NAME

SYNOPSIS

  SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, 
 *      LDVL, VR, LDVR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 JOBVL, JOBVR
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
  INTEGER N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  REAL WORK2(*)
 
  SUBROUTINE CGEGV_64( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, 
 *      VL, LDVL, VR, LDVR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 JOBVL, JOBVR
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
  INTEGER*8 N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  REAL WORK2(*)
 

F95 INTERFACE

  SUBROUTINE GEGV( JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHA, BETA, 
 *       VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: JOBVL, JOBVR
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VL, VR
  INTEGER :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  REAL, DIMENSION(:) :: WORK2
 
  SUBROUTINE GEGV_64( JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHA, 
 *       BETA, VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: JOBVL, JOBVR
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VL, VR
  INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  REAL, DIMENSION(:) :: WORK2
 

C INTERFACE

void cgegv(char jobvl, char jobvr, int n, complex *a, int lda, complex *b, int ldb, complex *alpha, complex *beta, complex *vl, int ldvl, complex *vr, int ldvr, int *info);

void cgegv_64(char jobvl, char jobvr, long n, complex *a, long lda, complex *b, long ldb, complex *alpha, complex *beta, complex *vl, long ldvl, complex *vr, long ldvr, long *info);

PURPOSE

CGEGV computes for a pair of N-by-N complex nonsymmetric matrices A and B, the generalized eigenvalues (alpha, beta), and optionally, the left and/or right generalized eigenvectors (VL and VR).

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)

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - w B) r = 0 . A left generalized eigenvector is a vector l such that l**H * (A - w B) = 0, where l**H is the

conjugate-transpose of l.

Note: this routine performs ``full balancing'' on A and B. See ``Further Details'', below.

ARGUMENTS

* JOBVL (input)

* JOBVR (input)

* N (input)

The order of the matrices A, B, VL, and VR. N >= 0.

* A (input/output)

On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed. On exit, the contents will have been destroyed. (For a description of the contents of A on exit, see "Further Details", below.)

* 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) generalized eigenvectors are to be computed. On exit, the contents will have been destroyed. (For a description of the contents of B on exit, see "Further Details", below.)

* LDB (input)

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

* ALPHA (output)

On exit, ALPHA(j)/VL(j), j=1,...,N, will be the generalized eigenvalues.

Note: the quotients ALPHA(j)/VL(j) may easily over- or underflow, and VL(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 VL always less than and usually comparable with norm(B).

* BETA (output)

If JOBVL = 'V', the left generalized eigenvectors. (See ``Purpose'', above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVL = 'N'.

* VL (output)

If JOBVL = 'V', the left generalized eigenvectors. (See ``Purpose'', above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVL = 'N'.

* LDVL (input)

The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.

* VR (output)

If JOBVR = 'V', the right generalized eigenvectors. (See ``Purpose'', above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVR = 'N'.

* LDVR (input)

The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= 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 MAX( 2*N, 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)