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Livre :
Expressions régulières,
Syntaxe et mise en oeuvre :
GNU/Linux |
CentOS 4.8 |
i386 |
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chgeqz(l) |
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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
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SUBROUTINE CHGEQZ( |
JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO ) | ||
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CHARACTER |
COMPQ, COMPZ, JOB | ||
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INTEGER |
IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N | ||
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REAL |
RWORK( * ) | ||
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COMPLEX |
A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) |
CHGEQZ implements 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. The diagonal elements of 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.
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 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 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 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 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 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.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ=’V’ or ’I’, then LDQ >= N.
Z (input/output) COMPLEX 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 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).
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) REAL 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.
We assume that complex ABS works as long as its value is less than overflow.
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chgeqz(l) | |