parameter specified by ISPEC ! < 0: if ILAENV = -k, the k-th argument had an illegal value. ! ! Further Details ! =============== ! ! The following conventions have been used when calling ILAENV from the ! LAPACK routines: ! 1) OPTS is a concatenation of all of the character options to ! subroutine NAME, in the same order that they appear in the ! argument list for NAME, even if they are not used in determining ! the value of the parameter specified by ISPEC. ! 2) The problem dimensions N1, N2, N3, N4 are specified in the order ! that they appear in the argument list for NAME. N1 is used ! first, N2 second, and so on, and unused problem dimensions are ! passed a value of -1. ! 3) The parameter value returned by ILAENV is checked for validity in ! the calling subroutine. For example, ILAENV is used to retrieve ! the optimal blocksize for STRTRI as follows: ! ! NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 ) ! IF( NB.LE.1 ) NB = MAX( 1, N ) ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL CNAME, SNAME CHARACTER*1 C1 CHARACTER*2 C2, C4 CHARACTER*3 C3 CHARACTER*6 SUBNAM INTEGER I, IC, IZ, NB, NBMIN, NX ! .. ! .. Executable Statements .. ! GO TO ( 100, 100, 100, 400, 500, 600, 700, 800 ) ISPEC ! ! Invalid value for ISPEC ! ILAENV = -1 RETURN ! 100 CONTINUE ! ! Convert NAME to upper case if the first character is lower case. ! ILAENV = 1 SUBNAM = NAME IC = ICHAR( SUBNAM( 1:1 ) ) IZ = ICHAR( 'Z' ) IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN ! ! ASCII character set ! IF( IC.GE.97 .AND. IC.LE.122 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 10 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.97 .AND. IC.LE.122 ) & SUBNAM( I:I ) = CHAR( IC-32 ) 10 CONTINUE END IF ! ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN ! ! EBCDIC character set ! IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. & ( IC.GE.145 .AND. IC.LE.153 ) .OR. & ( IC.GE.162 .AND. IC.LE.169 ) ) THEN SUBNAM( 1:1 ) = CHAR( IC+64 ) DO 20 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. & ( IC.GE.145 .AND. IC.LE.153 ) .OR. & ( IC.GE.162 .AND. IC.LE.169 ) ) & SUBNAM( I:I ) = CHAR( IC+64 ) 20 CONTINUE END IF ! ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN ! ! Prime machines: ASCII+128 ! IF( IC.GE.225 .AND. IC.LE.250 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 30 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.225 .AND. IC.LE.250 ) & SUBNAM( I:I ) = CHAR( IC-32 ) 30 CONTINUE END IF END IF ! C1 = SUBNAM( 1:1 ) SNAME = C1.EQ.'S' .OR. C1.EQ.'D' CNAME = C1.EQ.'C' .OR. C1.EQ.'Z' IF( .NOT.( CNAME .OR. SNAME ) ) & RETURN C2 = SUBNAM( 2:3 ) C3 = SUBNAM( 4:6 ) C4 = C3( 2:3 ) ! GO TO ( 110, 200, 300 ) ISPEC ! 110 CONTINUE ! ! ISPEC = 1: block size ! ! In these examples, separate code is provided for setting NB for ! real and complex. We assume that NB will take the same value in ! single or double precision. ! NB = 1 ! IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'PO' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NB = 1 ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRF' ) THEN NB = 64 ELSE IF( C3.EQ.'TRD' ) THEN NB = 1 ELSE IF( C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( C2.EQ.'GB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'PB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'TR' ) THEN IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'LA' ) THEN IF( C3.EQ.'UUM' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN IF( C3.EQ.'EBZ' ) THEN NB = 1 END IF END IF ILAENV = NB RETURN ! 200 CONTINUE ! ! ISPEC = 2: minimum block size ! NBMIN = 2 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NBMIN = 8 ELSE NBMIN = 8 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF END IF ILAENV = NBMIN RETURN ! 300 CONTINUE ! ! ISPEC = 3: crossover point ! NX = 0 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( SNAME .AND. C3.EQ.'TRD' ) THEN NX = 1 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NX = 1 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NX = 128 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NX = 128 END IF END IF END IF ILAENV = NX RETURN ! 400 CONTINUE ! ! ISPEC = 4: number of shifts (used by xHSEQR) ! ILAENV = 6 RETURN ! 500 CONTINUE ! ! ISPEC = 5: minimum column dimension (not used) ! ILAENV = 2 RETURN ! 600 CONTINUE ! ! ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD) ! ILAENV = INT( REAL( MIN( N1, N2 ) ,r8)*1.6E0_r8 ) RETURN ! 700 CONTINUE ! ! ISPEC = 7: number of processors (not used) ! ILAENV = 1 RETURN ! 800 CONTINUE ! ! ISPEC = 8: crossover point for multishift (used by xHSEQR) ! ILAENV = 50 RETURN ! ! End of ILAENV ! END FUNCTION ILAENV !--------1---------2---------3---------4---------5---------6---------7---------8 LOGICAL FUNCTION LSAME( CA, CB ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER CA, CB ! .. ! ! Purpose ! ======= ! ! LSAME returns .TRUE. if CA is the same letter as CB regardless of ! case. ! ! Arguments ! ========= ! ! CA (input) CHARACTER*1 ! CB (input) CHARACTER*1 ! CA and CB specify the single characters to be compared. ! ! ===================================================================== ! ! .. ! .. Local Scalars .. INTEGER INTA, INTB, ZCODE ! .. ! .. Executable Statements .. ! ! Test if the characters are equal ! LSAME = CA.EQ.CB IF( LSAME ) & RETURN ! ! Now test for equivalence if both characters are alphabetic. ! ZCODE = ICHAR( 'Z' ) ! ! Use 'Z' rather than 'A' so that ASCII can be detected on Prime ! machines, on which ICHAR returns a value with bit 8 set. ! ICHAR('A') on Prime machines returns 193 which is the same as ! ICHAR('A') on an EBCDIC machine. ! INTA = ICHAR( CA ) INTB = ICHAR( CB ) ! IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN ! ! ASCII is assumed - ZCODE is the ASCII code of either lower or ! upper case 'Z'. ! IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32 IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32 ! ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN ! ! EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or ! upper case 'Z'. ! IF( INTA.GE.129 .AND. INTA.LE.137 .OR. & INTA.GE.145 .AND. INTA.LE.153 .OR. & INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64 IF( INTB.GE.129 .AND. INTB.LE.137 .OR. & INTB.GE.145 .AND. INTB.LE.153 .OR. & INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64 ! ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN ! ! ASCII is assumed, on Prime machines - ZCODE is the ASCII code ! plus 128 of either lower or upper case 'Z'. ! IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32 IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32 END IF LSAME = INTA.EQ.INTB ! ! RETURN ! ! End of LSAME ! END FUNCTION LSAME !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SLAMCH( CMACH ) real(r8) slamch ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER CMACH ! .. ! ! Purpose ! ======= ! ! SLAMCH determines single precision machine parameters. ! ! Arguments ! ========= ! ! CMACH (input) CHARACTER*1 ! Specifies the value to be returned by SLAMCH: ! = 'E' or 'e', SLAMCH := eps ! = 'S' or 's , SLAMCH := sfmin ! = 'B' or 'b', SLAMCH := base ! = 'P' or 'p', SLAMCH := eps*base ! = 'N' or 'n', SLAMCH := t ! = 'R' or 'r', SLAMCH := rnd ! = 'M' or 'm', SLAMCH := emin ! = 'U' or 'u', SLAMCH := rmin ! = 'L' or 'l', SLAMCH := emax ! = 'O' or 'o', SLAMCH := rmax ! ! where ! ! eps = relative machine precision ! sfmin = safe minimum, such that 1/sfmin does not overflow ! base = base of the machine ! prec = eps*base ! t = number of (base) digits in the mantissa ! rnd = 1.0 when rounding occurs in addition, 0.0 otherwise ! emin = minimum exponent before (gradual) underflow ! rmin = underflow threshold - base**(emin-1) ! emax = largest exponent before overflow ! rmax = overflow threshold - (base**emax)*(1-eps) ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ONE, ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Local Scalars .. LOGICAL FIRST, LRND INTEGER BETA, IMAX, IMIN, IT REAL(r8) BASE, EMAX, EMIN, EPS, PREC, RMACH, RMAX, RMIN, & RND, SFMIN, SMALL, T ! .. ! .. Save statement .. SAVE FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN, & EMAX, RMAX, PREC ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. CALL SLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX ) BASE = BETA T = IT IF( LRND ) THEN RND = ONE EPS = ( BASE**( 1-IT ) ) / 2 ELSE RND = ZERO EPS = BASE**( 1-IT ) END IF PREC = EPS*BASE EMIN = IMIN EMAX = IMAX SFMIN = RMIN SMALL = ONE / RMAX IF( SMALL.GE.SFMIN ) THEN ! ! Use SMALL plus a bit, to avoid the possibility of rounding ! causing overflow when computing 1/sfmin. ! SFMIN = SMALL*( ONE+EPS ) END IF END IF ! IF( LSAME( CMACH, 'E' ) ) THEN RMACH = EPS ELSE IF( LSAME( CMACH, 'S' ) ) THEN RMACH = SFMIN ELSE IF( LSAME( CMACH, 'B' ) ) THEN RMACH = BASE ELSE IF( LSAME( CMACH, 'P' ) ) THEN RMACH = PREC ELSE IF( LSAME( CMACH, 'N' ) ) THEN RMACH = T ELSE IF( LSAME( CMACH, 'R' ) ) THEN RMACH = RND ELSE IF( LSAME( CMACH, 'M' ) ) THEN RMACH = EMIN ELSE IF( LSAME( CMACH, 'U' ) ) THEN RMACH = RMIN ELSE IF( LSAME( CMACH, 'L' ) ) THEN RMACH = EMAX ELSE IF( LSAME( CMACH, 'O' ) ) THEN RMACH = RMAX END IF ! SLAMCH = RMACH RETURN ! ! End of SLAMCH ! END FUNCTION SLAMCH ! !*********************************************************************** ! SUBROUTINE SLAMC1( BETA, T, RND, IEEE1 ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL IEEE1, RND INTEGER BETA, T ! .. ! ! Purpose ! ======= ! ! SLAMC1 determines the machine parameters given by BETA, T, RND, and ! IEEE1. ! ! Arguments ! ========= ! ! BETA (output) INTEGER ! The base of the machine. ! ! T (output) INTEGER ! The number of ( BETA ) digits in the mantissa. ! ! RND (output) LOGICAL ! Specifies whether proper rounding ( RND = .TRUE. ) or ! chopping ( RND = .FALSE. ) occurs in addition. This may not ! be a reliable guide to the way in which the machine performs ! its arithmetic. ! ! IEEE1 (output) LOGICAL ! Specifies whether rounding appears to be done in the IEEE ! 'round to nearest' style. ! ! Further Details ! =============== ! ! The routine is based on the routine ENVRON by Malcolm and ! incorporates suggestions by Gentleman and Marovich. See ! ! Malcolm M. A. (1972) Algorithms to reveal properties of ! floating-point arithmetic. Comms. of the ACM, 15, 949-951. ! ! Gentleman W. M. and Marovich S. B. (1974) More on algorithms ! that reveal properties of floating point arithmetic units. ! Comms. of the ACM, 17, 276-277. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL FIRST, LIEEE1, LRND INTEGER LBETA, LT REAL(r8) A, B, C, F, ONE, QTR, SAVEC, T1, T2 ! .. ! .. Save statement .. SAVE FIRST, LIEEE1, LBETA, LRND, LT ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. ONE = 1 ! ! LBETA, LIEEE1, LT and LRND are the local values of BETA, ! IEEE1, T and RND. ! ! Throughout this routine we use the function SLAMC3 to ensure ! that relevant values are stored and not held in registers, or ! are not affected by optimizers. ! ! Compute a = 2.0**m with the smallest positive integer m such ! that ! ! fl( a + 1.0 ) = a. ! A = 1 C = 1 ! !+ WHILE( C.EQ.ONE )LOOP 10 CONTINUE IF( C.EQ.ONE ) THEN A = 2*A C = SLAMC3( A, ONE ) C = SLAMC3( C, -A ) GO TO 10 END IF !+ END WHILE ! ! Now compute b = 2.0**m with the smallest positive integer m ! such that ! ! fl( a + b ) .gt. a. ! B = 1 C = SLAMC3( A, B ) ! !+ WHILE( C.EQ.A )LOOP 20 CONTINUE IF( C.EQ.A ) THEN B = 2*B C = SLAMC3( A, B ) GO TO 20 END IF !