SYNOPSIS

SUBROUTINE DGEGS(

JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO )

    

CHARACTER JOBVSL, JOBVSR

    

INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

    

DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

PURPOSE

This routine is deprecated and has been replaced by routine DGGES. DGEGS computes the eigenvalues, real Schur form, and, optionally, left and or/right Schur vectors of a real matrix pair (A,B). Given two square matrices A and B, the generalized real Schur factorization has the form

  A = Q*S*Z**T,  B = Q*T*Z**T
where Q and Z are orthogonal matrices, T is upper triangular, and S is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks, the 2-by-2 blocks corresponding to complex conjugate pairs of eigenvalues of (A,B). The columns of Q are the left Schur vectors and the columns of Z are the right Schur vectors.
If only the eigenvalues of (A,B) are needed, the driver routine DGEGV should be used instead. See DGEGV for a description of the eigenvalues of the generalized nonsymmetric eigenvalue problem (GNEP).

ARGUMENTS

JOBVSL (input) CHARACTER*1

= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors (returned in VSL).

JOBVSR (input) CHARACTER*1

= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors (returned in VSR).

N (input) INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

A (input/output) DOUBLE PRECISION array, dimension (LDA, N)

On entry, the matrix A. On exit, the upper quasi-triangular matrix S from the generalized real Schur factorization.

LDA (input) INTEGER

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

B (input/output) DOUBLE PRECISION array, dimension (LDB, N)

On entry, the matrix B. On exit, the upper triangular matrix T from the generalized real Schur factorization.

LDB (input) INTEGER

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

ALPHAR (output) DOUBLE PRECISION array, dimension (N)

The real parts of each scalar alpha defining an eigenvalue of GNEP.

ALPHAI (output) DOUBLE PRECISION array, dimension (N)

The imaginary parts of each scalar alpha defining an eigenvalue of GNEP. 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) = -ALPHAI(j).

BETA (output) DOUBLE PRECISION array, dimension (N)

The scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.

VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)

If JOBVSL = 'V', the matrix of left Schur vectors Q. Not referenced if JOBVSL = 'N'.

LDVSL (input) INTEGER

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

VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)

If JOBVSR = 'V', the matrix of right Schur vectors Z. Not referenced if JOBVSR = 'N'.

LDVSR (input) INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= max(1,4*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR The optimal LWORK is 2*N + N*(NB+1). 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.

INFO (output) INTEGER

= 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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems:
=N+1: error return from DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed iteration) =N+7: error return from DGGBAK (computing VSL)
=N+8: error return from DGGBAK (computing VSR)
=N+9: error return from DLASCL (various places)