chgeqz(3P) Sun Performance Library chgeqz(3P)NAMEchgeqz - implement a single-shift version of the QZ method for finding
the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation
det( A-w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously
reduced to Schur form (i.e., A and B are both upper triangular) by
applying one unitary tranformation (usually called Q) on the left and
another (usually called Z) on the right
SYNOPSIS
SUBROUTINE CHGEQZ(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
CHARACTER * 1 JOB, COMPQ, COMPZ
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*),
WORK(*)
INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL RWORK(*)
SUBROUTINE CHGEQZ_64(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
CHARACTER * 1 JOB, COMPQ, COMPZ
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*),
WORK(*)
INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL RWORK(*)
F95 INTERFACE
SUBROUTINE HGEQZ(JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B, [LDB],
ALPHA, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK], [RWORK], [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, Q, Z
INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL, DIMENSION(:) :: RWORK
SUBROUTINE HGEQZ_64(JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B,
[LDB], ALPHA, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK], [RWORK],
[INFO])
CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, Q, Z
INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL, DIMENSION(:) :: RWORK
C INTERFACE
#include <sunperf.h>
void chgeqz(char job, char compq, char compz, int n, int ilo, int ihi,
complex *a, int lda, complex *b, int ldb, complex *alpha,
complex *beta, complex *q, int ldq, complex *z, int ldz, int
*info);
void chgeqz_64(char job, char compq, char compz, long n, long ilo, long
ihi, complex *a, long lda, complex *b, long ldb, complex
*alpha, complex *beta, complex *q, long ldq, complex *z, long
ldz, long *info);
PURPOSEchgeqz implements a single-shift version of the QZ method for finding
the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are
then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary trans‐
formations used to reduce (A,B) are accumulated into the arrays Q and Z
s.t.:
(in) A(in)Z(in)* = Q(out)A(out)Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrixi‐
genvalue Problems", SIAM J. Numer. Anal., 10(1973),p. 241--256.
ARGUMENTS
JOB (input)
= 'E': compute only ALPHA and BETA. A and B will not neces‐
sarily be put into generalized Schur form. = 'S': put A and
B into generalized Schur form, as well as computing ALPHA and
BETA.
COMPQ (input)
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the conjugate
transpose of the unitary tranformation that is applied to the
left side of A and B to reduce them to Schur form. = 'I':
like COMPQ='V', except that Q will be initialized to the
identity first.
COMPZ (input)
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the unitary tran‐
formation that is applied to the right side of A and B to
reduce them to Schur form. = 'I': like COMPZ='V', except
that Z will be initialized to the identity first.
N (input) The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input)
It is assumed that A is already upper triangular in rows and
columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0;
ILO=1 and IHI=0, if N=0.
IHI (input)
It is assumed that A is already upper triangular in rows and
columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0;
ILO=1 and IHI=0, if N=0.
A (input) On entry, the N-by-N upper Hessenberg matrix A. Elements
below the subdiagonal must be zero. If JOB='S', then on exit
A and B will have been simultaneously reduced to upper trian‐
gular form. If JOB='E', then on exit A will have been
destroyed.
LDA (input)
The leading dimension of the array A. LDA >= max( 1, N ).
B (input) On entry, the N-by-N upper triangular matrix B. Elements
below the diagonal must be zero. If JOB='S', then on exit A
and B will have been simultaneously reduced to upper triangu‐
lar form. If JOB='E', then on exit B will have been
destroyed.
LDB (input)
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output)
The diagonal elements of A when the pair (A,B) has been
reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the
generalized eigenvalues.
BETA (output)
The diagonal elements of B when the pair (A,B) has been
reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the
generalized eigenvalues. A and B are normalized so that
BETA(1),...,BETA(N) are non-negative real numbers.
Q (input/output)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or
'I', then the conjugate transpose of the unitary transforma‐
tions which are applied to A and B on the left will be
applied to the array Q on the right.
LDQ (input)
The leading dimension of the array Q. LDQ >= 1. If
COMPQ='V' or 'I', then LDQ >= N.
Z (input/output)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or
'I', then the unitary transformations which are applied to A
and B on the right will be applied to the array Z on the
right.
LDZ (input)
The leading dimension of the array Z. LDZ >= 1. If
COMPZ='V' or 'I', then LDZ >= N.
WORK (workspace)
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= max(1,N).
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.
RWORK (workspace)
dimension(N)
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not
in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N
should be correct. = N+1,...,2*N: the shift calculation
failed. (A,B) is not in Schur form, but ALPHA(i) and
BETA(i), i=INFO-N+1,...,N should be correct. > 2*N: var‐
ious "impossible" errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is less than
overflow.
6 Mar 2009 chgeqz(3P)
