SGGEV - compute for a pair of N-by-N real nonsymmetric matrices (A,B)

SUBROUTINE SGGEV(

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( * )

SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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.

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j).

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

u(j)**H * A = lambda(j) * u(j)**H * B .

where u(j)**H is the conjugate-transpose of u(j).

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigenvectors;

= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized eigenvectors;

= 'V': compute the right generalized eigenvectors.

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 in the pair (A,B). On exit, A has been overwritten.

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 in the pair (A,B). On exit, B has been overwritten.

LDB (input) INTEGER

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

ALPHAR (output) REAL array, dimension (N)

ALPHAI (output) REAL array, dimension (N) BETA (output) REAL array, dimension (N) 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).

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

If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th 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 will be scaled so the largest component have abs(real part)+abs(imag. part)=1. 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 v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each eigenvector will be scaled so the largest component have abs(real part)+abs(imag. part)=1. 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 (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.

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: =N+1: other than QZ iteration failed in SHGEQZ.

=N+2: error return from STGEVC.