
DGEGS(l) ) DGEGS(l)
NAME
DGEGS  routine is deprecated and has been replaced by routine DGGES
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This routine is deprecated and has been replaced by routine DGGES.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 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 DGEGV 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE
PRECISION 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 trans
formations. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive,
then the jth 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 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 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 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 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)
LAPACK version 3.0 15 June 2000 DGEGS(l) 
