SYNOPSIS

SUBROUTINE SGEGV(

JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )

    

CHARACTER JOBVL, JOBVR

    

INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N

    

REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

This routine is deprecated and has been replaced by routine SGGEV. SGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).
Given two square matrices A and B,
the generalized nonsymmetric eigenvalue problem (GNEP) is to find the eigenvalues lambda and corresponding (non-zero) eigenvectors x such that

   A*x = lambda*B*x.
An alternate form is to find the eigenvalues mu and corresponding eigenvectors y such that

   mu*A*y = B*y.
These two forms are equivalent with mu = 1/lambda and x = y if neither lambda nor mu is zero. In order to deal with the case that lambda or mu is zero or small, two values alpha and beta are returned for each eigenvalue, such that lambda = alpha/beta and
mu = beta/alpha.
The vectors x and y in the above equations are right eigenvectors of the matrix pair (A,B). Vectors u and v satisfying

   u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B
are left eigenvectors of (A,B).
Note: this routine performs "full balancing" on A and B -- see "Further Details", below.

ARGUMENTS

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors (returned in VL).

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors (returned in VR).

N (input) INTEGER

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

A (input/output) REAL array, dimension (LDA, N)

On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on exit A contains the real Schur form of A from the generalized Schur factorization of the pair (A,B) after balancing. If no eigenvectors were computed, then only the diagonal blocks from the Schur form will be correct. See SGGHRD and SHGEQZ for details.

LDA (input) INTEGER

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

B (input/output) REAL array, dimension (LDB, N)

On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the upper triangular matrix obtained from B in the generalized Schur factorization of the pair (A,B) after balancing. If no eigenvectors were computed, then only those elements of B corresponding to the diagonal blocks from the Schur form of A will be correct. See SGGHRD and SHGEQZ for details.

LDB (input) INTEGER

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

ALPHAR (output) REAL array, dimension (N)

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

ALPHAI (output) REAL 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) REAL 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.

VL (output) REAL array, dimension (LDVL,N)

If JOBVL = 'V', the left eigenvectors u(j) are stored in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j). If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1). Each eigenvector is scaled so that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvectors corresponding to an eigenvalue with alpha = beta = 0, which are set to zero. Not referenced if JOBVL = 'N'.

LDVL (input) INTEGER

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

VR (output) REAL array, dimension (LDVR,N)

If JOBVR = 'V', the right eigenvectors x(j) are stored in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then x(j) = VR(:,j). If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then x(j) = VR(:,j) + i*VR(:,j+1) and x(j+1) = VR(:,j) - i*VR(:,j+1). Each eigenvector is scaled so that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvalues corresponding to an eigenvalue with alpha = beta = 0, which are set to zero. Not referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

WORK (workspace/output) REAL 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(1,8*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LWORK is: 2*N + MAX( 6*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. No eigenvectors have been calculated, 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 SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed iteration) =N+7: error return from STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)

FURTHER DETAILS

Balancing
---------
This driver calls SGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been computed, SGGBAK transforms the eigenvectors back to what they would have been (in perfect arithmetic) if they had not been balanced.
Contents of A and B on Exit
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the real Schur form[*] of the "balanced" versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct. [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
    by Golub & van Loan, pub. by Johns Hopkins U. Press.