
SGEGS(l) ) SGEGS(l)
NAME
SGEGS  routine is deprecated and has been replaced by routine SGGES
SYNOPSIS
SUBROUTINE SGEGS( 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
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VSL(
LDVSL, * ), VSR( LDVSR, * ), WORK( * )
PURPOSE
This routine is deprecated and has been replaced by routine SGGES. SGEGS computes for a
pair of NbyN 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 vec
tors (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
nonnegative diagonal and A is block upper triangular with 1by1 and 2by2 blocks.
1by1 blocks correspond to real generalized eigenvalues, while 2by2 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 2by2 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) CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input) CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output) REAL array, dimension (LDA, N)
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) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
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) 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. 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 2by2 diagonal
blocks of the real Schur form of (A,B) were further reduced to triangular form
using 2by2 complex unitary transformations. If ALPHAI(j) is zero, then the jth
eigenvalue is real; if positive, then the jth and (j+1)st eigenvalues are a com
plex 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 com
puting 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).
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors. (See "Purpose", above.)
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) REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors. (See "Purpose",
above.) 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) 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(1,4*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 +
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 ith 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 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 SGGBAK (computing VSL)
=N+8: error return from SGGBAK (computing VSR)
=N+9: error return from SLASCL (various places)
LAPACK version 3.0 15 June 2000 SGEGS(l) 
