SHGEQZ(3F) SHGEQZ(3F)
SHGEQZ  implement a single/doubleshift 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 SHGEQZ( 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
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
SHGEQZ implements a single/doubleshift 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 1by1 or 2by2,
the 2by2 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 2by2 uppertriangular
diagonal blocks of B corresponding to 2by2 blocks of A will be reduced
to positive diagonal matrices. (I.e., if A(j+1,j) is nonzero, 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. 241256.
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
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SHGEQZ(3F) SHGEQZ(3F)
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:ILO1 and IHI+1:N. 1 <= ILO <=
IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the NbyN 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 offdiagonal portion
will be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) REAL array, dimension (LDB, N)
On entry, the NbyN upper triangular matrix B. Elements below
the diagonal must be zero. 2by2 blocks in B corresponding to
2by2 blocks in A will be reduced to positive diagonal form.
(I.e., if A(j+1,j) is nonzero, 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
offdiagonal portion will be meaningless.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
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SHGEQZ(3F) SHGEQZ(3F)
ALPHAR (output) REAL 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 nonnegative real.
Thus, if A(j,j) is in a 1by1 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) REAL 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 nonnegative
real. Thus, if A(j,j) is in a 1by1 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) REAL 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 nonnegative real.
Thus, if A(j,j) is in a 1by1 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 nonnegative, and no BETAI is necessary.)
Q (input/output) REAL 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) REAL 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
SHGEQZ(3F) SHGEQZ(3F)
WORK (workspace/output) REAL 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 ith 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=INFON+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 1by1 or
2by2 block deflates off the bottom.
SHGEQZ(3F) SHGEQZ(3F)
SHGEQZ  implement a single/doubleshift 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 SHGEQZ( 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
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
SHGEQZ implements a single/doubleshift 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 1by1 or 2by2,
the 2by2 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 2by2 uppertriangular
diagonal blocks of B corresponding to 2by2 blocks of A will be reduced
to positive diagonal matrices. (I.e., if A(j+1,j) is nonzero, 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. 241256.
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
Page 1
SHGEQZ(3F) SHGEQZ(3F)
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:ILO1 and IHI+1:N. 1 <= ILO <=
IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the NbyN 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 offdiagonal portion
will be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) REAL array, dimension (LDB, N)
On entry, the NbyN upper triangular matrix B. Elements below
the diagonal must be zero. 2by2 blocks in B corresponding to
2by2 blocks in A will be reduced to positive diagonal form.
(I.e., if A(j+1,j) is nonzero, 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
offdiagonal portion will be meaningless.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
Page 2
SHGEQZ(3F) SHGEQZ(3F)
ALPHAR (output) REAL 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 nonnegative real.
Thus, if A(j,j) is in a 1by1 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) REAL 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 nonnegative
real. Thus, if A(j,j) is in a 1by1 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) REAL 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 nonnegative real.
Thus, if A(j,j) is in a 1by1 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 nonnegative, and no BETAI is necessary.)
Q (input/output) REAL 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) REAL 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
SHGEQZ(3F) SHGEQZ(3F)
WORK (workspace/output) REAL 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 ith 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=INFON+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 1by1 or
2by2 block deflates off the bottom.
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