ZHGEQZ(3F) ZHGEQZ(3F)
ZHGEQZ - 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 ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA,
BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
LDQ, * ), WORK( * ), Z( LDZ, * )
ZHGEQZ 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
transformations used to reduce (A,B) are accumulated into the arrays Q
and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
JOB (input) CHARACTER*1
= '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) CHARACTER*1
= '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.
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ZHGEQZ(3F) ZHGEQZ(3F)
COMPZ (input) CHARACTER*1
= '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 initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER 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/output) COMPLEX*16 array, dimension (LDA, N)
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 destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
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 destroyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output) COMPLEX*16 array, dimension (N)
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) COMPLEX*16 array, dimension (N)
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) COMPLEX*16 array, dimension (LDQ, N)
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.
Page 2
ZHGEQZ(3F) ZHGEQZ(3F)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or
'I', then LDQ >= N.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
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) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or
'I', then LDZ >= N.
WORK (workspace/output) COMPLEX*16 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,N).
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 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: various "impossible" errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is less than
overflow.
ZHGEQZ(3F) ZHGEQZ(3F)
ZHGEQZ - 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 ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA,
BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
LDQ, * ), WORK( * ), Z( LDZ, * )
ZHGEQZ 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
transformations used to reduce (A,B) are accumulated into the arrays Q
and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
JOB (input) CHARACTER*1
= '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) CHARACTER*1
= '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.
Page 1
ZHGEQZ(3F) ZHGEQZ(3F)
COMPZ (input) CHARACTER*1
= '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 initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER 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/output) COMPLEX*16 array, dimension (LDA, N)
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 destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
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 destroyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output) COMPLEX*16 array, dimension (N)
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) COMPLEX*16 array, dimension (N)
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) COMPLEX*16 array, dimension (LDQ, N)
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.
Page 2
ZHGEQZ(3F) ZHGEQZ(3F)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or
'I', then LDQ >= N.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
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) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or
'I', then LDZ >= N.
WORK (workspace/output) COMPLEX*16 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,N).
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 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: various "impossible" errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is less than
overflow.
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