--- rpl/lapack/lapack/dggevx.f 2010/08/07 13:22:14 1.5 +++ rpl/lapack/lapack/dggevx.f 2012/12/14 14:22:30 1.12 @@ -1,12 +1,400 @@ +*> \brief DGGEVX 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 DGGEVX + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DGGEVX( 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 +* DOUBLE PRECISION ABNRM, BBNRM +* .. +* .. Array Arguments .. +* LOGICAL BWORK( * ) +* INTEGER IWORK( * ) +* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), +* $ B( LDB, * ), BETA( * ), LSCALE( * ), +* $ RCONDE( * ), RCONDV( * ), RSCALE( * ), +* $ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DGGEVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 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 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 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION +*> The one-norm of the balanced matrix A. +*> \endverbatim +*> +*> \param[out] BBNRM +*> \verbatim +*> BBNRM is DOUBLE PRECISION +*> The one-norm of the balanced matrix B. +*> \endverbatim +*> +*> \param[out] RCONDE +*> \verbatim +*> RCONDE is DOUBLE PRECISION 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 DOUBLE PRECISION 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 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(1,2*N). +*> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', +*> LWORK >= max(1,6*N). +*> If SENSE = 'E' or 'B', 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 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 +* +*> \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 DGGEVX( 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 3.2) -- +* -- LAPACK driver routine (version 3.4.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 +* April 2012 * * .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE @@ -22,245 +410,6 @@ $ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) * .. * -* Purpose -* ======= -* -* DGGEVX 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). -* -* -* 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) DOUBLE PRECISION 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. -* -* LDA (input) INTEGER -* The leading dimension of A. LDA >= max(1,N). -* -* B (input/output) DOUBLE PRECISION 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. -* -* LDB (input) INTEGER -* The leading dimension of B. LDB >= max(1,N). -* -* ALPHAR (output) DOUBLE PRECISION array, dimension (N) -* ALPHAI (output) DOUBLE PRECISION array, dimension (N) -* BETA (output) 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). -* -* VL (output) 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 will be scaled so the largest component 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) 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 will be scaled so the largest component 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 (output) INTEGER -* 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION -* The one-norm of the balanced matrix A. -* -* BBNRM (output) DOUBLE PRECISION -* The one-norm of the balanced matrix B. -* -* RCONDE (output) DOUBLE PRECISION 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. -* -* RCONDV (output) DOUBLE PRECISION 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. -* -* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', -* LWORK >= max(1,6*N). -* If SENSE = 'E' or 'B', 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. -* -* IWORK (workspace) INTEGER array, dimension (N+6) -* 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 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. -* -* 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 .. @@ -700,6 +849,8 @@ * * Undo scaling if necessary * + 130 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 ) @@ -709,9 +860,7 @@ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF * - 130 CONTINUE WORK( 1 ) = MAXWRK -* RETURN * * End of DGGEVX