--- rpl/lapack/lapack/zhgeqz.f 2010/12/21 13:53:47 1.7
+++ rpl/lapack/lapack/zhgeqz.f 2011/11/21 20:43:12 1.8
@@ -1,11 +1,293 @@
+*> \brief \b ZHGEQZ
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZHGEQZ + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
+* ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
+* RWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER COMPQ, COMPZ, JOB
+* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION RWORK( * )
+* COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
+* $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
+* $ Z( LDZ, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZHGEQZ 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 = Q1*H*Z1**H, B = Q1*T*Z1**H,
+*>
+*> as computed by ZGGHRD.
+*>
+*> 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 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 ZGGHRD that reduced
+*> the matrix pair (A,B) to generalized 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 complex values
+*> (alpha,beta). 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.
+*> 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOB
+*> \verbatim
+*> JOB is CHARACTER*1
+*> = 'E': Compute eigenvalues only;
+*> = 'S': Computer eigenvalues and the Schur form.
+*> \endverbatim
+*>
+*> \param[in] COMPQ
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] COMPZ
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrices H, T, Q, and Z. N >= 0.
+*> \endverbatim
+*>
+*> \param[in] ILO
+*> \verbatim
+*> ILO is INTEGER
+*> \endverbatim
+*>
+*> \param[in] IHI
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in,out] H
+*> \verbatim
+*> H is COMPLEX*16 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.
+*> \endverbatim
+*>
+*> \param[in] LDH
+*> \verbatim
+*> LDH is INTEGER
+*> The leading dimension of the array H. LDH >= max( 1, N ).
+*> \endverbatim
+*>
+*> \param[in,out] T
+*> \verbatim
+*> T is COMPLEX*16 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.
+*> \endverbatim
+*>
+*> \param[in] LDT
+*> \verbatim
+*> LDT is INTEGER
+*> The leading dimension of the array T. LDT >= max( 1, N ).
+*> \endverbatim
+*>
+*> \param[out] ALPHA
+*> \verbatim
+*> ALPHA is COMPLEX*16 array, dimension (N)
+*> The complex scalars alpha that define the eigenvalues of
+*> GNEP. ALPHA(i) = S(i,i) in the generalized Schur
+*> factorization.
+*> \endverbatim
+*>
+*> \param[out] BETA
+*> \verbatim
+*> BETA is COMPLEX*16 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.
+*> \endverbatim
+*>
+*> \param[in,out] Q
+*> \verbatim
+*> Q is COMPLEX*16 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'.
+*> \endverbatim
+*>
+*> \param[in] LDQ
+*> \verbatim
+*> LDQ is INTEGER
+*> The leading dimension of the array Q. LDQ >= 1.
+*> If COMPQ='V' or 'I', then LDQ >= N.
+*> \endverbatim
+*>
+*> \param[in,out] Z
+*> \verbatim
+*> Z is COMPLEX*16 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'.
+*> \endverbatim
+*>
+*> \param[in] LDZ
+*> \verbatim
+*> LDZ is INTEGER
+*> The leading dimension of the array Z. LDZ >= 1.
+*> If COMPZ='V' or 'I', then LDZ >= N.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
+*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension (N)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> 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.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup complex16GEcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> We assume that complex ABS works as long as its value is less than
+*> overflow.
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
$ RWORK, INFO )
*
-* -- LAPACK routine (version 3.2) --
+* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
@@ -18,172 +300,6 @@
$ Z( LDZ, * )
* ..
*
-* Purpose
-* =======
-*
-* ZHGEQZ 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 = Q1*H*Z1**H, B = Q1*T*Z1**H,
-*
-* as computed by ZGGHRD.
-*
-* 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 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 ZGGHRD that reduced
-* the matrix pair (A,B) to generalized 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 complex values
-* (alpha,beta). 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.
-* 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.
-*
-* Arguments
-* =========
-*
-* JOB (input) CHARACTER*1
-* = 'E': Compute eigenvalues only;
-* = 'S': Computer 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 a unitary matrix Q1 on entry and
-* the product Q1*Q is returned.
-*
-* COMPZ (input) 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 (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) COMPLEX*16 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 (input) INTEGER
-* The leading dimension of the array H. LDH >= max( 1, N ).
-*
-* T (input/output) COMPLEX*16 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 (input) INTEGER
-* The leading dimension of the array T. LDT >= max( 1, N ).
-*
-* ALPHA (output) COMPLEX*16 array, dimension (N)
-* The complex scalars alpha that define the eigenvalues of
-* GNEP. ALPHA(i) = S(i,i) in the generalized Schur
-* factorization.
-*
-* BETA (output) COMPLEX*16 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 (input/output) COMPLEX*16 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 (input) INTEGER
-* The leading dimension of the array Q. LDQ >= 1.
-* If COMPQ='V' or 'I', then LDQ >= N.
-*
-* Z (input/output) COMPLEX*16 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 (input) INTEGER
-* The leading dimension of the array Z. LDZ >= 1.
-* If COMPZ='V' or 'I', then LDZ >= N.
-*
-* WORK (workspace/output) COMPLEX*16 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 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) DOUBLE PRECISION 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. (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.
-*
-* Further Details
-* ===============
-*
-* We assume that complex ABS works as long as its value is less than
-* overflow.
-*
* =====================================================================
*
* .. Parameters ..