Contents
chgeqz - 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
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, com-
plex *q, long ldq, complex *z, long ldz, long
*info);
chgeqz 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 uni-
tary transformations 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 General-
ized Matrixigenvalue Problems", SIAM J. Numer. Anal.,
10(1973),p. 241--256.
JOB (input)
= 'E': compute only ALPHA and BETA. A and B will
not necessarily 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 tranformation 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 ini-
tialized 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 triangular form.
If JOB='E', then on exit A will have been des-
troyed.
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 triangular form.
If JOB='E', then on exit B will have been des-
troyed.
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 transformations 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 ille-
gal 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:
various "impossible" errors.
We assume that complex ABS works as long as its value is
less than overflow.