+ END WHILE ! ! Now compute the base. a and c are neighbouring floating point ! numbers in the interval ( beta**t, beta**( t + 1 ) ) and so ! their difference is beta. Adding 0.25 to c is to ensure that it ! is truncated to beta and not ( beta - 1 ). ! QTR = ONE / 4 SAVEC = C C = SLAMC3( C, -A ) LBETA = C + QTR ! ! Now determine whether rounding or chopping occurs, by adding a ! bit less than beta/2 and a bit more than beta/2 to a. ! B = LBETA F = SLAMC3( B / 2, -B / 100 ) C = SLAMC3( F, A ) IF( C.EQ.A ) THEN LRND = .TRUE. ELSE LRND = .FALSE. END IF F = SLAMC3( B / 2, B / 100 ) C = SLAMC3( F, A ) IF( ( LRND ) .AND. ( C.EQ.A ) ) & LRND = .FALSE. ! ! Try and decide whether rounding is done in the IEEE 'round to ! nearest' style. B/2 is half a unit in the last place of the two ! numbers A and SAVEC. Furthermore, A is even, i.e. has last bit ! zero, and SAVEC is odd. Thus adding B/2 to A should not change ! A, but adding B/2 to SAVEC should change SAVEC. ! T1 = SLAMC3( B / 2, A ) T2 = SLAMC3( B / 2, SAVEC ) LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND ! ! Now find the mantissa, t. It should be the integer part of ! log to the base beta of a, however it is safer to determine t ! by powering. So we find t as the smallest positive integer for ! which ! ! fl( beta**t + 1.0 ) = 1.0. ! LT = 0 A = 1 C = 1 ! !+ WHILE( C.EQ.ONE )LOOP 30 CONTINUE IF( C.EQ.ONE ) THEN LT = LT + 1 A = A*LBETA C = SLAMC3( A, ONE ) C = SLAMC3( C, -A ) GO TO 30 END IF !+ END WHILE ! END IF ! BETA = LBETA T = LT RND = LRND IEEE1 = LIEEE1 RETURN ! ! End of SLAMC1 ! END SUBROUTINE SLAMC1 ! !*********************************************************************** ! SUBROUTINE SLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL RND INTEGER BETA, EMAX, EMIN, T REAL(r8) EPS, RMAX, RMIN ! .. ! ! Purpose ! ======= ! ! SLAMC2 determines the machine parameters specified in its argument ! list. ! ! Arguments ! ========= ! ! BETA (output) INTEGER ! The base of the machine. ! ! T (output) INTEGER ! The number of ( BETA ) digits in the mantissa. ! ! RND (output) LOGICAL ! Specifies whether proper rounding ( RND = .TRUE. ) or ! chopping ( RND = .FALSE. ) occurs in addition. This may not ! be a reliable guide to the way in which the machine performs ! its arithmetic. ! ! EPS (output) REAL ! The smallest positive number such that ! ! fl( 1.0 - EPS ) .LT. 1.0, ! ! where fl denotes the computed value. ! ! EMIN (output) INTEGER ! The minimum exponent before (gradual) underflow occurs. ! ! RMIN (output) REAL ! The smallest normalized number for the machine, given by ! BASE**( EMIN - 1 ), where BASE is the floating point value ! of BETA. ! ! EMAX (output) INTEGER ! The maximum exponent before overflow occurs. ! ! RMAX (output) REAL ! The largest positive number for the machine, given by ! BASE**EMAX * ( 1 - EPS ), where BASE is the floating point ! value of BETA. ! ! Further Details ! =============== ! ! The computation of EPS is based on a routine PARANOIA by ! W. Kahan of the University of California at Berkeley. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL FIRST, IEEE, IWARN, LIEEE1, LRND INTEGER GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT, & NGNMIN, NGPMIN REAL(r8) A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE, & SIXTH, SMALL, THIRD, TWO, ZERO ! .. ! .. Save statement .. SAVE FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX, & LRMIN, LT ! .. ! .. Data statements .. DATA FIRST / .TRUE. / , IWARN / .FALSE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. ZERO = 0 ONE = 1 TWO = 2 ! ! LBETA, LT, LRND, LEPS, LEMIN and LRMIN are the local values of ! BETA, T, RND, EPS, EMIN and RMIN. ! ! Throughout this routine we use the function SLAMC3 to ensure ! that relevant values are stored and not held in registers, or ! are not affected by optimizers. ! ! SLAMC1 returns the parameters LBETA, LT, LRND and LIEEE1. ! CALL SLAMC1( LBETA, LT, LRND, LIEEE1 ) ! ! Start to find EPS. ! B = LBETA A = B**( -LT ) LEPS = A ! ! Try some tricks to see whether or not this is the correct EPS. ! B = TWO / 3 HALF = ONE / 2 SIXTH = SLAMC3( B, -HALF ) THIRD = SLAMC3( SIXTH, SIXTH ) B = SLAMC3( THIRD, -HALF ) B = SLAMC3( B, SIXTH ) B = ABS( B ) IF( B.LT.LEPS ) & B = LEPS ! LEPS = 1 ! !+ WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP 10 CONTINUE IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN LEPS = B C = SLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) ) C = SLAMC3( HALF, -C ) B = SLAMC3( HALF, C ) C = SLAMC3( HALF, -B ) B = SLAMC3( HALF, C ) GO TO 10 END IF !+ END WHILE ! IF( A.LT.LEPS ) & LEPS = A ! ! Computation of EPS complete. ! ! Now find EMIN. Let A = + or - 1, and + or - (1 + BASE**(-3)). ! Keep dividing A by BETA until (gradual) underflow occurs. This ! is detected when we cannot recover the previous A. ! RBASE = ONE / LBETA SMALL = ONE DO 20 I = 1, 3 SMALL = SLAMC3( SMALL*RBASE, ZERO ) 20 CONTINUE A = SLAMC3( ONE, SMALL ) CALL SLAMC4( NGPMIN, ONE, LBETA ) CALL SLAMC4( NGNMIN, -ONE, LBETA ) CALL SLAMC4( GPMIN, A, LBETA ) CALL SLAMC4( GNMIN, -A, LBETA ) IEEE = .FALSE. ! IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN IF( NGPMIN.EQ.GPMIN ) THEN LEMIN = NGPMIN ! ( Non twos-complement machines, no gradual underflow; ! e.g., VAX ) ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN LEMIN = NGPMIN - 1 + LT IEEE = .TRUE. ! ( Non twos-complement machines, with gradual underflow; ! e.g., IEEE standard followers ) ELSE LEMIN = MIN( NGPMIN, GPMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN LEMIN = MAX( NGPMIN, NGNMIN ) ! ( Twos-complement machines, no gradual underflow; ! e.g., CYBER 205 ) ELSE LEMIN = MIN( NGPMIN, NGNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND. & ( GPMIN.EQ.GNMIN ) ) THEN IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT ! ( Twos-complement machines with gradual underflow; ! no known machine ) ELSE LEMIN = MIN( NGPMIN, NGNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF !** ! Comment out this if block if EMIN is ok IF( IWARN ) THEN FIRST = .TRUE. write(iulog, FMT = 9999 )LEMIN END IF !** ! ! Assume IEEE arithmetic if we found denormalised numbers above, ! or if arithmetic seems to round in the IEEE style, determined ! in routine SLAMC1. A true IEEE machine should have both things ! true; however, faulty machines may have one or the other. ! IEEE = IEEE .OR. LIEEE1 ! ! Compute RMIN by successive division by BETA. We could compute ! RMIN as BASE**( EMIN - 1 ), but some machines underflow during ! this computation. ! LRMIN = 1 DO 30 I = 1, 1 - LEMIN LRMIN = SLAMC3( LRMIN*RBASE, ZERO ) 30 CONTINUE ! ! Finally, call SLAMC5 to compute EMAX and RMAX. ! CALL SLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX ) END IF ! BETA = LBETA T = LT RND = LRND EPS = LEPS EMIN = LEMIN RMIN = LRMIN EMAX = LEMAX RMAX = LRMAX ! RETURN ! 