SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, $ B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, $ LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK, $ IWORK, LIWORK, BWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SENSE, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, $ SDIM * .. * .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) DOUBLE PRECISION RCONDE( 2 ), RCONDV( 2 ), RWORK( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), $ WORK( * ) * .. * .. Function Arguments .. LOGICAL SELCTG EXTERNAL SELCTG * .. * * Purpose * ======= * * ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices * (A,B), the generalized eigenvalues, the complex Schur form (S,T), * and, optionally, the left and/or right matrices of Schur vectors (VSL * and VSR). This gives the generalized Schur factorization * * (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H ) * * where (VSR)**H is the conjugate-transpose of VSR. * * Optionally, it also orders the eigenvalues so that a selected cluster * of eigenvalues appears in the leading diagonal blocks of the upper * triangular matrix S and the upper triangular matrix T; computes * a reciprocal condition number for the average of the selected * eigenvalues (RCONDE); and computes a reciprocal condition number for * the right and left deflating subspaces corresponding to the selected * eigenvalues (RCONDV). The leading columns of VSL and VSR then form * an orthonormal basis for the corresponding left and right eigenspaces * (deflating subspaces). * * A generalized eigenvalue for a pair of matrices (A,B) is a scalar w * or a ratio alpha/beta = w, such that A - w*B is singular. It is * usually represented as the pair (alpha,beta), as there is a * reasonable interpretation for beta=0 or for both being zero. * * A pair of matrices (S,T) is in generalized complex Schur form if T is * upper triangular with non-negative diagonal and S is upper * triangular. * * Arguments * ========= * * JOBVSL (input) CHARACTER*1 * = 'N': do not compute the left Schur vectors; * = 'V': compute the left Schur vectors. * * JOBVSR (input) CHARACTER*1 * = 'N': do not compute the right Schur vectors; * = 'V': compute the right Schur vectors. * * SORT (input) CHARACTER*1 * Specifies whether or not to order the eigenvalues on the * diagonal of the generalized Schur form. * = 'N': Eigenvalues are not ordered; * = 'S': Eigenvalues are ordered (see SELCTG). * * SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX*16 arguments * SELCTG must be declared EXTERNAL in the calling subroutine. * If SORT = 'N', SELCTG is not referenced. * If SORT = 'S', SELCTG is used to select eigenvalues to sort * to the top left of the Schur form. * Note that a selected complex eigenvalue may no longer satisfy * SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since * ordering may change the value of complex eigenvalues * (especially if the eigenvalue is ill-conditioned), in this * case INFO is set to N+3 see INFO below). * * SENSE (input) CHARACTER*1 * Determines which reciprocal condition numbers are computed. * = 'N' : None are computed; * = 'E' : Computed for average of selected eigenvalues only; * = 'V' : Computed for selected deflating subspaces only; * = 'B' : Computed for both. * If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. * * N (input) INTEGER * The order of the matrices A, B, VSL, and VSR. N >= 0. * * A (input/output) COMPLEX*16 array, dimension (LDA, N) * On entry, the first of the pair of matrices. * On exit, A has been overwritten by its generalized Schur * form S. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) COMPLEX*16 array, dimension (LDB, N) * On entry, the second of the pair of matrices. * On exit, B has been overwritten by its generalized Schur * form T. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * SDIM (output) INTEGER * If SORT = 'N', SDIM = 0. * If SORT = 'S', SDIM = number of eigenvalues (after sorting) * for which SELCTG is true. * * ALPHA (output) COMPLEX*16 array, dimension (N) * BETA (output) COMPLEX*16 array, dimension (N) * On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the * generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are * the diagonals of the complex Schur form (S,T). BETA(j) will * be non-negative real. * * 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). * * VSL (output) COMPLEX*16 array, dimension (LDVSL,N) * If JOBVSL = 'V', VSL will contain the left Schur vectors. * Not referenced if JOBVSL = 