*> \brief DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGGEV + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, * BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVL, JOBVR * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), * \$ B( LDB, * ), BETA( * ), VL( LDVL, * ), * \$ VR( LDVR, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGGEV computes for a pair of N-by-N real 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 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). *> *> \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 DOUBLE PRECISION 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 DOUBLE PRECISION 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] ALPHAR *> \verbatim *> ALPHAR is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] ALPHAI *> \verbatim *> ALPHAI is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is DOUBLE PRECISION array, dimension (N) *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will *> be the generalized eigenvalues. If ALPHAI(j) is zero, then *> the j-th eigenvalue is real; if positive, then the j-th and *> (j+1)-st eigenvalues are a complex conjugate pair, with *> ALPHAI(j+1) negative. *> *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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 DOUBLE PRECISION 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 the j-th eigenvalue is real, then *> u(j) = VL(:,j), the j-th column of VL. If the j-th and *> (j+1)-th 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). *> 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 DOUBLE PRECISION 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 the j-th eigenvalue is real, then *> v(j) = VR(:,j), the j-th column of VR. If the j-th and *> (j+1)-th 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). *> 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 DOUBLE PRECISION 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. LWORK >= max(1,8*N). *> For good performance, LWORK must generally be larger. *> *> 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] 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 ALPHAR(j), ALPHAI(j), and BETA(j) *> should be correct for j=INFO+1,...,N. *> > N: =N+1: other than QZ iteration failed in DHGEQZ. *> =N+2: error return from DTGEVC. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date April 2012 * *> \ingroup doubleGEeigen * * ===================================================================== SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, \$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * April 2012 * * .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), \$ B( LDB, * ), BETA( * ), VL( LDVL, * ), \$ VR( LDVR, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO, \$ IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK, \$ MINWRK DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, \$ SMLNUM, TEMP * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD, \$ DLACPY,DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, \$ XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT * .. * .. 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 = -12 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -14 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.) * IF( INFO.EQ.0 ) THEN MINWRK = MAX( 1, 8*N ) MAXWRK = MAX( 1, N*( 7 + \$ ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) ) MAXWRK = MAX( MAXWRK, N*( 7 + \$ ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) ) IF( ILVL ) THEN MAXWRK = MAX( MAXWRK, N*( 7 + \$ ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) ) END IF WORK( 1 ) = MAXWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) \$ INFO = -16 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGEV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Get machine constants * EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) 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 DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) 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 DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) * * Permute the matrices A, B to isolate eigenvalues if possible * (Workspace: need 6*N) * ILEFT = 1 IRIGHT = N + 1 IWRK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), WORK( IWRK ), IERR ) * * Reduce B to triangular form (QR decomposition of B) * (Workspace: need N, prefer N*NB) * IROWS = IHI + 1 - ILO IF( ILV ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = IWRK IWRK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), \$ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the orthogonal transformation to matrix A * (Workspace: need N, prefer N*NB) * CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, \$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), \$ LWORK+1-IWRK, IERR ) * * Initialize VL * (Workspace: need N, prefer N*NB) * IF( ILVL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) IF( IROWS.GT.1 ) THEN CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, \$ VL( ILO+1, ILO ), LDVL ) END IF CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, \$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Initialize VR * IF( ILVR ) \$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * (Workspace: none needed) * IF( ILV ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, IERR ) ELSE CALL DGGHRD( '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) * (Workspace: need N) * IWRK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, \$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, \$ WORK( IWRK ), LWORK+1-IWRK, 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 110 END IF * * Compute Eigenvectors * (Workspace: need 6*N) * IF( ILV ) THEN IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, \$ VR, LDVR, N, IN, WORK( IWRK ), IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 110 END IF * * Undo balancing on VL and VR and normalization * (Workspace: none needed) * IF( ILVL ) THEN CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), N, VL, LDVL, IERR ) DO 50 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 50 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 10 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 10 CONTINUE ELSE DO 20 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ \$ ABS( VL( JR, JC+1 ) ) ) 20 CONTINUE END IF IF( TEMP.LT.SMLNUM ) \$ GO TO 50 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 30 CONTINUE ELSE DO 40 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 40 CONTINUE END IF 50 CONTINUE END IF IF( ILVR ) THEN CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), \$ WORK( IRIGHT ), N, VR, LDVR, IERR ) DO 100 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 100 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 60 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 60 CONTINUE ELSE DO 70 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ \$ ABS( VR( JR, JC+1 ) ) ) 70 CONTINUE END IF IF( TEMP.LT.SMLNUM ) \$ GO TO 100 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 80 CONTINUE ELSE DO 90 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 90 CONTINUE END IF 100 CONTINUE END IF * * End of eigenvector calculation * END IF * * Undo scaling if necessary * 110 CONTINUE * IF( ILASCL ) THEN CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF * IF( ILBSCL ) THEN CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF * WORK( 1 ) = MAXWRK RETURN * * End of DGGEV * END