NAME

SYNOPSIS

  SUBROUTINE CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, 
 *      VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL WORK2(*)
 
  SUBROUTINE CGEGS_64( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, 
 *      VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER*8 N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL WORK2(*)
 

F95 INTERFACE

  SUBROUTINE GEGS( JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHA, 
 *       BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [WORK2], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOBVSL, JOBVSR
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
  INTEGER :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL, DIMENSION(:) :: WORK2
 
  SUBROUTINE GEGS_64( JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHA, 
 *       BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [WORK2], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOBVSL, JOBVSR
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
  INTEGER(8) :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL, DIMENSION(:) :: WORK2
 

C INTERFACE

void cgegs(char jobvsl, char jobvsr, int n, complex *a, int lda, complex *b, int ldb, complex *alpha, complex *beta, complex *vsl, int ldvsl, complex *vsr, int ldvsr, int *info);

void cgegs_64(char jobvsl, char jobvsr, long n, complex *a, long lda, complex *b, long ldb, complex *alpha, complex *beta, complex *vsl, long ldvsl, complex *vsr, long ldvsr, long *info);

PURPOSE

CGEGS computes for a pair of N-by-N complex nonsymmetric matrices A, B: the generalized eigenvalues (alpha, beta), the complex Schur form (A, B), and optionally left and/or right Schur vectors (VSL and VSR).

(If only the generalized eigenvalues are needed, use the driver CGEGV instead.)

A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, 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 good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press)

The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one unitary matrix and both on the right by another unitary matrix, these two unitary matrices being chosen so as to bring the pair of matrices into upper triangular form with the diagonal elements of B being non-negative real numbers (this is also called complex Schur form.)

The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the unitary matrices

which reduce A and B to Schur form:

Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) )

ARGUMENTS

* JOBVSL (input)

* JOBVSR (input)

* 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 whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of A.

* LDA (input)

The leading dimension of A. LDA >= max(1,N).

* B (input/output)

On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) Schur vectors are to be computed. On exit, the generalized Schur form of B.

* LDB (input)

The leading dimension of B. LDB >= max(1,N).

* ALPHA (output)

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 CGEGS. 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).

* BETA (output)

See the description of ALPHA.

* VSL (input)

If JOBVSL = 'V', VSL will contain the left Schur vectors. (See ``Purpose'', above.) Not referenced if JOBVSL = 'N'.

* LDVSL (input)

The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N.

* VSR (input)

If JOBVSR = 'V', VSR will contain the right Schur vectors. (See ``Purpose'', above.) 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 LDWORK.

* LDWORK (input)

The dimension of the array WORK. LDWORK >= max(1,2*N). For good performance, LDWORK must generally be larger. To compute the optimal value of LDWORK, call ILAENV to get blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute: NB as the MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR; the optimal LDWORK is N*(NB+1).

If LDWORK = -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 LDWORK is issued by XERBLA.

* WORK2 (workspace)

* INFO (output)