/lapack/double/dggevx.f
FORTRAN Legacy | 718 lines | 320 code | 0 blank | 398 comment | 0 complexity | bc76a1ae782b553ed7cee0f8de7911cb MD5 | raw file
Possible License(s): CC-BY-SA-3.0
- 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 3.1) --
- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
- * November 2006
- *
- * .. 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( * )
- * ..
- *
- * 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 ..
- DOUBLE PRECISION ZERO, ONE
- PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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
- 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,
- $ DTGSNA, 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
- *
- 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.( 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 = -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 .OR. WANTSB ) THEN
- MINWRK = 10*N
- END IF
- IF( WANTSV .OR. WANTSB ) THEN
- MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
- END IF
- MAXWRK = MINWRK
- MAXWRK = MAX( MAXWRK,
- $ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
- MAXWRK = MAX( MAXWRK,
- $ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
- IF( ILVL ) THEN
- MAXWRK = MAX( MAXWRK, N +
- $ N*ILAENV( 1, 'DORGQR', ' ', 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( 'DGGEVX', -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 and/or balance the matrix pair (A,B)
- * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
- *
- CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
- $ WORK, IERR )
- *
- * Compute ABNRM and BBNRM
- *
- ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
- IF( ILASCL ) THEN
- WORK( 1 ) = ABNRM
- CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
- $ IERR )
- ABNRM = WORK( 1 )
- END IF
- *
- BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
- IF( ILBSCL ) THEN
- WORK( 1 ) = BBNRM
- CALL DLASCL( '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 DGEQRF( 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 DORMQR( '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 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
- *
- IF( ILVR )
- $ CALL DLASET( '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 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)
- *
- IF( ILV .OR. .NOT.WANTSN ) 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,
- $ 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: DTGEVC: need 6*N
- * DTGSNA: 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 DTGEVC( 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 (DTGEVC) and estimate condition
- * numbers (DTGSNA). 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 DTGEVC( '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 DTGSNA( 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 DGGBAK( 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 DGGBAK( 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
- *
- 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
- *
- 130 CONTINUE
- WORK( 1 ) = MAXWRK
- *
- RETURN
- *
- * End of DGGEVX
- *
- END