DHGEQZ(3F) DHGEQZ(3F)
DHGEQZ - implement a single-/double-shift version of the QZ method for
finding the generalized eigenvalues w(j)=(ALPHAR(j) +
i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In
addition, the pair A,B may be reduced to generalized Schur form
SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, *
)
DHGEQZ implements a single-/double-shift version of the QZ method for
finding the generalized eigenvalues B is upper triangular, and A is block
upper triangular, where the diagonal blocks are either 1-by-1 or 2-by-2,
the 2-by-2 blocks having complex generalized eigenvalues (see the
description of the argument JOB.)
If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form
by applying one orthogonal tranformation (usually called Q) on the left
and another (usually called Z) on the right. The 2-by-2 upper-triangular
diagonal blocks of B corresponding to 2-by-2 blocks of A will be reduced
to positive diagonal matrices. (I.e., if A(j+1,j) is non-zero, then
B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.)
If JOB='E', then at each iteration, the same transformations are
computed, but they are only applied to those parts of A and B which are
needed to compute ALPHAR, ALPHAI, and BETAR.
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
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.
Page 1
DHGEQZ(3F) DHGEQZ(3F)
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHAR, ALPHAI, 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 ALPHAR,
ALPHAI, and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the transpose of the
orthogonal 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) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the orthogonal
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) DOUBLE PRECISION 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 generalized Schur form.
If JOB='E', then on exit A will have been destroyed. The
diagonal blocks will be correct, but the off-diagonal portion
will be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. Elements below
the diagonal must be zero. 2-by-2 blocks in B corresponding to
2-by-2 blocks in A will be reduced to positive diagonal form.
(I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and
B(j,j) and B(j+1,j+1) will be positive.) If JOB='S', then on
exit A and B will have been simultaneously reduced to Schur form.
If JOB='E', then on exit B will have been destroyed. Elements
corresponding to diagonal blocks of A will be correct, but the
off-diagonal portion will be meaningless.
Page 2
DHGEQZ(3F) DHGEQZ(3F)
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAR(1:N) will be set to real parts of the diagonal elements of
A that would result from reducing A and B to Schur form and then
further reducing them both to triangular form using unitary
transformations s.t. the diagonal of B was non-negative real.
Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
then ALPHAR(j)=A(j,j). Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
eigenvalues of the matrix pencil A - wB.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
ALPHAI(1:N) will be set to imaginary parts of the diagonal
elements of A that would result from reducing A and B to Schur
form and then further reducing them both to triangular form using
unitary transformations s.t. the diagonal of B was non-negative
real. Thus, if A(j,j) is in a 1-by-1 block (i.e.,
A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. Note that the (real or
complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are
the generalized eigenvalues of the matrix pencil A - wB.
BETA (output) DOUBLE PRECISION array, dimension (N)
BETA(1:N) will be set to the (real) diagonal elements of B that
would result from reducing A and B to Schur form and then further
reducing them both to triangular form using unitary
transformations s.t. the diagonal of B was non-negative real.
Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
then BETA(j)=B(j,j). Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
eigenvalues of the matrix pencil A - wB. (Note that BETA(1:N)
will always be non-negative, and no BETAI is necessary.)
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or
'I', then the transpose of the orthogonal transformations which
are applied to A and B on the left will be applied to the array Q
on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or
'I', then LDQ >= N.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or
'I', then the orthogonal 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.
Page 3
DHGEQZ(3F) DHGEQZ(3F)
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,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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct. > 2*N: various
"impossible" errors.
FURTHER DETAILS
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
DHGEQZ(3F) DHGEQZ(3F)
DHGEQZ - implement a single-/double-shift version of the QZ method for
finding the generalized eigenvalues w(j)=(ALPHAR(j) +
i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In
addition, the pair A,B may be reduced to generalized Schur form
SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, *
)
DHGEQZ implements a single-/double-shift version of the QZ method for
finding the generalized eigenvalues B is upper triangular, and A is block
upper triangular, where the diagonal blocks are either 1-by-1 or 2-by-2,
the 2-by-2 blocks having complex generalized eigenvalues (see the
description of the argument JOB.)
If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form
by applying one orthogonal tranformation (usually called Q) on the left
and another (usually called Z) on the right. The 2-by-2 upper-triangular
diagonal blocks of B corresponding to 2-by-2 blocks of A will be reduced
to positive diagonal matrices. (I.e., if A(j+1,j) is non-zero, then
B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.)
If JOB='E', then at each iteration, the same transformations are
computed, but they are only applied to those parts of A and B which are
needed to compute ALPHAR, ALPHAI, and BETAR.
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
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.
Page 1
DHGEQZ(3F) DHGEQZ(3F)
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHAR, ALPHAI, 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 ALPHAR,
ALPHAI, and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the transpose of the
orthogonal 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) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the orthogonal
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) DOUBLE PRECISION 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 generalized Schur form.
If JOB='E', then on exit A will have been destroyed. The
diagonal blocks will be correct, but the off-diagonal portion
will be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. Elements below
the diagonal must be zero. 2-by-2 blocks in B corresponding to
2-by-2 blocks in A will be reduced to positive diagonal form.
(I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and
B(j,j) and B(j+1,j+1) will be positive.) If JOB='S', then on
exit A and B will have been simultaneously reduced to Schur form.
If JOB='E', then on exit B will have been destroyed. Elements
corresponding to diagonal blocks of A will be correct, but the
off-diagonal portion will be meaningless.
Page 2
DHGEQZ(3F) DHGEQZ(3F)
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAR(1:N) will be set to real parts of the diagonal elements of
A that would result from reducing A and B to Schur form and then
further reducing them both to triangular form using unitary
transformations s.t. the diagonal of B was non-negative real.
Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
then ALPHAR(j)=A(j,j). Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
eigenvalues of the matrix pencil A - wB.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
ALPHAI(1:N) will be set to imaginary parts of the diagonal
elements of A that would result from reducing A and B to Schur
form and then further reducing them both to triangular form using
unitary transformations s.t. the diagonal of B was non-negative
real. Thus, if A(j,j) is in a 1-by-1 block (i.e.,
A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. Note that the (real or
complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are
the generalized eigenvalues of the matrix pencil A - wB.
BETA (output) DOUBLE PRECISION array, dimension (N)
BETA(1:N) will be set to the (real) diagonal elements of B that
would result from reducing A and B to Schur form and then further
reducing them both to triangular form using unitary
transformations s.t. the diagonal of B was non-negative real.
Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
then BETA(j)=B(j,j). Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
eigenvalues of the matrix pencil A - wB. (Note that BETA(1:N)
will always be non-negative, and no BETAI is necessary.)
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or
'I', then the transpose of the orthogonal transformations which
are applied to A and B on the left will be applied to the array Q
on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or
'I', then LDQ >= N.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or
'I', then the orthogonal 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.
Page 3
DHGEQZ(3F) DHGEQZ(3F)
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,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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct. > 2*N: various
"impossible" errors.
FURTHER DETAILS
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
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