cgges - 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)
SUBROUTINE CGGES(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER*1 JOBVSL, JOBVSR, SORT COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*) INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL SELCTG LOGICAL BWORK(*) REAL RWORK(*) SUBROUTINE CGGES_64(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER*1 JOBVSL, JOBVSR, SORT COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*) INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL*8 SELCTG LOGICAL*8 BWORK(*) REAL RWORK(*) F95 INTERFACE SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL :: SELCTG LOGICAL, DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: RWORK SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL(8) :: SELCTG LOGICAL(8), DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: RWORK C INTERFACE #include <sunperf.h> void cgges(char jobvsl, char jobvsr, char sort, int(*selctg)(com- plex,complex), int n, complex *a, int lda, complex *b, int ldb, int *sdim, complex *alpha, complex *beta, complex *vsl, int ldvsl, complex *vsr, int ldvsr, int *info); void cgges_64(char jobvsl, char jobvsr, char sort, long(*selctg)(com- plex,complex), long n, complex *a, long lda, complex *b, long ldb, long *sdim, complex *alpha, complex *beta, complex *vsl, long ldvsl, complex *vsr, long ldvsr, long *info);
Oracle Solaris Studio Performance Library cgges(3P) NAME cgges - compute 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) SYNOPSIS SUBROUTINE CGGES(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER*1 JOBVSL, JOBVSR, SORT COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*) INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL SELCTG LOGICAL BWORK(*) REAL RWORK(*) SUBROUTINE CGGES_64(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER*1 JOBVSL, JOBVSR, SORT COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*) INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL*8 SELCTG LOGICAL*8 BWORK(*) REAL RWORK(*) F95 INTERFACE SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL :: SELCTG LOGICAL, DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: RWORK SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO LOGICAL(8) :: SELCTG LOGICAL(8), DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: RWORK C INTERFACE #include <sunperf.h> void cgges(char jobvsl, char jobvsr, char sort, int(*selctg)(com- plex,complex), int n, complex *a, int lda, complex *b, int ldb, int *sdim, complex *alpha, complex *beta, complex *vsl, int ldvsl, complex *vsr, int ldvsr, int *info); void cgges_64(char jobvsl, char jobvsr, char sort, long(*selctg)(com- plex,complex), long n, complex *a, long lda, complex *b, long ldb, long *sdim, complex *alpha, complex *beta, complex *vsl, long ldvsl, complex *vsr, long ldvsr, long *info); 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 tri- angular 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 inter- pretation 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) = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors. JOBVSR (input) = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors. SORT (input) 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 (input) 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 (espe- cially if the eigenvalue is ill-conditioned), in this case INFO is set to N+2 (See INFO below). N (input) The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. LDA (input) The leading dimension of A. LDA >= max(1,N). B (input/output) On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. LDB (input) The leading dimension of B. LDB >= max(1,N). SDIM (output) If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei- genvalues (after sorting) for which SELCTG is true. ALPHA (output) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the general- ized 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. How- ever, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usu- ally comparable with norm(B). BETA (output) See description of ALPHA. VSL (output) If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. LDVSL (input) The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N. VSR (output) If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. LDVSR (input) The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) 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) dimension(8*N) BWORK (workspace) dimension(N) Not referenced if SORT = 'N'. INFO (output) = 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 com- plex eigenvalues so that leading eigenvalues in the General- ized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling. =N+3: reordering falied in CTGSEN. 7 Nov 2015 cgges(3P)