SUBROUTINE CGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, $ SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, $ LWORK, RWORK, BWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM * .. * .. Array Arguments .. LOGICAL BWORK( * ) REAL RWORK( * ) COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), $ WORK( * ) * .. * .. Function Arguments .. LOGICAL SELCTG EXTERNAL SELCTG * .. * * Purpose * ======= * * CGGES computes for a pair of N-by-N complex nonsymmetric matrices * (A,B), the generalized eigenvalues, the generalized complex Schur * form (S, T), and optionally left and/or right 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. The leading * columns of VSL and VSR then form an unitary basis for the * corresponding left and right eigenspaces (deflating subspaces). * * (If only the generalized eigenvalues are needed, use the driver * CGGEV instead, which is faster.) * * 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, and even for both being zero. * * A pair of matrices (S,T) is in generalized complex Schur form if S * and T are upper triangular and, in addition, the diagonal elements * of T are non-negative real numbers. * * 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 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. * An eigenvalue ALPHA(j)/BETA(j) is selected if * SELCTG(ALPHA(j),BETA(j)) is true. * * 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+2 (See INFO below). * * N (input) INTEGER * The order of the matrices A, B, VSL, and VSR. N >= 0. * * A (input/output) COMPLEX 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 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 array, dimension (N) * BETA (output) COMPLEX array, dimension (N) * On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the * generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), * j=1,...,N are the diagonals of the complex Schur form (A,B) * output by CGGES. The 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 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 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. * * WORK (workspace/output) COMPLEX 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). * 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. * * RWORK (workspace) REAL array, dimension (8*N) * * 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 CHGEQZ * =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 falied in CTGSEN. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ), $ CONE = ( 1.0E0, 0.0E0 ) ) * .. * .. Local Scalars .. LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL, $ LQUERY, WANTST INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, $ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKMIN, $ LWKOPT REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL, $ PVSR, SMLNUM * .. * .. Local Arrays .. INTEGER IDUM( 1 ) REAL DIF( 2 ) * .. * .. External Subroutines .. EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY, $ CLASCL, CLASET, CTGSEN, CUNGQR, CUNMQR, SLABAD, $ XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL CLANGE, SLAMCH EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH * .. * .. 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' ) * * 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( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -3 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( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -14 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.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.) * IF( INFO.EQ.0 ) THEN LWKMIN = MAX( 1, 2*N ) LWKOPT = MAX( 1, N + N*ILAENV( 1, 'CGEQRF', ' ', N, 1, N, 0 ) ) LWKOPT = MAX( LWKOPT, N + $ N*ILAENV( 1, 'CUNMQR', ' ', N, 1, N, -1 ) ) IF( ILVSL ) THEN LWKOPT = MAX( LWKOPT, N + $ N*ILAENV( 1, 'CUNGQR', ' ', N, 1, N, -1 ) ) END IF WORK( 1 ) = LWKOPT * IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) $ INFO = -18 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGGES ', -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 = 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 = CLANGE( '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 CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = CLANGE( '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 CLASCL( '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 CGGBAL( '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 CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), $ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the orthogonal transformation to matrix A * (Complex Workspace: need N, prefer N*NB) * CALL CUNMQR( '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 CLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL ) IF( IROWS.GT.1 ) THEN CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, $ VSL( ILO+1, ILO ), LDVSL ) END IF CALL CUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Initialize VSR * IF( ILVSR ) $ CALL CLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR ) * * Reduce to generalized Hessenberg form * (Workspace: none needed) * CALL CGGHRD( 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 CHGEQZ( '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 30 END IF * * Sort eigenvalues ALPHA/BETA if desired * (Workspace: none needed) * IF( WANTST ) THEN * * Undo scaling on eigenvalues before selecting * IF( ILASCL ) $ CALL CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, 1, ALPHA, N, IERR ) IF( ILBSCL ) $ CALL CLASCL( 'G', 0, 0, BNRM, BNRMTO, N, 1, BETA, N, IERR ) * * Select eigenvalues * DO 10 I = 1, N BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) ) 10 CONTINUE * CALL CTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHA, $ BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL, PVSR, $ DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1, IERR ) IF( IERR.EQ.1 ) $ INFO = N + 3 * END IF * * Apply back-permutation to VSL and VSR * (Workspace: none needed) * IF( ILVSL ) $ CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VSL, LDVSL, IERR ) IF( ILVSR ) $ CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), $ RWORK( IRIGHT ), N, VSR, LDVSR, IERR ) * * Undo scaling * IF( ILASCL ) THEN CALL CLASCL( 'U', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR ) CALL CLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) END IF * IF( ILBSCL ) THEN CALL CLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR ) CALL CLASCL( '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 20 I = 1, N CURSL = SELCTG( ALPHA( I ), BETA( I ) ) IF( CURSL ) $ SDIM = SDIM + 1 IF( CURSL .AND. .NOT.LASTSL ) $ INFO = N + 2 LASTSL = CURSL 20 CONTINUE * END IF * 30 CONTINUE * WORK( 1 ) = LWKOPT * RETURN * * End of CGGES * END