NAME

SYNOPSIS

  SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, 
 *      BETA, VL, LDVL, VR, LDVR, WORK, LDWORK, INFO)
  CHARACTER * 1 JOBVL, JOBVR
  INTEGER N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
 
  SUBROUTINE DGEGV_64( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, 
 *      ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LDWORK, INFO)
  CHARACTER * 1 JOBVL, JOBVR
  INTEGER*8 N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
  DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
 

F95 INTERFACE

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

C INTERFACE

void dgegv(char jobvl, char jobvr, int n, double *a, int lda, double *b, int ldb, double *alphar, double *alphai, double *beta, double *vl, int ldvl, double *vr, int ldvr, int *info);

void dgegv_64(char jobvl, char jobvr, long n, double *a, long lda, double *b, long ldb, double *alphar, double *alphai, double *beta, double *vl, long ldvl, double *vr, long ldvr, long *info);

PURPOSE

SGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B, the generalized eigenvalues (alphar +/- alphai*i, 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).

* ALPHAR (output)

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

* ALPHAI (output)

See the description of ALPHAR.

* BETA (output)

See the description of ALPHAR.

* VL (output)

If JOBVL = 'V', the left generalized eigenvectors. (See ``Purpose'', above.) Real eigenvectors take one column, complex take two columns, the first for the real part and the second for the imaginary part. Complex eigenvectors correspond to an eigenvalue with positive imaginary part. 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.) Real eigenvectors take one column, complex take two columns, the first for the real part and the second for the imaginary part. Complex eigenvectors correspond to an eigenvalue with positive imaginary part. 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(1,8*N). For good performance, LDWORK must generally be larger. To compute the optimal value of LDWORK, call ILAENV to get blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LDWORK is: 2*N + MAX( 6*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.

* INFO (output)