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# claqz0.f - Man Page

SRC/claqz0.f

## Synopsis

### Functions/Subroutines

recursive subroutine claqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)
CLAQZ0

## Function/Subroutine Documentation

### recursive subroutine claqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex, dimension( * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta, complex, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex, dimension( * ), intent(inout) work, integer, intent(in) lwork, real, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info)

CLAQZ0 Ā

Purpose:

CLAQZ0 computes the eigenvalues of a matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a matrix pair (A,B):

A = Q1*H*Z1**H,  B = Q1*T*Z1**H,

as computed by CGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,

H = Q*S*Z**H,  T = Q*P*Z**H,

where Q and Z are unitary matrices, P and S are an upper triangular
matrices.

Optionally, the unitary matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
unitary matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the unitary matrices from CGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the unitary factors from the
generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
Eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ
Algorithm with Aggressive Early Deflation', SIAM J. Numer.
Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift,
multipole rational QZ method with aggressive early deflation'
Parameters

WANTS

WANTS is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.

WANTQ

WANTQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (A,B) is returned;
= 'V': Q must contain an unitary matrix Q1 on entry and
the product Q1*Q is returned.

WANTZ

WANTZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (A,B) is returned;
= 'V': Z must contain an unitary matrix Z1 on entry and
the product Z1*Z is returned.

N

N is INTEGER
The order of the matrices A, B, Q, and Z.  N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI mark the rows and columns of A which are in
Hessenberg form.  It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

A

A is COMPLEX array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A.
On exit, if JOB = 'S', A contains the upper triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal of A matches that of S, but
the rest of A is unspecified.

LDA

LDA is INTEGER
The leading dimension of the array A.  LDA >= max( 1, N ).

B

B is COMPLEX array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, if JOB = 'S', B contains the upper triangular
matrix P from the generalized Schur factorization.
If JOB = 'E', the diagonal of B matches that of P, but
the rest of B is unspecified.

LDB

LDB is INTEGER
The leading dimension of the array B.  LDB >= max( 1, N ).

ALPHA

ALPHA is COMPLEX array, dimension (N)
Each scalar alpha defining an eigenvalue
of GNEP.

BETA

BETA is COMPLEX array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = ALPHA(j) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha.  Since either lambda or mu may overflow,
they should not, in general, be computed.

Q

Q is COMPLEX array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the unitary matrix of left Schur
vectors of (A,B), and if COMPQ = 'V', the unitary matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.

Z

Z is COMPLEX array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the unitary matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
unitary matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.

LDZ

LDZ is INTEGER
The leading dimension of the array Z.  LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is 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

RWORK is REAL array, dimension (N)

REC

REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.

INFO

INFO is 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.
Author

Thijs Steel, KU Leuven

Date

May 2020

Definition at line 280 of file claqz0.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page claqz0(3) is an alias of claqz0.f(3).

Tue Nov 28 2023 12:08:41 Version 3.12.0 LAPACK