9999 FORMAT( / / ' WARNING. The value EMIN may be incorrect:-', & ' EMIN = ', I8, / & ' If, after inspection, the value EMIN looks', & ' acceptable please comment out ', & / ' the IF block as marked within the code of routine', & ' SLAMC2,', / ' otherwise supply EMIN explicitly.', / ) ! ! End of SLAMC2 ! END SUBROUTINE SLAMC2 ! !*********************************************************************** ! FUNCTION SLAMC3( A, B ) real(r8) slamc3 ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. REAL(r8) A, B ! .. ! ! Purpose ! ======= ! ! SLAMC3 is intended to force A and B to be stored prior to doing ! the addition of A and B , for use in situations where optimizers ! might hold one of these in a register. ! ! Arguments ! ========= ! ! A, B (input) REAL ! The values A and B. ! ! ===================================================================== ! ! .. Executable Statements .. ! SLAMC3 = A + B ! RETURN ! ! End of SLAMC3 ! END FUNCTION SLAMC3 ! !*********************************************************************** ! SUBROUTINE SLAMC4( EMIN, START, BASE ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER BASE, EMIN REAL(r8) START ! .. ! ! Purpose ! ======= ! ! SLAMC4 is a service routine for SLAMC2. ! ! Arguments ! ========= ! ! EMIN (output) EMIN ! The minimum exponent before (gradual) underflow, computed by ! setting A = START and dividing by BASE until the previous A ! can not be recovered. ! ! START (input) REAL ! The starting point for determining EMIN. ! ! BASE (input) INTEGER ! The base of the machine. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I REAL(r8) A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO ! .. ! .. Executable Statements .. ! A = START ONE = 1 RBASE = ONE / BASE ZERO = 0 EMIN = 1 B1 = SLAMC3( A*RBASE, ZERO ) C1 = A C2 = A D1 = A D2 = A !+ WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND. ! $ ( D1.EQ.A ).AND.( D2.EQ.A ) )LOOP 10 CONTINUE IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND. & ( D2.EQ.A ) ) THEN EMIN = EMIN - 1 A = B1 B1 = SLAMC3( A / BASE, ZERO ) C1 = SLAMC3( B1*BASE, ZERO ) D1 = ZERO DO 20 I = 1, BASE D1 = D1 + B1 20 CONTINUE B2 = SLAMC3( A*RBASE, ZERO ) C2 = SLAMC3( B2 / RBASE, ZERO ) D2 = ZERO DO 30 I = 1, BASE D2 = D2 + B2 30 CONTINUE GO TO 10 END IF !+ END WHILE ! RETURN ! ! End of SLAMC4 ! END SUBROUTINE SLAMC4 ! !*********************************************************************** ! SUBROUTINE SLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL IEEE INTEGER BETA, EMAX, EMIN, P REAL(r8) RMAX ! .. ! ! Purpose ! ======= ! ! SLAMC5 attempts to compute RMAX, the largest machine floating-point ! number, without overflow. It assumes that EMAX + abs(EMIN) sum ! approximately to a power of 2. It will fail on machines where this ! assumption does not hold, for example, the Cyber 205 (EMIN = -28625, ! EMAX = 28718). It will also fail if the value supplied for EMIN is ! too large (i.e. too close to zero), probably with overflow. ! ! Arguments ! ========= ! ! BETA (input) INTEGER ! The base of floating-point arithmetic. ! ! P (input) INTEGER ! The number of base BETA digits in the mantissa of a ! floating-point value. ! ! EMIN (input) INTEGER ! The minimum exponent before (gradual) underflow. ! ! IEEE (input) LOGICAL ! A logical flag specifying whether or not the arithmetic ! system is thought to comply with the IEEE standard. ! ! EMAX (output) INTEGER ! The largest exponent before overflow ! ! RMAX (output) REAL ! The largest machine floating-point number. ! ! ===================================================================== ! ! .. Parameters .. REAL(r8) ZERO, ONE PARAMETER ( ZERO = 0.0E0_r8, ONE = 1.0E0_r8 ) ! .. ! .. Local Scalars .. INTEGER EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP REAL(r8) OLDY, RECBAS, Y, Z ! .. ! .. Executable Statements .. ! ! First compute LEXP and UEXP, two powers of 2 that bound ! abs(EMIN). We then assume that EMAX + abs(EMIN) will sum ! approximately to the bound that is closest to abs(EMIN). ! (EMAX is the exponent of the required number RMAX). ! LEXP = 1 EXBITS = 1 10 CONTINUE TRY = LEXP*2 IF( TRY.LE.( -EMIN ) ) THEN LEXP = TRY EXBITS = EXBITS + 1 GO TO 10 END IF IF( LEXP.EQ.-EMIN ) THEN UEXP = LEXP ELSE UEXP = TRY EXBITS = EXBITS + 1 END IF ! ! Now -LEXP is less than or equal to EMIN, and -UEXP is greater ! than or equal to EMIN. EXBITS is the number of bits needed to ! store the exponent. ! IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN EXPSUM = 2*LEXP ELSE EXPSUM = 2*UEXP END IF ! ! EXPSUM is the exponent range, approximately equal to ! EMAX - EMIN + 1 . ! EMAX = EXPSUM + EMIN - 1 NBITS = 1 + EXBITS + P ! ! NBITS is the total number of bits needed to store a ! floating-point number. ! IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN ! ! Either there are an odd number of bits used to store a ! floating-point number, which is unlikely, or some bits are ! not used in the representation of numbers, which is possible, ! (e.g. Cray machines) or the mantissa has an implicit bit, ! (e.g. IEEE machines, Dec Vax machines), which is perhaps the ! most likely. We have to assume the last alternative. ! If this is true, then we need to reduce EMAX by one because ! there must be some way of representing zero in an implicit-bit ! system. On machines like Cray, we are reducing EMAX by one ! unnecessarily. ! EMAX = EMAX - 1 END IF ! IF( IEEE ) THEN ! ! Assume we are on an IEEE machine which reserves one exponent ! for infinity and NaN. ! EMAX = EMAX - 1 END IF ! ! Now create RMAX, the largest machine number, which should ! be equal to (1.0 - BETA**(-P)) * BETA**EMAX . ! ! First compute 1.0 - BETA**(-P), being careful that the ! result is less than 1.0 . ! RECBAS = ONE / BETA Z = BETA - ONE Y = ZERO DO 20 I = 1, P Z = Z*RECBAS IF( Y.LT.ONE ) & OLDY = Y Y = SLAMC3( Y, Z ) 20 CONTINUE IF( Y.GE.ONE ) & Y = OLDY ! ! Now multiply by BETA**EMAX to get RMAX. ! DO 30 I = 1, EMAX Y = SLAMC3( Y*BETA, ZERO ) 30 CONTINUE ! RMAX = Y RETURN ! ! End of SLAMC5 ! END SUBROUTINE SLAMC5 !