NAME

SYNOPSIS

  SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, 
 *      BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR
  INTEGER N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
 
  SUBROUTINE SGEGS_64( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, 
 *      ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR
  INTEGER*8 N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
 

F95 INTERFACE

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

C INTERFACE

void sgegs(char jobvsl, char jobvsr, int n, float *a, int lda, float *b, int ldb, float *alphar, float *alphai, float *beta, float *vsl, int ldvsl, float *vsr, int ldvsr, int *info);

void sgegs_64(char jobvsl, char jobvsr, long n, float *a, long lda, float *b, long ldb, float *alphar, float *alphai, float *beta, float *vsl, long ldvsl, float *vsr, long ldvsr, long *info);

PURPOSE

SGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: the generalized eigenvalues (alphar +/- alphai*i, beta), the real 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 SGEGV 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 orthogonal matrix and both on the right by another orthogonal matrix, these two orthogonal matrices being chosen so as to bring the pair of matrices into (real) Schur form.

A pair of matrices A, B is in generalized real Schur form if B is upper triangular with non-negative diagonal and A is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of A will be ``standardized'' by making the corresponding elements of B have the form:

        [  a  0  ]
        [  0  b  ]

and the pair of corresponding 2-by-2 blocks in A and B will have a complex conjugate pair of generalized eigenvalues.

The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the orthogonal matrices which reduce A and B to Schur form:

Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T 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. Note: to avoid overflow, the Frobenius norm of the matrix A should be less than the overflow threshold.

* 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. Note: to avoid overflow, the Frobenius norm of the matrix B should be less than the overflow threshold.

* 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. ALPHAR(j) + ALPHAI(j)*i, j=1,...,N and BETA(j),j=1,...,N are the diagonals of the complex Schur form (A,B) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. 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 for ALPHAR.

* BETA (output)

See the description for ALPHAR.

* 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(1,4*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 + 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)