chgeqz-f(3)

chgeqz.f –

Synopsis

Functions/Subroutines

subroutine chgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
CHGEQZ

Function/Subroutine Documentation

subroutine chgeqz (characterJOB, characterCOMPQ, characterCOMPZ, integerN, integerILO, integerIHI, complex, dimension( ldh, * )H, integerLDH, complex, dimension( ldt, * )T, integerLDT, complex, dimension( * )ALPHA, complex, dimension( * )BETA, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( ldz, * )Z, integerLDZ, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integerINFO)

CHGEQZ

Purpose:

 CHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
 where H is an upper Hessenberg matrix and T is upper triangular,
 using the single-shift QZ method.
 Matrix pairs of this type are produced by the reduction to
 generalized upper Hessenberg form of a complex matrix pair (A,B):
A = Q1HZ1H, B = Q1TZ1H,
as computed by CGGHRD.
If JOB=‘S’, then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = QSZH, T = QPZH,
where Q and Z are unitary matrices and S and P are upper triangular.
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 Hessenberg form, then the output
matrices Q1Q and Z1Z are the unitary factors from the generalized
Schur factorization of (A,B):
A = (Q1Q)S(Z1Z)**H, B = (Q1Q)P(Z1Z)**H.
To avoid overflow, eigenvalues of the matrix pair (H,T)
(equivalently, of (A,B)) are computed as a pair of complex values
(alpha,beta). If beta is nonzero, lambda = alpha / beta is an
eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
Ax = lambdaBx
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
muAy = By.
The values of alpha and beta for the i-th eigenvalue 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.

Parameters:

JOB

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

COMPQ

 COMPQ 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 (H,T) is returned;
 = 'V': Q must contain a unitary matrix Q1 on entry and
 the product Q1*Q is returned.

COMPZ

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

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

ILO

 ILO is INTEGER

IHI

 IHI is INTEGER
 ILO and IHI mark the rows and columns of H 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.

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

LDH

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

 T is COMPLEX array, dimension (LDT, N)
 On entry, the N-by-N upper triangular matrix T.
 On exit, if JOB = 'S', T contains the upper triangular
 matrix P from the generalized Schur factorization.
 If JOB = 'E', the diagonal of T matches that of P, but
 the rest of T is unspecified.

LDT

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

ALPHA

 ALPHA is COMPLEX array, dimension (N)
 The complex scalars alpha that define the eigenvalues of
 GNEP. ALPHA(i) = S(i,i) in the generalized Schur
 factorization.

BETA

 BETA is COMPLEX array, dimension (N)
 The real non-negative scalars beta that define the
 eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
 Schur factorization.
 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 is COMPLEX array, dimension (LDQ, N)
 On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
 reduction of (A,B) to generalized Hessenberg form.
 On exit, if COMPZ = 'I', the unitary matrix of left Schur
 vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
 left Schur vectors of (A,B).
 Not referenced if COMPZ = 'N'.

LDQ

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

 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)

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. (H,T) 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. (H,T) is not
 in Schur form, but ALPHA(i) and BETA(i),
 i=INFO-N+1,...,N should be correct.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Further Details:

 We assume that complex ABS works as long as its value is less than
 overflow.

Definition at line 283 of file chgeqz.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.