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE XERBLA( SRNAME, INFO ) ! ! -- LAPACK auxiliary routine (version 2.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER*6 SRNAME INTEGER INFO ! .. ! ! Purpose ! ======= ! ! XERBLA is an error handler for the LAPACK routines. ! It is called by an LAPACK routine if an input parameter has an ! invalid value. A message is printed and execution stops. ! ! Installers may consider modifying the STOP statement in order to ! call system-specific exception-handling facilities. ! ! Arguments ! ========= ! ! SRNAME (input) CHARACTER*6 ! The name of the routine which called XERBLA. ! ! INFO (input) INTEGER ! The position of the invalid parameter in the parameter list ! of the calling routine. ! ! ===================================================================== ! ! .. Executable Statements .. ! write(iulog, FMT = 9999 )SRNAME, INFO ! call endrun('XERBLA') ! 9999 FORMAT( ' ** On entry to ', A6, ' parameter number ', I2, ' had ', & 'an illegal value' ) ! ! End of XERBLA ! END SUBROUTINE XERBLA !--------1---------2---------3---------4---------5---------6---------7---------8 subroutine scopy(n,sx,incx,sy,incy) ! ! copies a vector, x, to a vector, y. ! uses unrolled loops for increments equal to 1. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! real(r8) sx(*),sy(*) integer i,incx,incy,ix,iy,m,mp1,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments ! not equal to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n sy(iy) = sx(ix) ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,7) if( m .eq. 0 ) go to 40 do 30 i = 1,m sy(i) = sx(i) 30 continue if( n .lt. 7 ) return 40 mp1 = m + 1 do 50 i = mp1,n,7 sy(i) = sx(i) sy(i + 1) = sx(i + 1) sy(i + 2) = sx(i + 2) sy(i + 3) = sx(i + 3) sy(i + 4) = sx(i + 4) sy(i + 5) = sx(i + 5) sy(i + 6) = sx(i + 6) 50 continue return end subroutine scopy !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGER ( M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) ! .. Scalar Arguments .. REAL(r8) ALPHA INTEGER INCX, INCY, LDA, M, N ! .. Array Arguments .. REAL(r8) A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! SGER performs the rank 1 operation ! ! A := alpha*x*y' + A, ! ! where alpha is a scalar, x is an m element vector, y is an n element ! vector and A is an m by n matrix. ! ! Parameters ! ========== ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix A. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - REAL . ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - REAL array of dimension at least ! ( 1 + ( m - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the m ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! Y - REAL array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. ! Unchanged on exit. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! A - REAL array of DIMENSION ( LDA, n ). ! Before entry, the leading m by n part of the array A must ! contain the matrix of coefficients. On exit, A is ! overwritten by the updated matrix. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E+0_r8 ) ! .. Local Scalars .. REAL(r8) TEMP INTEGER I, INFO, IX, J, JY, KX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( M.LT.0 )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGER ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( INCY.GT.0 )THEN JY = 1 ELSE JY = 1 - ( N - 1 )*INCY END IF IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) DO 10, I = 1, M A( I, J ) = A( I, J ) + X( I )*TEMP 10 CONTINUE END IF JY = JY + INCY 20 CONTINUE ELSE IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( M - 1 )*INCX END IF DO 40, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) IX = KX DO 30, I = 1, M A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JY = JY + INCY 40 CONTINUE END IF ! RETURN ! ! End of SGER . ! END SUBROUTINE SGER !--------1---------2---------3---------4---------5---------6---------7---------8 FUNCTION SNRM2 ( N, X, INCX ) real(r8) snrm2 ! .. Scalar Arguments .. INTEGER INCX, N ! .. Array Arguments .. REAL(r8) X( * ) ! .. ! ! SNRM2 returns the euclidean norm of a vector via the function ! name, so that ! ! SNRM2 := sqrt( x'*x ) ! ! ! ! -- This version written on 25-October-1982. ! Modified on 14-October-1993 to inline the call to SLASSQ. ! Sven Hammarling, Nag Ltd. ! ! ! .. Parameters .. REAL(r8) ONE , ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. Local Scalars .. INTEGER IX REAL(r8) ABSXI, NORM, SCALE, SSQ ! .. ! .. Executable Statements .. IF( N.LT.1 .OR. INCX.LT.1 )THEN NORM = ZERO ELSE IF( N.EQ.1 )THEN NORM = ABS( X( 1 ) ) ELSE SCALE = ZERO SSQ = ONE ! The following loop is equivalent to this call to the LAPACK ! auxiliary routine: ! CALL SLASSQ( N, X, INCX, SCALE, SSQ ) ! DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX IF( X( IX ).NE.ZERO )THEN ABSXI = ABS( X( IX ) ) IF( SCALE.LT.ABSXI )THEN SSQ = ONE + SSQ*( SCALE/ABSXI )**2 SCALE = ABSXI ELSE SSQ = SSQ + ( ABSXI/SCALE )**2 END IF END IF 10 CONTINUE NORM = SCALE * SQRT( SSQ ) END IF ! SNRM2 = NORM RETURN ! ! End of SNRM2. ! END FUNCTION SNRM2 !--------1---------2---------3---------4---------5---------6---------7---------8 subroutine srot (n,sx,incx,sy,incy,c,s) ! ! applies a plane rotation. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! real(r8) sx(*),sy(*),stemp,c,s integer i,incx,incy,ix,iy,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments not equal ! to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n stemp = c*sx(ix) + s*sy(iy) sy(iy) = c*sy(iy) - s*sx(ix) sx(ix) = stemp ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! 20 do 30 i = 1,n stemp = c*sx(i) + s*sy(i) sy(i) = c*sy(i) - s*sx(i) sx(i) = stemp 30 continue return end subroutine srot !