SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, $ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, $ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, $ WORK, LWORK, RWORK, IWORK, BWORK, INFO ) * * -- LAPACK driver routine (version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * June 30, 1999 * * .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N REAL ABNRM, BBNRM * .. * .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) REAL LSCALE( * ), RCONDE( * ), RCONDV( * ), $ RSCALE( * ), RWORK( * ) COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), $ WORK( * ) * .. * * Purpose * ======= * * CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices * (A,B) the generalized eigenvalues, and optionally, the left and/or * right generalized eigenvectors. * * Optionally, it also computes a balancing transformation to improve * the conditioning of the eigenvalues and eigenvectors (ILO, IHI, * LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for * the eigenvalues (RCONDE), and reciprocal condition numbers for the * right eigenvectors (RCONDV). * * A generalized eigenvalue for a pair of matrices (A,B) is a scalar * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is * singular. It is usually represented as the pair (alpha,beta), as * there is a reasonable interpretation for beta=0, and even for both * being zero. * * The right eigenvector v(j) corresponding to the eigenvalue lambda(j) * of (A,B) satisfies * A * v(j) = lambda(j) * B * v(j) . * The left eigenvector u(j) corresponding to the eigenvalue lambda(j) * of (A,B) satisfies * u(j)**H * A = lambda(j) * u(j)**H * B. * where u(j)**H is the conjugate-transpose of u(j). * * * Arguments * ========= * * BALANC (input) CHARACTER*1 * Specifies the balance option to be performed: * = 'N': do not diagonally scale or permute; * = 'P': permute only; * = 'S': scale only; * = 'B': both permute and scale. * Computed reciprocal condition numbers will be for the * matrices after permuting and/or balancing. Permuting does * not change condition numbers (in exact arithmetic), but * balancing does. * * JOBVL (input) CHARACTER*1 * = 'N': do not compute the left generalized eigenvectors; * = 'V': compute the left generalized eigenvectors. * * JOBVR (input) CHARACTER*1 * = 'N': do not compute the right generalized eigenvectors; * = 'V': compute the right generalized eigenvectors. * * SENSE (input) CHARACTER*1 * Determines which reciprocal condition numbers are computed. * = 'N': none are computed; * = 'E': computed for eigenvalues only; * = 'V': computed for eigenvectors only; * = 'B': computed for eigenvalues and eigenvectors. * * N (input) INTEGER * The order of the matrices A, B, VL, and VR. N >= 0. * * A (input/output) COMPLEX array, dimension (LDA, N) * On entry, the matrix A in the pair (A,B). * On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' * or both, then A contains the first part of the complex Schur * form of the "balanced" versions of the input A and B. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) COMPLEX array, dimension (LDB, N) * On entry, the matrix B in the pair (A,B). * On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' * or both, then B contains the second part of the complex * Schur form of the "balanced" versions of the input A and B. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * ALPHA (output) COMPLEX array, dimension (N) * BETA (output) COMPLEX array, dimension (N) * On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized * eigenvalues. * * Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or * underflow, and BETA(j) may even be zero. Thus, the user * should avoid naively computing the ratio ALPHA/BETA. * However, ALPHA will be always less than and usually * comparable with norm(A) in magnitude, and BETA always less * than and usually comparable with norm(B). * * VL (output) COMPLEX array, dimension (LDVL,N) * If JOBVL = 'V', the left generalized eigenvectors u(j) are * stored one after another in the columns of VL, in the same * order as their eigenvalues. * Each eigenvector will be scaled so the largest component * will have abs(real part) + abs(imag. part) = 1. * Not referenced if JOBVL = 'N'. * * LDVL (input) INTEGER * The leading dimension of the matrix VL. LDVL >= 1, and * if JOBVL = 'V', LDVL >= N. * * VR (output) COMPLEX array, dimension (LDVR,N) * If JOBVR = 'V', the right generalized eigenvectors v(j) are * stored one after another in the columns of VR, in the same * order as their eigenvalues. * Each eigenvector will be scaled so the largest component * will have abs(real part) + abs(imag. part) = 1. * Not referenced if JOBVR = 'N'. * * LDVR (input) INTEGER * The leading dimension of the matrix VR. LDVR >= 1, and * if JOBVR = 'V', LDVR >= N. * * ILO,IHI (output) INTEGER * ILO and IHI are integer values such that on exit * A(i,j) = 0 and B(i,j) = 0 if i > j and * j = 1,...,ILO-1 or i = IHI+1,...,N. * If BALANC = 'N' or 'S', ILO = 1 and IHI = N. * * LSCALE (output) REAL array, dimension (N) * Details of the permutations and scaling factors applied * to the left side of A and B. If PL(j) is the index of the * row interchanged with row