'N'. * * LDVSL (input) INTEGER * The leading dimension of the matrix VSL. LDVSL >=1, and * if JOBVSL = 'V', LDVSL >= N. * * VSR (output) COMPLEX*16 array, dimension (LDVSR,N) * If JOBVSR = 'V', VSR will contain the right Schur vectors. * Not referenced if JOBVSR = 'N'. * * LDVSR (input) INTEGER * The leading dimension of the matrix VSR. LDVSR >= 1, and * if JOBVSR = 'V', LDVSR >= N. * * RCONDE (output) DOUBLE PRECISION array, dimension ( 2 ) * If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the * reciprocal condition numbers for the average of the selected * eigenvalues. * Not referenced if SENSE = 'N' or 'V'. * * RCONDV (output) DOUBLE PRECISION array, dimension ( 2 ) * If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the * reciprocal condition number for the selected deflating * subspaces. * Not referenced if SENSE = 'N' or 'E'. * * WORK (workspace/output) COMPLEX*16 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. * If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', * LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else * LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2. * Note also that an error is only returned if * LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may * not be large enough. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the bound on the optimal size of the WORK * array and the minimum size of the IWORK array, returns these * values as the first entries of the WORK and IWORK arrays, and * no error message related to LWORK or LIWORK is issued by * XERBLA. * * RWORK (workspace) DOUBLE PRECISION array, dimension ( 8*N ) * Real workspace. * * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. * If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise * LIWORK >= N+2. * * If LIWORK = -1, then a workspace query is assumed; the * routine only calculates the bound on the optimal size of the * WORK array and the minimum size of the IWORK array, returns * these values as the first entries of the WORK and IWORK * arrays, and no error message related to LWORK or LIWORK is * issued by XERBLA. * * BWORK (workspace) LOGICAL array, dimension (N) * Not referenced if SORT = 'N'. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * = 1,...,N: * The QZ iteration failed. (A,B) are not in Schur * form, but ALPHA(j) and BETA(j) should be correct for * j=INFO+1,...,N. * > N: =N+1: other than QZ iteration failed in ZHGEQZ * =N+2: after reordering, roundoff changed values of * some complex eigenvalues so that leading * eigenvalues in the Generalized Schur form no * longer satisfy SELCTG=.TRUE. This could also * be caused due to scaling. * =N+3: reordering failed in ZTGSEN. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL, $ LQUERY, WANTSB, WANTSE, WANTSN, WANTST, WANTSV INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR, $ ILEFT, ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, $ LIWMIN, LWRK, MAXWRK, MINWRK DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL, $ PR, SMLNUM * .. * .. Local Arrays .. DOUBLE PRECISION DIF( 2 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, $ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGSEN, ZUNGQR, $ ZUNMQR * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Decode the input arguments * IF( LSAME( JOBVSL, 'N' ) ) THEN IJOBVL = 1 ILVSL = .FALSE. ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN IJOBVL = 2 ILVSL = .TRUE. ELSE IJOBVL = -1 ILVSL = .FALSE. END IF * IF( LSAME( JOBVSR, 'N' ) ) THEN IJOBVR = 1 ILVSR = .FALSE. ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN IJOBVR = 2 ILVSR = .TRUE. ELSE IJOBVR = -1 ILVSR = .FALSE. END IF * WANTST = LSAME( SORT, 'S' ) WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) IF( WANTSN ) THEN IJOB = 0 ELSE IF( WANTSE ) THEN IJOB = 1 ELSE IF( WANTSV ) THEN IJOB = 2 ELSE IF( WANTSB ) THEN IJOB = 4 END IF * * Test the input arguments * INFO = 0 IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -3 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. $ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN INFO = -5 ELSE IF( N.