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, & BETA, C, LDC ) ! .. Scalar Arguments .. CHARACTER*1 TRANSA, TRANSB INTEGER M, N, K, LDA, LDB, LDC REAL(r8) ALPHA, BETA ! .. Array Arguments .. REAL(r8) A( LDA, * ), B( LDB, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! SGEMM performs one of the matrix-matrix operations ! ! C := alpha*op( A )*op( B ) + beta*C, ! ! where op( X ) is one of ! ! op( X ) = X or op( X ) = X', ! ! alpha and beta are scalars, and A, B and C are matrices, with op( A ) ! an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. ! ! Parameters ! ========== ! ! TRANSA - CHARACTER*1. ! On entry, TRANSA specifies the form of op( A ) to be used in ! the matrix multiplication as follows: ! ! TRANSA = 'N' or 'n', op( A ) = A. ! ! TRANSA = 'T' or 't', op( A ) = A'. ! ! TRANSA = 'C' or 'c', op( A ) = A'. ! ! Unchanged on exit. ! ! TRANSB - CHARACTER*1. ! On entry, TRANSB specifies the form of op( B ) to be used in ! the matrix multiplication as follows: ! ! TRANSB = 'N' or 'n', op( B ) = B. ! ! TRANSB = 'T' or 't', op( B ) = B'. ! ! TRANSB = 'C' or 'c', op( B ) = B'. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix ! op( A ) and of the matrix C. M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix ! op( B ) and the number of columns of the matrix C. N must be ! at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry, K specifies the number of columns of the matrix ! op( A ) and the number of rows of the matrix op( B ). K must ! be at least zero. ! Unchanged on exit. ! ! ALPHA - REAL . ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - REAL array of DIMENSION ( LDA, ka ), where ka is ! k when TRANSA = 'N' or 'n', and is m otherwise. ! Before entry with TRANSA = 'N' or 'n', the leading m by k ! part of the array A must contain the matrix A, otherwise ! the leading k by m part of the array A must contain the ! matrix A. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When TRANSA = 'N' or 'n' then ! LDA must be at least max( 1, m ), otherwise LDA must be at ! least max( 1, k ). ! Unchanged on exit. ! ! B - REAL array of DIMENSION ( LDB, kb ), where kb is ! n when TRANSB = 'N' or 'n', and is k otherwise. ! Before entry with TRANSB = 'N' or 'n', the leading k by n ! part of the array B must contain the matrix B, otherwise ! the leading n by k part of the array B must contain the ! matrix B. ! Unchanged on exit. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. When TRANSB = 'N' or 'n' then ! LDB must be at least max( 1, k ), otherwise LDB must be at ! least max( 1, n ). ! Unchanged on exit. ! ! BETA - REAL . ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then C need not be set on input. ! Unchanged on exit. ! ! C - REAL array of DIMENSION ( LDC, n ). ! Before entry, the leading m by n part of the array C must ! contain the matrix C, except when beta is zero, in which ! case C need not be set on entry. ! On exit, the array C is overwritten by the m by n matrix ! ( alpha*op( A )*op( B ) + beta*C ). ! ! LDC - INTEGER. ! On entry, LDC specifies the first dimension of C as declared ! in the calling (sub) program. LDC must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. Local Scalars .. LOGICAL NOTA, NOTB INTEGER I, INFO, J, L, NCOLA, NROWA, NROWB REAL(r8) TEMP ! .. Parameters .. REAL(r8) ONE , ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Executable Statements .. ! ! Set NOTA and NOTB as true if A and B respectively are not ! transposed and set NROWA, NCOLA and NROWB as the number of rows ! and columns of A and the number of rows of B respectively. ! NOTA = LSAME( TRANSA, 'N' ) NOTB = LSAME( TRANSB, 'N' ) IF( NOTA )THEN NROWA = M NCOLA = K ELSE NROWA = K NCOLA = M END IF IF( NOTB )THEN NROWB = K ELSE NROWB = N END IF ! ! Test the input parameters. ! INFO = 0 IF( ( .NOT.NOTA ).AND. & ( .NOT.LSAME( TRANSA, 'C' ) ).AND. & ( .NOT.LSAME( TRANSA, 'T' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.NOTB ).AND. & ( .NOT.LSAME( TRANSB, 'C' ) ).AND. & ( .NOT.LSAME( TRANSB, 'T' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( K .LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 8 ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN INFO = 10 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 13 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGEMM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! And if alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF ! ! Start the operations. ! IF( NOTB )THEN IF( NOTA )THEN ! ! Form C := alpha*A*B + beta*C. ! DO 90, J = 1, N IF( BETA.EQ.ZERO )THEN DO 50, I = 1, M C( I, J ) = ZERO 50 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 60, I = 1, M C( I, J ) = BETA*C( I, J ) 60 CONTINUE END IF DO 80, L = 1, K IF( B( L, J ).NE.ZERO )THEN TEMP = ALPHA*B( L, J ) DO 70, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 70 CONTINUE END IF 80 CONTINUE 90 CONTINUE ELSE ! ! Form C := alpha*A'*B + beta*C ! DO 120, J = 1, N DO 110, I = 1, M TEMP = ZERO DO 100, L = 1, K TEMP = TEMP + A( L, I )*B( L, J ) 100 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 110 CONTINUE 120 CONTINUE END IF ELSE IF( NOTA )THEN ! ! Form C := alpha*A*B' + beta*C ! DO 170, J = 1, N IF( BETA.EQ.ZERO )THEN DO 130, I = 1, M C( I, J ) = ZERO 130 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 140, I = 1, M C( I, J ) = BETA*C( I, J ) 140 CONTINUE END IF DO 160, L = 1, K IF( B( J, L ).NE.ZERO )THEN TEMP = ALPHA*B( J, L ) DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 150 CONTINUE END IF 160 CONTINUE 170 CONTINUE ELSE ! ! Form C := alpha*A'*B' + beta*C ! DO 200, J = 1, N DO 190, I = 1, M TEMP = ZERO DO 180, L = 1, K TEMP = TEMP + A( L, I )*B( J, L ) 180 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 190 CONTINUE 200 CONTINUE END IF END IF ! RETURN ! ! End of SGEMM . ! END SUBROUTINE SGEMM !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE SGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX, & BETA, Y, INCY ) ! .. Scalar Arguments .. REAL(r8) ALPHA, BETA INTEGER INCX, INCY, LDA, M, N CHARACTER*1 TRANS ! .. Array Arguments .. REAL(r8) A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! SGEMV performs one of the matrix-vector operations ! ! y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, ! ! where alpha and beta are scalars, x and y are vectors and A is an ! m by n matrix. ! ! Parameters ! ========== ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' y := alpha*A*x + beta*y. ! ! TRANS = 'T' or 't' y := alpha*A'*x + beta*y. ! ! TRANS = 'C' or 'c' y := alpha*A'*x + beta*y. