*> \brief SGGEVX 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 SGGEVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, * ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, * IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, * RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) * * .. 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ), * \$ B( LDB, * ), BETA( * ), LSCALE( * ), * \$ RCONDE( * ), RCONDV( * ), RSCALE( * ), * \$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGGEVX 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. *> *> Optionally also, it 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). *> *> \endverbatim * * Arguments: * ========== * *> \param[in] BALANC *> \verbatim *> BALANC is 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. *> \endverbatim *> *> \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] SENSE *> \verbatim *> SENSE is 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. *> \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 REAL 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 real Schur *> form of the "balanced" versions of the input A and B. *> \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 REAL 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 real Schur *> form of the "balanced" versions of the input A and B. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] ALPHAR *> \verbatim *> ALPHAR is REAL array, dimension (N) *> \endverbatim *> *> \param[out] ALPHAI *> \verbatim *> ALPHAI is REAL array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is REAL 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 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 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 will be scaled so the largest component have *> 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 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 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 will be scaled so the largest component have *> 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] ILO *> \verbatim *> ILO is INTEGER *> \endverbatim *> *> \param[out] IHI *> \verbatim *> IHI is 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. *> \endverbatim *> *> \param[out] LSCALE *> \verbatim *> LSCALE is 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. *> \endverbatim *> *> \param[out] RSCALE *> \verbatim *> RSCALE is 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. *> \endverbatim *> *> \param[out] ABNRM *> \verbatim *> ABNRM is REAL *> The one-norm of the balanced matrix A. *> \endverbatim *> *> \param[out] BBNRM *> \verbatim *> BBNRM is REAL *> The one-norm of the balanced matrix B. *> \endverbatim *> *> \param[out] RCONDE *> \verbatim *> RCONDE is REAL array, dimension (N) *> If SENSE = 'E' or 'B', the reciprocal condition numbers of *> the eigenvalues, stored in consecutive elements of the array. *> For a complex conjugate pair of eigenvalues two consecutive *> elements of RCONDE are set to the same value. Thus RCONDE(j), *> RCONDV(j), and the j-th columns of VL and VR all correspond *> to the j-th eigenpair. *> If SENSE = 'N' or 'V', RCONDE is not referenced. *> \endverbatim *> *> \param[out] RCONDV *> \verbatim *> RCONDV is REAL array, dimension (N) *> If SENSE = 'V' or 'B', the estimated reciprocal condition *> numbers of the eigenvectors, stored in consecutive elements *> of the array. For a complex eigenvector two consecutive *> elements of RCONDV are set to the same value. 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 = 'N' or 'E', RCONDV is not referenced. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL 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(1,2*N). *> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', *> LWORK >= max(1,6*N). *> If SENSE = 'E', LWORK >= max(1,10*N). *> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. *> *> 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] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N+6) *> If SENSE = 'E', IWORK is not referenced. *> \endverbatim *> *> \param[out] BWORK *> \verbatim *> BWORK is LOGICAL array, dimension (N) *> If SENSE = 'N', BWORK is not referenced. *> \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 SHGEQZ. *> =N+2: error return from STGEVC. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date April 2012 * *> \ingroup realGEeigen * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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. *> \endverbatim *> * ===================================================================== SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, \$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, \$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, \$ RCONDV, WORK, LWORK, IWORK, BWORK, 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 BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N REAL ABNRM, BBNRM * .. * .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), \$ B( LDB, * ), BETA( * ), LSCALE( * ), \$ RCONDE( * ), RCONDV( * ), RSCALE( * ), \$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL, \$ PAIR, WANTSB, WANTSE, WANTSN, WANTSV CHARACTER CHTEMP INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS, \$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, \$ MINWRK, MM REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, \$ SMLNUM, TEMP * .. * .. Local Arrays .. LOGICAL LDUMMA( 1 ) * .. * .. External Subroutines .. EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD, \$ SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGEVC, \$ STGSNA, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL SLAMCH, SLANGE EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE * .. * .. 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 * NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' ) 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.