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -15 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN INFO = -17 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.) * IF( INFO.EQ.0 ) THEN IF( N.GT.0) THEN MINWRK = 2*N MAXWRK = N*(1 + ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) ) MAXWRK = MAX( MAXWRK, N*( 1 + $ ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, -1 ) ) ) IF( ILVSL ) THEN MAXWRK = MAX( MAXWRK, N*( 1 + $ ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) ) ) END IF LWRK = MAXWRK IF( IJOB.GE.1 ) $ LWRK = MAX( LWRK, N*N/2 ) ELSE MINWRK = 1 MAXWRK = 1 LWRK = 1 END IF WORK( 1 ) = LWRK IF( WANTSN .OR. N.EQ.0 ) THEN LIWMIN = 1 ELSE LIWMIN = N + 2 END IF IWORK( 1 ) = LIWMIN * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -21 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY) THEN INFO = -24 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGGESX', -INFO ) RETURN ELSE IF (LQUERY) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF * * 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 = ZLANGE( '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 ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = ZLANGE( '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 ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) * * Permute the matrix to make it more nearly triangular * (Real Workspace: need 6*N) * ILEFT = 1 IRIGHT = N + 1 IRWRK = IRIGHT + N CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), RWORK( IRWRK ), IERR ) * * Reduce B to triangular form (QR decomposition of B) * (Complex Workspace: need N, prefer N*NB) * IROWS = IHI + 1 - ILO ICOLS = N + 1 - ILO ITAU = 1 IWRK = ITAU + IROWS CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), $ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the unitary transformation to matrix A * (Complex Workspace: need N, prefer N*NB) * CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), $ LWORK+1-IWRK, IERR ) * * Initialize VSL * (Complex Workspace: need N, prefer N*NB) * IF( ILVSL ) THEN CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL ) IF( IROWS.GT.1 ) THEN CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, $ VSL( ILO+1, ILO ), LDVSL ) END IF CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Initialize VSR * IF( ILVSR ) $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR ) * * Reduce to generalized Hessenberg form * (Workspace: none needed) * CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, $ LDVSL, VSR, LDVSR, IERR ) * SDIM = 0 * * Perform QZ algorithm, computing Schur vectors if desired * (Complex Workspace: need N) * (Real Workspace: need N) * IWRK = ITAU CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, $ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWRK ), $ LWORK+1-IWRK, RWORK( IRWRK ), 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 40 END IF * * Sort eigenvalues ALPHA/BETA and compute the reciprocal of * condition number(s) * IF( WANTST ) THEN * * Undo scaling on eigenvalues before SELCTGing * IF( ILASCL ) $ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) IF( ILBSCL ) $ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) * * Select eigenvalues * DO 10 I = 1, N BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) ) 10 CONTINUE * * Reorder eigenvalues, transform Generalized Schur vectors, and * compute reciprocal condition numbers * (Complex Workspace: If IJOB >= 1, need MAX(1, 2*SDIM*(N-SDIM)) * otherwise, need 1 ) * CALL ZTGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, $ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PL, PR, $ DIF, WORK( IWRK ), LWORK-IWRK+1, IWORK, LIWORK, $ IERR ) * IF( IJOB.GE.1 ) $ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) ) IF( IERR.EQ.-21 ) THEN * * not enough complex workspace * INFO = -21 ELSE IF( IJOB.EQ.1 .OR. IJOB.EQ.4 ) THEN RCONDE( 1 ) = PL RCONDE( 2 ) = PR END IF IF( IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN RCONDV( 1 ) = DIF( 1 ) RCONDV( 2 ) = DIF( 2 ) END IF IF( IERR.EQ.1 ) $ INFO = N + 3 END IF * END IF * * Apply permutation to VSL and VSR * (Workspace: none needed) * IF( ILVSL ) $ CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VSL, LDVSL, IERR ) * IF( ILVSR ) $ CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VSR, LDVSR, IERR ) * * Undo scaling * IF( ILASCL ) THEN CALL ZLASCL( 'U', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR ) CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) END IF * IF( ILBSCL ) THEN CALL ZLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR ) CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF * IF( WANTST ) THEN * * Check if reordering is correct * LASTSL = .TRUE. SDIM = 0 DO 30 I = 1, N CURSL = SELCTG( ALPHA( I ), BETA( I ) ) IF( CURSL ) $ SDIM = SDIM + 1 IF( CURSL .AND. .NOT.LASTSL ) $ INFO = N + 2 LASTSL = CURSL 30 CONTINUE * END IF * 40 CONTINUE * WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWMIN * RETURN * * End of ZGGESX * END