--- rpl/lapack/lapack/ztrsen.f 2011/07/22 07:38:21 1.8
+++ rpl/lapack/lapack/ztrsen.f 2011/11/21 20:43:23 1.9
@@ -1,12 +1,273 @@
+*> \brief \b ZTRSEN
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZTRSEN + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
+* SEP, WORK, LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER COMPQ, JOB
+* INTEGER INFO, LDQ, LDT, LWORK, M, N
+* DOUBLE PRECISION S, SEP
+* ..
+* .. Array Arguments ..
+* LOGICAL SELECT( * )
+* COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZTRSEN reorders the Schur factorization of a complex matrix
+*> A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
+*> the leading positions on the diagonal of the upper triangular matrix
+*> T, and the leading columns of Q form an orthonormal basis of the
+*> corresponding right invariant subspace.
+*>
+*> Optionally the routine computes the reciprocal condition numbers of
+*> the cluster of eigenvalues and/or the invariant subspace.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOB
+*> \verbatim
+*> JOB is CHARACTER*1
+*> Specifies whether condition numbers are required for the
+*> cluster of eigenvalues (S) or the invariant subspace (SEP):
+*> = 'N': none;
+*> = 'E': for eigenvalues only (S);
+*> = 'V': for invariant subspace only (SEP);
+*> = 'B': for both eigenvalues and invariant subspace (S and
+*> SEP).
+*> \endverbatim
+*>
+*> \param[in] COMPQ
+*> \verbatim
+*> COMPQ is CHARACTER*1
+*> = 'V': update the matrix Q of Schur vectors;
+*> = 'N': do not update Q.
+*> \endverbatim
+*>
+*> \param[in] SELECT
+*> \verbatim
+*> SELECT is LOGICAL array, dimension (N)
+*> SELECT specifies the eigenvalues in the selected cluster. To
+*> select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix T. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] T
+*> \verbatim
+*> T is COMPLEX*16 array, dimension (LDT,N)
+*> On entry, the upper triangular matrix T.
+*> On exit, T is overwritten by the reordered matrix T, with the
+*> selected eigenvalues as the leading diagonal elements.
+*> \endverbatim
+*>
+*> \param[in] LDT
+*> \verbatim
+*> LDT is INTEGER
+*> The leading dimension of the array T. LDT >= max(1,N).
+*> \endverbatim
+*>
+*> \param[in,out] Q
+*> \verbatim
+*> Q is COMPLEX*16 array, dimension (LDQ,N)
+*> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
+*> On exit, if COMPQ = 'V', Q has been postmultiplied by the
+*> unitary transformation matrix which reorders T; the leading M
+*> columns of Q form an orthonormal basis for the specified
+*> invariant subspace.
+*> If COMPQ = 'N', Q is not referenced.
+*> \endverbatim
+*>
+*> \param[in] LDQ
+*> \verbatim
+*> LDQ is INTEGER
+*> The leading dimension of the array Q.
+*> LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
+*> \endverbatim
+*>
+*> \param[out] W
+*> \verbatim
+*> W is COMPLEX*16 array, dimension (N)
+*> The reordered eigenvalues of T, in the same order as they
+*> appear on the diagonal of T.
+*> \endverbatim
+*>
+*> \param[out] M
+*> \verbatim
+*> M is INTEGER
+*> The dimension of the specified invariant subspace.
+*> 0 <= M <= N.
+*> \endverbatim
+*>
+*> \param[out] S
+*> \verbatim
+*> S is DOUBLE PRECISION
+*> If JOB = 'E' or 'B', S is a lower bound on the reciprocal
+*> condition number for the selected cluster of eigenvalues.
+*> S cannot underestimate the true reciprocal condition number
+*> by more than a factor of sqrt(N). If M = 0 or N, S = 1.
+*> If JOB = 'N' or 'V', S is not referenced.
+*> \endverbatim
+*>
+*> \param[out] SEP
+*> \verbatim
+*> SEP is DOUBLE PRECISION
+*> If JOB = 'V' or 'B', SEP is the estimated reciprocal
+*> condition number of the specified invariant subspace. If
+*> M = 0 or N, SEP = norm(T).
+*> If JOB = 'N' or 'E', SEP is not referenced.
+*> \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.
+*> If JOB = 'N', LWORK >= 1;
+*> if JOB = 'E', LWORK = max(1,M*(N-M));
+*> if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
+*>
+*> 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
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup complex16OTHERcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> ZTRSEN first collects the selected eigenvalues by computing a unitary
+*> transformation Z to move them to the top left corner of T. In other
+*> words, the selected eigenvalues are the eigenvalues of T11 in:
+*>
+*> Z**H * T * Z = ( T11 T12 ) n1
+*> ( 0 T22 ) n2
+*> n1 n2
+*>
+*> where N = n1+n2. The first
+*> n1 columns of Z span the specified invariant subspace of T.
+*>
+*> If T has been obtained from the Schur factorization of a matrix
+*> A = Q*T*Q**H, then the reordered Schur factorization of A is given by
+*> A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the
+*> corresponding invariant subspace of A.
+*>
+*> The reciprocal condition number of the average of the eigenvalues of
+*> T11 may be returned in S. S lies between 0 (very badly conditioned)
+*> and 1 (very well conditioned). It is computed as follows. First we
+*> compute R so that
+*>
+*> P = ( I R ) n1
+*> ( 0 0 ) n2
+*> n1 n2
+*>
+*> is the projector on the invariant subspace associated with T11.
+*> R is the solution of the Sylvester equation:
+*>
+*> T11*R - R*T22 = T12.
