*> \brief CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm) * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CGGEV3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, * $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVL, JOBVR * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. * REAL RWORK( * ) * COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), * $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), * $ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGGEV3 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. *> *> 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 generalized eigenvector v(j) corresponding to the *> generalized eigenvalue lambda(j) of (A,B) satisfies *> *> A * v(j) = lambda(j) * B * v(j). *> *> The left generalized eigenvector u(j) corresponding to the *> generalized eigenvalues 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). *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVL *> \verbatim *> JOBVL is CHARACTER*1 *> = 'N': do not compute the left generalized eigenvectors; *> = 'V': compute the left generalized eigenvectors. *> \endverbatim *> *> \param[in] JOBVR *> \verbatim *> JOBVR is CHARACTER*1 *> = 'N': do not compute the right generalized eigenvectors; *> = 'V': compute the right generalized eigenvectors. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A, B, VL, and VR. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX array, dimension (LDA, N) *> On entry, the matrix A in the pair (A,B). *> On exit, A has been overwritten. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A. LDA >= max(1,N). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX array, dimension (LDB, N) *> On entry, the matrix B in the pair (A,B). *> On exit, B has been overwritten. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is COMPLEX array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is COMPLEX array, dimension (N) *> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the *> generalized eigenvalues. *> *> Note: the quotients 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). *> \endverbatim *> *> \param[out] VL *> \verbatim *> VL is 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 is scaled so the largest component has *> abs(real part) + abs(imag. part) = 1. *> Not referenced if JOBVL = 'N'. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of the matrix VL. LDVL >= 1, and *> if JOBVL = 'V', LDVL >= N. *> \endverbatim *> *> \param[out] VR *> \verbatim *> VR is 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 is scaled so the largest component has *> abs(real part) + abs(imag. part) = 1. *> Not referenced if JOBVR = 'N'. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the matrix VR. LDVR >= 1, and *> if JOBVR = 'V', LDVR >= N. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> *> 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. *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is REAL array, dimension (8*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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 then QZ iteration failed in CHGEQZ, *> =N+2: error return from CTGEVC. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complexGEeigen * * ===================================================================== SUBROUTINE CGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) * * -- LAPACK driver routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. REAL RWORK( * ) COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), $ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ), $ CONE = ( 1.0E0, 0.0E0 ) ) * .. * .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO, $ IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR, $ LWKOPT REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, $ SMLNUM, TEMP COMPLEX X * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHD3, CLAQZ0, CLACPY, $ CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, SLABAD, $ XERBLA * .. * .. External Functions .. LOGICAL LSAME REAL CLANGE, SLAMCH EXTERNAL LSAME, 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 * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -11 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -13 ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN INFO = -15 END IF * * Compute workspace * IF( INFO.EQ.0 ) THEN CALL CGEQRF( N, N, B, LDB, WORK, WORK, -1, IERR ) LWKOPT = MAX( N, N+INT( WORK( 1 ) ) ) CALL CUNMQR( 'L', 'C', N, N, N, B, LDB, WORK, A, LDA, WORK, $ -1, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) IF( ILVL ) THEN CALL CUNGQR( N, N, N, VL, LDVL, WORK, WORK, -1, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) END IF IF( ILV ) THEN CALL CGGHD3( JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, WORK, -1, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) CALL CLAQZ0( 'S', JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB, $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, -1, $ RWORK, 0, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) ELSE CALL CGGHD3( 'N', 'N', N, 1, N, A, LDA, B, LDB, VL, LDVL, $ VR, LDVR, WORK, -1, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) CALL CLAQZ0( 'E', JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB, $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, -1, $ RWORK, 0, IERR ) LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) ) END IF WORK( 1 ) = CMPLX( LWKOPT ) END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGGEV3 ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Get machine constants * EPS = SLAMCH( 'E' )*SLAMCH( 'B' ) 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 the matrices A, B to isolate eigenvalues if possible * ILEFT = 1 IRIGHT = N + 1 IRWRK = IRIGHT + N CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), RWORK( IRWRK ), IERR ) * * Reduce B to triangular form (QR decomposition of B) * IROWS = IHI + 1 - ILO IF( ILV ) 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 orthogonal transformation to matrix A * 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 * IF( ILVL ) THEN CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL ) IF( IROWS.GT.1 ) THEN CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, $ VL( ILO+1, ILO ), LDVL ) END IF CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Initialize VR * IF( ILVR ) $ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * IF( ILV ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL CGGHD3( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, WORK( IWRK ), LWORK+1-IWRK, $ IERR ) ELSE CALL CGGHD3( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, $ WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Perform QZ algorithm (Compute eigenvalues, and optionally, the * Schur form and Schur vectors) * IWRK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL CLAQZ0( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ), $ LWORK+1-IWRK, RWORK( IRWRK ), 0, 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 70 END IF * * Compute Eigenvectors * 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( IRWRK ), $ IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 70 END IF * * Undo balancing on VL and VR and normalization * IF( ILVL ) THEN CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VL, LDVL, IERR ) DO 30 JC = 1, N TEMP = ZERO DO 10 JR = 1, N TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) ) 10 CONTINUE IF( TEMP.LT.SMLNUM ) $ GO TO 30 TEMP = ONE / TEMP DO 20 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 20 CONTINUE 30 CONTINUE END IF IF( ILVR ) THEN CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VR, LDVR, IERR ) DO 60 JC = 1, N TEMP = ZERO DO 40 JR = 1, N TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) ) 40 CONTINUE IF( TEMP.LT.SMLNUM ) $ GO TO 60 TEMP = ONE / TEMP DO 50 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 50 CONTINUE 60 CONTINUE END IF END IF * * Undo scaling if necessary * 70 CONTINUE * 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 ) * WORK( 1 ) = CMPLX( LWKOPT ) RETURN * * End of CGGEV3 * END