j, and DL(j) is the scaling * factor applied to row j, then * LSCALE(j) = PL(j) for j = 1,...,ILO-1 * = DL(j) for j = ILO,...,IHI * = PL(j) for j = IHI+1,...,N. * The order in which the interchanges are made is N to IHI+1, * then 1 to ILO-1. * * RSCALE (output) REAL array, dimension (N) * Details of the permutations and scaling factors applied * to the right side of A and B. If PR(j) is the index of the * column interchanged with column j, and DR(j) is the scaling * factor applied to column j, then * RSCALE(j) = PR(j) for j = 1,...,ILO-1 * = DR(j) for j = ILO,...,IHI * = PR(j) for j = IHI+1,...,N * The order in which the interchanges are made is N to IHI+1, * then 1 to ILO-1. * * ABNRM (output) REAL * The one-norm of the balanced matrix A. * * BBNRM (output) REAL * The one-norm of the balanced matrix B. * * RCONDE (output) REAL array, dimension (N) * If SENSE = 'E' or 'B', the reciprocal condition numbers of * the selected eigenvalues, stored in consecutive elements of * the array. * If SENSE = 'V', RCONDE is not referenced. * * RCONDV (output) REAL array, dimension (N) * If JOB = 'V' or 'B', the estimated reciprocal condition * numbers of the selected eigenvectors, stored in consecutive * elements of the array. If the eigenvalues cannot be reordered * to compute RCONDV(j), RCONDV(j) is set to 0; this can only * occur when the true value would be very small anyway. * If SENSE = 'E', RCONDV is not referenced. * Not referenced if JOB = 'E'. * * WORK (workspace/output) COMPLEX 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(1,2*N). * If SENSE = 'N' or 'E', LWORK >= 2*N. * If SENSE = 'V' or 'B', LWORK >= 2*N*N+2*N. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * RWORK (workspace) REAL array, dimension (6*N) * Real workspace. * * IWORK (workspace) INTEGER array, dimension (N+2) * If SENSE = 'E', IWORK is not referenced. * * BWORK (workspace) LOGICAL array, dimension (N) * If SENSE = 'N', BWORK is not referenced. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * = 1,...,N: * The QZ iteration failed. No eigenvectors have been * calculated, but ALPHA(j) and BETA(j) should be correct * for j=INFO+1,...,N. * > N: =N+1: other than QZ iteration failed in CHGEQZ. * =N+2: error return from CTGEVC. * * Further Details * =============== * * Balancing a matrix pair (A,B) includes, first, permuting rows and * columns to isolate eigenvalues, second, applying diagonal similarity * transformation to the rows and columns to make the rows and columns * as close in norm as possible. The computed reciprocal condition * numbers correspond to the balanced matrix. Permuting rows and columns * will not change the condition numbers (in exact arithmetic) but * diagonal scaling will. For further explanation of balancing, see * section 4.11.1.2 of LAPACK Users' Guide. * * An approximate error bound on the chordal distance between the i-th * computed generalized eigenvalue w and the corresponding exact * eigenvalue lambda is * * chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) * * An approximate error bound for the angle between the i-th computed * eigenvector VL(i) or VR(i) is given by * * EPS * norm(ABNRM, BBNRM) / DIF(i). * * For further explanation of the reciprocal condition numbers RCONDE * and RCONDV, see section 4.11 of LAPACK User's Guide. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), $ CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, $ WANTSB, WANTSE, WANTSN, WANTSV CHARACTER CHTEMP INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS, $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, MINWRK REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, $ SMLNUM, TEMP COMPLEX X * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY, $ CLASCL, CLASET, CTGEVC, CTGSNA, CUNGQR, CUNMQR, $ SLABAD, SLASCL, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL CLANGE, SLAMCH EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, MAX, REAL, SQRT * .. * .. Statement Functions .. REAL ABS1 * .. * .. Statement Function definitions .. ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) ) * .. * .. Executable Statements .. * * Decode the input arguments * IF( LSAME( JOBVL, 'N' ) ) THEN IJOBVL = 1 ILVL = .FALSE. ELSE IF( LSAME( JOBVL, 'V' ) ) THEN IJOBVL = 2 ILVL = .TRUE. ELSE IJOBVL = -1 ILVL = .FALSE. END IF * IF( LSAME( JOBVR, 'N' ) ) THEN IJOBVR = 1 ILVR = .FALSE. ELSE IF( LSAME( JOBVR, 'V' ) ) THEN IJOBVR = 2 ILVR = .TRUE. ELSE IJOBVR = -1 ILVR = .FALSE. END IF ILV = ILVL .OR. ILVR * WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) $ THEN INFO = -1 ELSE IF( IJOBVL.LE.0 ) THEN INFO = -2 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -3 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) ) $ THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -13 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -15 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. The workspace is * computed assuming ILO = 1 and IHI = N, the worst case.) * MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = N + N*ILAENV( 1, 'CGEQRF', ' ', N, 1, N, 0 ) IF( WANTSE ) THEN MINWRK = MAX( 1, 2*N ) ELSE IF( WANTSV .OR. WANTSB ) THEN MINWRK = 2*N*N + 2*N MAXWRK = MAX( MAXWRK, 2*N*N+2*N ) END IF WORK( 1 ) = MAXWRK END IF * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -25 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGGEVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN 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 = CLANGE( 'M', N, N, A, LDA, RWORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF IF( ILASCL ) $ CALL CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = CLANGE( 'M', N, N, B, LDB, RWORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF IF( ILBSCL ) $ CALL CLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) * * Permute and/or balance the matrix pair (A,B) * (Real Workspace: need 6*N) * CALL CGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, $ RWORK, IERR ) * * Compute ABNRM and BBNRM * ABNRM = CLANGE( '1', N, N, A, LDA, RWORK( 1 ) ) IF( ILASCL ) THEN RWORK( 1 ) = ABNRM CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, RWORK( 1 ), 1, $ IERR ) ABNRM = RWORK( 1 ) END IF * BBNRM = CLANGE( '1', N, N, B, LDB, RWORK( 1 ) ) IF( ILBSCL ) THEN RWORK( 1 ) = BBNRM CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, RWORK( 1 ), 1, $ IERR ) BBNRM = RWORK( 1 ) END IF * * Reduce B to triangular form (QR decomposition of B) * (Complex Workspace: need N, prefer N*NB ) * IROWS = IHI + 1 - ILO IF( ILV .OR. .NOT.WANTSN ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = 1 IWRK = ITAU + IROWS CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), $ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the unitary transformation to A * (Complex Workspace: need N, prefer N*NB) * CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), $ LWORK+1-IWRK, IERR ) * * Initialize VL and/or VR * (Workspace: need N, prefer N*NB) * IF( ILVL ) THEN CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL ) CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, $ VL( ILO+1, ILO ), LDVL ) CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * IF( ILVR ) $ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * (Workspace: none needed) * IF( ILV .OR. .NOT.WANTSN ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, IERR ) ELSE CALL CGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) END IF * * Perform QZ algorithm (Compute eigenvalues, and optionally, the * Schur forms and Schur vectors) * (Complex Workspace: need N) * (Real Workspace: need N) * IWRK = ITAU IF( ILV .OR. .NOT.WANTSN ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF * CALL CHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ), $ LWORK+1-IWRK, RWORK, IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 90 END IF * * Compute Eigenvectors and estimate condition numbers if desired * CTGEVC: (Complex Workspace: need 2*N ) * (Real Workspace: need 2*N ) * CTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B') * (Integer Workspace: need N+2 ) * IF( ILV .OR. .NOT.WANTSN ) THEN IF( ILV ) THEN IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF * CALL CTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, N, IN, WORK( IWRK ), RWORK, $ IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 90 END IF END IF * IF( .NOT.WANTSN ) THEN * * compute eigenvectors (STGEVC) and estimate condition * numbers (STGSNA). Note that the definition of the condition * number is not invariant under transformation (u,v) to * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized * Schur form (S,T), Q and Z are orthogonal matrices. In order * to avoid using extra 2*N*N workspace, we have to * re-calculate eigenvectors and estimate the condition numbers * one at a time. * DO 20 I = 1, N * DO 10 J = 1, N BWORK( J ) = .FALSE. 10 CONTINUE BWORK( I ) = .TRUE. * IWRK = N + 1 IWRK1 = IWRK + N * IF( WANTSE .OR. WANTSB ) THEN CALL CTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB, $ WORK( 1 ), N, WORK( IWRK ), N, 1, M, $ WORK( IWRK1 ), RWORK, IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 90 END IF END IF * CALL CTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB, $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ), $ RCONDV( I ), 1, M, WORK( IWRK1 ), $ LWORK-IWRK1+1, IWORK, IERR ) * 20 CONTINUE END IF END IF * * Undo balancing on VL and VR and normalization * (Workspace: none needed) * IF( ILVL ) THEN CALL CGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL, $ LDVL, IERR ) * DO 50 JC = 1, N TEMP = ZERO DO 30 JR = 1, N TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) ) 30 CONTINUE IF( TEMP.LT.SMLNUM ) $ GO TO 50 TEMP = ONE / TEMP DO 40 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 40 CONTINUE 50 CONTINUE END IF * IF( ILVR ) THEN CALL CGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR, $ LDVR, IERR ) DO 80 JC = 1, N TEMP = ZERO DO 60 JR = 1, N TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) ) 60 CONTINUE IF( TEMP.LT.SMLNUM ) $ GO TO 80 TEMP = ONE / TEMP DO 70 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 70 CONTINUE 80 CONTINUE END IF * * Undo scaling if necessary * IF( ILASCL ) $ CALL CLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) * IF( ILBSCL ) $ CALL CLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) * 90 CONTINUE WORK( 1 ) = MAXWRK * RETURN * * End of CGGEVX * END