shgeqz
shgeqz - 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 SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
CHARACTER * 1 JOB, COMPQ, COMPZ
INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*)
SUBROUTINE SHGEQZ_64( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B,
* LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
CHARACTER * 1 JOB, COMPQ, COMPZ
INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*)
SUBROUTINE HGEQZ( JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B,
* [LDB], ALPHAR, ALPHAI, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK],
* [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, Q, Z
SUBROUTINE HGEQZ_64( JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B,
* [LDB], ALPHAR, ALPHAI, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK],
* [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, Q, Z
#include <sunperf.h>
void shgeqz(char job, char compq, char compz, int n, int ilo, int ihi, float *a, int lda, float *b, int ldb, float *alphar, float *alphai, float *beta, float *q, int ldq, float *z, int ldz, int *info);
void shgeqz_64(char job, char compq, char compz, long n, long ilo, long ihi, float *a, long lda, float *b, long ldb, float *alphar, float *alphai, float *beta, float *q, long ldq, float *z, long ldz, long *info);
shgeqz 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.:
(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, ``An Algorithm for Generalized Matrixigenvalue Problems'', SIAM J. Numer. Anal., 10(1973),p. 241--256.
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* JOB (input)
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* COMPQ (input)
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* COMPZ (input)
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* N (input)
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The order of the matrices A, B, Q, and Z. N >= 0.
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* ILO (input)
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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.
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* IHI (input)
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See the description of ILO.
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* A (input)
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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.
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* LDA (input)
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The leading dimension of the array A. LDA >= max( 1, N ).
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* B (input)
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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.
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* LDB (input)
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The leading dimension of the array B. LDB >= max( 1, N ).
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* ALPHAR (output)
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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.
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* ALPHAI (output)
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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.
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* BETA (output)
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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.)
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* Q (input/output)
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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.
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* LDQ (input)
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The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
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* Z (input/output)
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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.
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* LDZ (input)
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The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
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* WORK (workspace)
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On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
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* LWORK (input)
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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.
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* INFO (output)
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