dhgeqz man page on Scientific

Man page or keyword search:  
man Server   26626 pages
apropos Keyword Search (all sections)
Output format
Scientific logo
[printable version]

DHGEQZ(1)		 LAPACK routine (version 3.2)		     DHGEQZ(1)

NAME
       DHGEQZ - computes the eigenvalues of a real matrix pair (H,T),

SYNOPSIS
       SUBROUTINE DHGEQZ( JOB,	COMPQ,	COMPZ,	N,  ILO,  IHI, H, LDH, T, LDT,
			  ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ,	 WORK,	LWORK,
			  INFO )

	   CHARACTER	  COMPQ, COMPZ, JOB

	   INTEGER	  IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N

	   DOUBLE	  PRECISION  ALPHAI(  *	 ), ALPHAR( * ), BETA( * ), H(
			  LDH, * ), Q( LDQ, * ), T( LDT, * ), WORK(  *	),  Z(
			  LDZ, * )

PURPOSE
       DHGEQZ 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 DGGHRD.
       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	corre‐
       sponding	 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 factor‐
       ization may be postmultiplied into an input matrix Q1, and the orthogo‐
       nal  matrix Z may be postmultiplied into an input matrix Z1.  If Q1 and
       Z1 are the orthogonal matrices from DGGHRD 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 gen‐
       eralized 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.

ARGUMENTS
       JOB     (input) CHARACTER*1
	       = 'E': Compute eigenvalues only;
	       = 'S': Compute eigenvalues and the Schur form.

       COMPQ   (input) 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   (input) 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       (input) INTEGER
	       The order of the matrices H, T, Q, and Z.  N >= 0.

       ILO     (input) INTEGER
	       IHI	(input)	 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       (input/output) DOUBLE PRECISION 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;	2-by-2 diagonal blocks
	       (corresponding to complex conjugate pairs of  eigenvalues)  are
	       returned	 in  standard  form,  with  H(i,i)  =  H(i+1,i+1)  and
	       H(i+1,i)*H(i,i+1) < 0.  If JOB = 'E', the diagonal blocks of  H
	       match those of S, but the rest of H is unspecified.

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

       T       (input/output) DOUBLE PRECISION 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     (input) INTEGER
	       The leading dimension of the array T.  LDT >= max( 1, N ).

       ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
	       The real parts of each scalar alpha defining an	eigenvalue  of
	       GNEP.

       ALPHAI  (output) DOUBLE PRECISION 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    (output) DOUBLE PRECISION 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       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
	       On  entry, if COMPZ = 'V', the orthogonal matrix Q1 used in the
	       reduction of (A,B) to generalized Hessenberg form.  On exit, if
	       COMPZ  =	 'I',  the  orthogonal matrix of left Schur vectors of
	       (H,T), and if COMPZ = 'V', the orthogonal matrix of left	 Schur
	       vectors of (A,B).  Not referenced if COMPZ = 'N'.

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

       Z       (input/output) DOUBLE PRECISION 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     (input) INTEGER
	       The leading dimension of the array Z.  LDZ >= 1.	 If  COMPZ='V'
	       or 'I', then LDZ >= N.

       WORK	  (workspace/output)   DOUBLE	PRECISION   array,   dimension
       (MAX(1,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 cal‐
	       culates 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    (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.  (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.

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.

 LAPACK routine (version 3.2)	 November 2008			     DHGEQZ(1)
[top]

List of man pages available for Scientific

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]
Tweet
Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
Vote for polarhome
Free Shell Accounts :: the biggest list on the net