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix A. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - REAL . ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - REAL array of DIMENSION ( LDA, n ). ! Before entry, the leading m by n part of the array A must ! contain the matrix of coefficients. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, m ). ! Unchanged on exit. ! ! X - REAL array of DIMENSION at least ! ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. ! Before entry, the incremented array X must contain the ! vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - REAL . ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then Y need not be set on input. ! Unchanged on exit. ! ! Y - REAL array of DIMENSION at least ! ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. ! Before entry with BETA non-zero, the incremented array Y ! must contain the vector y. On exit, Y is overwritten by the ! updated vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. REAL(r8) ONE , ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. Local Scalars .. REAL(r8) TEMP INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 1 ELSE IF( M.LT.0 )THEN INFO = 2 ELSE IF( N.LT.0 )THEN INFO = 3 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 ELSE IF( INCY.EQ.0 )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGEMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set LENX and LENY, the lengths of the vectors x and y, and set ! up the start points in X and Y. ! IF( LSAME( TRANS, 'N' ) )THEN LENX = N LENY = M ELSE LENX = M LENY = N END IF IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( LENX - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( LENY - 1 )*INCY END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, LENY Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, LENY Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, LENY Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, LENY Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN IF( LSAME( TRANS, 'N' ) )THEN ! ! Form y := alpha*A*x + y. ! JX = KX IF( INCY.EQ.1 )THEN DO 60, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) DO 50, I = 1, M Y( I ) = Y( I ) + TEMP*A( I, J ) 50 CONTINUE END IF JX = JX + INCX 60 CONTINUE ELSE DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IY = KY DO 70, I = 1, M Y( IY ) = Y( IY ) + TEMP*A( I, J ) IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF ELSE ! ! Form y := alpha*A'*x + y. ! JY = KY IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = ZERO DO 90, I = 1, M TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 100 CONTINUE ELSE DO 120, J = 1, N TEMP = ZERO IX = KX DO 110, I = 1, M TEMP = TEMP + A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 120 CONTINUE END IF END IF ! RETURN ! ! End of SGEMV . ! END SUBROUTINE SGEMV !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE STRMM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, & B, LDB ) ! .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO, TRANSA, DIAG INTEGER M, N, LDA, LDB REAL(r8) ALPHA ! .. Array Arguments .. REAL(r8) A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! STRMM performs one of the matrix-matrix operations ! ! B := alpha*op( A )*B, or B := alpha*B*op( A ), ! ! where alpha is a scalar, B is an m by n matrix, A is a unit, or ! non-unit, upper or lower triangular matrix and op( A ) is one of ! ! op( A ) = A or op( A ) = A'. ! ! Parameters ! ========== ! ! SIDE - CHARACTER*1. ! On entry, SIDE specifies whether op( A ) multiplies B from ! the left or right as follows: ! ! SIDE = 'L' or 'l' B := alpha*op( A )*B. ! ! SIDE = 'R' or 'r' B := alpha*B*op( A ). ! ! Unchanged on exit. ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix A is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANSA - CHARACTER*1. ! On entry, TRANSA specifies the form of op( A ) to be used in ! the matrix multiplication as follows: ! ! TRANSA = 'N' or 'n' op( A ) = A. ! ! TRANSA = 'T' or 't' op( A ) = A'. ! ! TRANSA = 'C' or 'c' op( A ) = A'. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit triangular ! as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of B. M must be at ! least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of B. N must be ! at least zero. ! Unchanged on exit. ! ! ALPHA - REAL . ! On entry, ALPHA specifies the scalar alpha. When alpha is ! zero then A is not referenced and B need not be set before ! entry. ! Unchanged on exit. ! ! A - REAL array of DIMENSION ( LDA, k ), where k is m ! when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. ! Before entry with UPLO = 'U' or 'u', the leading k by k ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading k by k ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When SIDE = 'L' or 'l' then ! LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' ! then LDA must be at least max( 1, n ). ! Unchanged on exit. ! ! B - REAL array of DIMENSION ( LDB, n ). ! Before entry, the leading m by n part of the array B must ! contain the matrix B, and on exit is overwritten by the ! transformed matrix. