--- rpl/lapack/lapack/dhgeqz.f 2011/07/22 07:38:06 1.8
+++ rpl/lapack/lapack/dhgeqz.f 2011/11/21 20:42:53 1.9
@@ -1,11 +1,313 @@
+*> \brief \b DHGEQZ
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DHGEQZ + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
+* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
+* LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER COMPQ, COMPZ, JOB
+* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
+* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
+* $ WORK( * ), Z( LDZ, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> 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
+*> 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 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
+*> 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOB
+*> \verbatim
+*> JOB is CHARACTER*1
+*> = 'E': Compute eigenvalues only;
+*> = 'S': Compute 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 an orthogonal 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': 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.
+*> \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 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.
+*> If JOB = 'E', the diagonal blocks of H match those 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 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.
+*> \endverbatim
+*>
+*> \param[in] LDT
+*> \verbatim
+*> LDT is INTEGER
+*> The leading dimension of the array T. LDT >= max( 1, N ).
+*> \endverbatim
+*>
+*> \param[out] ALPHAR
+*> \verbatim
+*> ALPHAR is DOUBLE PRECISION array, dimension (N)
+*> The real parts of each scalar alpha defining an eigenvalue
+*> of GNEP.
+*> \endverbatim
+*>
+*> \param[out] ALPHAI
+*> \verbatim
+*> ALPHAI is 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).
+*> \endverbatim
+*>
+*> \param[out] BETA
+*> \verbatim
+*> BETA is 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.
+*> \endverbatim
+*>
+*> \param[in,out] Q
+*> \verbatim
+*> Q is 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'.
+*> \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 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'.
+*> \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 DOUBLE PRECISION 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] 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 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.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup doubleGEcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> 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.
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
*
-* -- LAPACK routine (version 3.3.1) --
+* -- 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..--
-* -- April 2009 --
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
@@ -17,193 +319,6 @@
$ 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
-* 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 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
-* 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.
-*
-* 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.
-* 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 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 (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.
-*
* =====================================================================
*
* .. Parameters ..