shgeqz.f - Online Linux Manual PageSection : 3
Updated : Tue Nov 14 2017
Source : Version 3.8.0
Note : LAPACK
NAMEshgeqz.f
SYNOPSIS
Functions/Subroutinessubroutine shgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ
Function/Subroutine Documentation
subroutine shgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)SHGEQZ Purpose: SHGEQZ computes the eigenvalues of a real 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 real matrix pair (A,B):
A = Q1*H*Z1**T, B = Q1*T*Z1**T,
as computed by SGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = Q*S*Z**T, T = Q*P*Z**T,
where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.
The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.
Additionally, the 2-by-2 upper triangular diagonal blocks of P
corresponding to 2-by-2 blocks of S are reduced to positive diagonal
form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
P(j,j) > 0, and P(j+1,j+1) > 0.
Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
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.
Real 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.Parameters: JOB JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute 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 an orthogonal 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': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.
N 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 H is REAL array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of H match those 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 T is REAL 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;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = 'E', the diagonal blocks of T match those of P, but
the rest of T is unspecified.
LDT LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.
ALPHAI ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the orthogonal 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 REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
orthogonal 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 REAL 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.
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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(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: June 2016 Further Details: Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.Definition at line 306 of file shgeqz.f.
AuthorGenerated automatically by Doxygen for LAPACK from the source code. 0
Johanes Gumabo
Data Size : 28,085 byte
man-shgeqz.f.3Build : 2024-12-05, 20:55 :
Visitor Screen : x
Visitor Counter ( page / site ) : 2 / 184,234
Visitor ID : :
Visitor IP : 3.144.123.61 :
Visitor Provider : AMAZON-02 :
Provider Position ( lat x lon ) : 39.962500 x -83.006100 : x
Provider Accuracy Radius ( km ) : 1000 :
Provider City : Columbus :
Provider Province : Ohio , : ,
Provider Country : United States :
Provider Continent : North America :
Visitor Recorder : Version :
Visitor Recorder : Library :
Online Linux Manual Page : Version : Online Linux Manual Page - Fedora.40 - march=x86-64 - mtune=generic - 24.12.05
Online Linux Manual Page : Library : lib_c - 24.10.03 - march=x86-64 - mtune=generic - Fedora.40
Online Linux Manual Page : Library : lib_m - 24.10.03 - march=x86-64 - mtune=generic - Fedora.40
Data Base : Version : Online Linux Manual Page Database - 24.04.13 - march=x86-64 - mtune=generic - fedora-38
Data Base : Library : lib_c - 23.02.07 - march=x86-64 - mtune=generic - fedora.36
Very long time ago, I have the best tutor, Wenzel Svojanovsky . If someone knows the email address of Wenzel Svojanovsky , please send an email to johanes_gumabo@yahoo.co.id .
If error, please print screen and send to johanes_gumabo@yahoo.co.id
Under development. Support me via PayPal.