( NOSCL .OR. LSAME( BALANC, 'S' ) .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 = -14 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -16 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 IF( N.EQ.0 ) THEN MINWRK = 1 MAXWRK = 1 ELSE IF( NOSCL .AND. .NOT.ILV ) THEN MINWRK = 2*N ELSE MINWRK = 6*N END IF IF( WANTSE ) THEN MINWRK = 10*N ELSE IF( WANTSV .OR. WANTSB ) THEN MINWRK = 2*N*( N + 4 ) + 16 END IF MAXWRK = MINWRK MAXWRK = MAX( MAXWRK, \$ N + N*ILAENV( 1, 'SGEQRF', ' ', N, 1, N, 0 ) ) MAXWRK = MAX( MAXWRK, \$ N + N*ILAENV( 1, 'SORMQR', ' ', N, 1, N, 0 ) ) IF( ILVL ) THEN MAXWRK = MAX( MAXWRK, N + \$ N*ILAENV( 1, 'SORGQR', ' ', N, 1, N, 0 ) ) END IF END IF WORK( 1 ) = MAXWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -26 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGGEVX', -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 = SLANGE( '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 SLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = SLANGE( '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 SLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) * * Permute and/or balance the matrix pair (A,B) * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise) * CALL SGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, \$ WORK, IERR ) * * Compute ABNRM and BBNRM * ABNRM = SLANGE( '1', N, N, A, LDA, WORK( 1 ) ) IF( ILASCL ) THEN WORK( 1 ) = ABNRM CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1, \$ IERR ) ABNRM = WORK( 1 ) END IF * BBNRM = SLANGE( '1', N, N, B, LDB, WORK( 1 ) ) IF( ILBSCL ) THEN WORK( 1 ) = BBNRM CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1, \$ IERR ) BBNRM = WORK( 1 ) END IF * * Reduce B to triangular form (QR decomposition of B) * (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 SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), \$ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the orthogonal transformation to A * (Workspace: need N, prefer N*NB) * CALL SORMQR( 'L', 'T', 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 SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) IF( IROWS.GT.1 ) THEN CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, \$ VL( ILO+1, ILO ), LDVL ) END IF CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, \$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * IF( ILVR ) \$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) * * Reduce to generalized Hessenberg form * (Workspace: none needed) * IF( ILV .OR. .NOT.WANTSN ) THEN * * Eigenvectors requested -- work on whole matrix. * CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, IERR ) ELSE CALL SGGHRD( '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) * IF( ILV .OR. .NOT.WANTSN ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF * CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, \$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, \$ LWORK, 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 130 END IF * * Compute Eigenvectors and estimate condition numbers if desired * (Workspace: STGEVC: need 6*N * STGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B', * need N otherwise ) * 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 STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, N, IN, WORK, IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 130 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 recalculate * eigenvectors and estimate one condition numbers at a time. * PAIR = .FALSE. DO 20 I = 1, N * IF( PAIR ) THEN PAIR = .FALSE. GO TO 20 END IF MM = 1 IF( I.LT.N ) THEN IF( A( I+1, I ).NE.ZERO ) THEN PAIR = .TRUE. MM = 2 END IF END IF * DO 10 J = 1, N BWORK( J ) = .FALSE. 10 CONTINUE IF( MM.EQ.1 ) THEN BWORK( I ) = .TRUE. ELSE IF( MM.EQ.2 ) THEN BWORK( I ) = .TRUE. BWORK( I+1 ) = .TRUE. END IF * IWRK = MM*N + 1 IWRK1 = IWRK + MM*N * * Compute a pair of left and right eigenvectors. * (compute workspace: need up to 4*N + 6*N) * IF( WANTSE .OR. WANTSB ) THEN CALL STGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB, \$ WORK( 1 ), N, WORK( IWRK ), N, MM, M, \$ WORK( IWRK1 ), IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 130 END IF END IF * CALL STGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB, \$ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ), \$ RCONDV( I ), MM, 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 SGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL, \$ LDVL, IERR ) * DO 70 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 70 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 30 CONTINUE ELSE DO 40 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ \$ ABS( VL( JR, JC+1 ) ) ) 40 CONTINUE END IF IF( TEMP.LT.SMLNUM ) \$ GO TO 70 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 50 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 50 CONTINUE ELSE DO 60 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 60 CONTINUE END IF 70 CONTINUE END IF IF( ILVR ) THEN CALL SGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR, \$ LDVR, IERR ) DO 120 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) \$ GO TO 120 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 80 CONTINUE ELSE DO 90 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ \$ ABS( VR( JR, JC+1 ) ) ) 90 CONTINUE END IF IF( TEMP.LT.SMLNUM ) \$ GO TO 120 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 100 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 100 CONTINUE ELSE DO 110 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 110 CONTINUE END IF 120 CONTINUE END IF * * Undo scaling if necessary * 130 CONTINUE * IF( ILASCL ) THEN CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF * IF( ILBSCL ) THEN CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF * WORK( 1 ) = MAXWRK RETURN * * End of SGGEVX * END