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. LDB must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. Local Scalars .. LOGICAL LSIDE, NOUNIT, UPPER INTEGER I, INFO, J, K, NROWA REAL(r8) TEMP ! .. Parameters .. REAL(r8) ONE , ZERO PARAMETER ( ONE = 1.0E+0_r8, ZERO = 0.0E+0_r8 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! LSIDE = LSAME( SIDE , 'L' ) IF( LSIDE )THEN NROWA = M ELSE NROWA = N END IF NOUNIT = LSAME( DIAG , 'N' ) UPPER = LSAME( UPLO , 'U' ) ! INFO = 0 IF( ( .NOT.LSIDE ).AND. & ( .NOT.LSAME( SIDE , 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 2 ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND. & ( .NOT.LSAME( TRANSA, 'T' ) ).AND. & ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN INFO = 3 ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND. & ( .NOT.LSAME( DIAG , 'N' ) ) )THEN INFO = 4 ELSE IF( M .LT.0 )THEN INFO = 5 ELSE IF( N .LT.0 )THEN INFO = 6 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'STRMM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE RETURN END IF ! ! Start the operations. ! IF( LSIDE )THEN IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*A*B. ! IF( UPPER )THEN DO 50, J = 1, N DO 40, K = 1, M IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) DO 30, I = 1, K - 1 B( I, J ) = B( I, J ) + TEMP*A( I, K ) 30 CONTINUE IF( NOUNIT ) & TEMP = TEMP*A( K, K ) B( K, J ) = TEMP END IF 40 CONTINUE 50 CONTINUE ELSE DO 80, J = 1, N DO 70 K = M, 1, -1 IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) B( K, J ) = TEMP IF( NOUNIT ) & B( K, J ) = B( K, J )*A( K, K ) DO 60, I = K + 1, M B( I, J ) = B( I, J ) + TEMP*A( I, K ) 60 CONTINUE END IF 70 CONTINUE 80 CONTINUE END IF ELSE ! ! Form B := alpha*A'*B. ! IF( UPPER )THEN DO 110, J = 1, N DO 100, I = M, 1, -1 TEMP = B( I, J ) IF( NOUNIT ) & TEMP = TEMP*A( I, I ) DO 90, K = 1, I - 1 TEMP = TEMP + A( K, I )*B( K, J ) 90 CONTINUE B( I, J ) = ALPHA*TEMP 100 CONTINUE 110 CONTINUE ELSE DO 140, J = 1, N DO 130, I = 1, M TEMP = B( I, J ) IF( NOUNIT ) & TEMP = TEMP*A( I, I ) DO 120, K = I + 1, M TEMP = TEMP + A( K, I )*B( K, J ) 120 CONTINUE B( I, J ) = ALPHA*TEMP 130 CONTINUE 140 CONTINUE END IF END IF ELSE IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*B*A. ! IF( UPPER )THEN DO 180, J = N, 1, -1 TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 150, I = 1, M B( I, J ) = TEMP*B( I, J ) 150 CONTINUE DO 170, K = 1, J - 1 IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 160, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE ELSE DO 220, J = 1, N TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 190, I = 1, M B( I, J ) = TEMP*B( I, J ) 190 CONTINUE DO 210, K = J + 1, N IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 200, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 200 CONTINUE END IF 210 CONTINUE 220 CONTINUE END IF ELSE ! ! Form B := alpha*B*A'. ! IF( UPPER )THEN DO 260, K = 1, N DO 240, J = 1, K - 1 IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 230, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 230 CONTINUE END IF 240 CONTINUE TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 250, I = 1, M B( I, K ) = TEMP*B( I, K ) 250 CONTINUE END IF 260 CONTINUE ELSE DO 300, K = N, 1, -1 DO 280, J = K + 1, N IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 270, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 270 CONTINUE END IF 280 CONTINUE TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 290, I = 1, M B( I, K ) = TEMP*B( I, K ) 290 CONTINUE END IF 300 CONTINUE END IF END IF END IF ! RETURN ! ! End of STRMM . ! END SUBROUTINE STRMM !--------1---------2---------3---------4---------5---------6---------7---------8 SUBROUTINE STRMV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, LDA, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. REAL(r8) A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! STRMV performs one of the matrix-vector operations ! ! x := A*x, or x := A'*x, ! ! where x is an n element vector and A is an n by n unit, or non-unit, ! upper or lower triangular matrix. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' x := A*x. ! ! TRANS = 'T' or 't' x := A'*x. ! ! TRANS = 'C' or 'c' x := A'*x. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! A - REAL array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! X - REAL array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. On exit, X is overwritten with the ! tranformed vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. REAL(r8) ZERO PARAMETER ( ZERO = 0.0E+0_r8 ) ! .. Local Scalars .. REAL(r8) TEMP INTEGER I, INFO, IX, J, JX, KX LOGICAL NOUNIT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'STRMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := A*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 10, I = 1, J - 1 X( I ) = X( I ) + TEMP*A( I, J ) 10 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( J, J ) END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 30, I = 1, J - 1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX + INCX 30 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( J, J ) END IF JX = JX + INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 50, I = N, J + 1, -1 X( I ) = X( I ) + TEMP*A( I, J ) 50 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( J, J ) END IF 60 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 80, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 70, I = N, J + 1, -1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX - INCX 70 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( J, J ) END IF JX = JX - INCX 80 CONTINUE END IF END IF ELSE ! ! Form x := A'*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 100, J = N, 1, -1 TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 90, I = J - 1, 1, -1 TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE X( J ) = TEMP 100 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 120, J = N, 1, -1 TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 110, I = J - 1, 1, -1 IX = IX - INCX TEMP = TEMP + A( I, J )*X( IX ) 110 CONTINUE X( JX ) = TEMP JX = JX - INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = 1, N TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 130, I = J + 1, N TEMP = TEMP + A( I, J )*X( I ) 130 CONTINUE X( J ) = TEMP 140 CONTINUE ELSE JX = KX DO 160, J = 1, N TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 150, I = J + 1, N IX = IX + INCX TEMP = TEMP + A( I, J )*X( IX ) 150 CONTINUE X( JX ) = TEMP JX = JX + INCX 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of STRMV . ! END SUBROUTINE STRMV !--------1---------2---------3---------4---------5---------6---------7---------8 end module sgexx