+*>
+*> Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
+*> the two-norm of M. Then S is computed as the lower bound
+*>
+*> (1 + F-norm(R)**2)**(-1/2)
+*>
+*> on the reciprocal of 2-norm(P), the true reciprocal condition number.
+*> S cannot underestimate 1 / 2-norm(P) by more than a factor of
+*> sqrt(N).
+*>
+*> An approximate error bound for the computed average of the
+*> eigenvalues of T11 is
+*>
+*> EPS * norm(T) / S
+*>
+*> where EPS is the machine precision.
+*>
+*> The reciprocal condition number of the right invariant subspace
+*> spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
+*> SEP is defined as the separation of T11 and T22:
+*>
+*> sep( T11, T22 ) = sigma-min( C )
+*>
+*> where sigma-min(C) is the smallest singular value of the
+*> n1*n2-by-n1*n2 matrix
+*>
+*> C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
+*>
+*> I(m) is an m by m identity matrix, and kprod denotes the Kronecker
+*> product. We estimate sigma-min(C) by the reciprocal of an estimate of
+*> the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
+*> cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
+*>
+*> When SEP is small, small changes in T can cause large changes in
+*> the invariant subspace. An approximate bound on the maximum angular
+*> error in the computed right invariant subspace is
+*>
+*> EPS * norm(T) / SEP
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
$ SEP, 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 2011 --
-*
-* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPQ, JOB
@@ -18,171 +279,6 @@
COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* ZTRSEN reorders the Schur factorization of a complex matrix
-* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
-* the leading positions on the diagonal of the upper triangular matrix
-* T, and the leading columns of Q form an orthonormal basis of the
-* corresponding right invariant subspace.
-*
-* Optionally the routine computes the reciprocal condition numbers of
-* the cluster of eigenvalues and/or the invariant subspace.
-*
-* Arguments
-* =========
-*
-* JOB (input) CHARACTER*1
-* Specifies whether condition numbers are required for the
-* cluster of eigenvalues (S) or the invariant subspace (SEP):
-* = 'N': none;
-* = 'E': for eigenvalues only (S);
-* = 'V': for invariant subspace only (SEP);
-* = 'B': for both eigenvalues and invariant subspace (S and
-* SEP).
-*
-* COMPQ (input) CHARACTER*1
-* = 'V': update the matrix Q of Schur vectors;
-* = 'N': do not update Q.
-*
-* SELECT (input) LOGICAL array, dimension (N)
-* SELECT specifies the eigenvalues in the selected cluster. To
-* select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
-*
-* N (input) INTEGER
-* The order of the matrix T. N >= 0.
-*
-* T (input/output) COMPLEX*16 array, dimension (LDT,N)
-* On entry, the upper triangular matrix T.
-* On exit, T is overwritten by the reordered matrix T, with the
-* selected eigenvalues as the leading diagonal elements.
-*
-* LDT (input) INTEGER
-* The leading dimension of the array T. LDT >= max(1,N).
-*
-* Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
-* On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
-* On exit, if COMPQ = 'V', Q has been postmultiplied by the
-* unitary transformation matrix which reorders T; the leading M
-* columns of Q form an orthonormal basis for the specified
-* invariant subspace.
-* If COMPQ = 'N', Q is not referenced.
-*
-* LDQ (input) INTEGER
-* The leading dimension of the array Q.
-* LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
-*
-* W (output) COMPLEX*16 array, dimension (N)
-* The reordered eigenvalues of T, in the same order as they
-* appear on the diagonal of T.
-*
-* M (output) INTEGER
-* The dimension of the specified invariant subspace.
-* 0 <= M <= N.
-*
-* S (output) DOUBLE PRECISION
-* If JOB = 'E' or 'B', S is a lower bound on the reciprocal
-* condition number for the selected cluster of eigenvalues.
-* S cannot underestimate the true reciprocal condition number
-* by more than a factor of sqrt(N). If M = 0 or N, S = 1.
-* If JOB = 'N' or 'V', S is not referenced.
-*
-* SEP (output) DOUBLE PRECISION
-* If JOB = 'V' or 'B', SEP is the estimated reciprocal
-* condition number of the specified invariant subspace. If
-* M = 0 or N, SEP = norm(T).
-* If JOB = 'N' or 'E', SEP is not referenced.
-*
-* 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.
-* If JOB = 'N', LWORK >= 1;
-* if JOB = 'E', LWORK = max(1,M*(N-M));
-* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
-*
-* 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
-*
-* Further Details
-* ===============
-*
-* ZTRSEN first collects the selected eigenvalues by computing a unitary
-* transformation Z to move them to the top left corner of T. In other
-* words, the selected eigenvalues are the eigenvalues of T11 in:
-*
-* Z**H * T * Z = ( T11 T12 ) n1
-* ( 0 T22 ) n2
-* n1 n2
-*
-* where N = n1+n2. The first
-* n1 columns of Z span the specified invariant subspace of T.
-*
-* If T has been obtained from the Schur factorization of a matrix
-* A = Q*T*Q**H, then the reordered Schur factorization of A is given by
-* A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the
-* corresponding invariant subspace of A.
-*
-* The reciprocal condition number of the average of the eigenvalues of
-* T11 may be returned in S. S lies between 0 (very badly conditioned)
-* and 1 (very well conditioned). It is computed as follows. First we
-* compute R so that
-*
-* P = ( I R ) n1
-* ( 0 0 ) n2
-* n1 n2
-*
-* is the projector on the invariant subspace associated with T11.
-* R is the solution of the Sylvester equation:
-*
-* T11*R - R*T22 = T12.
-*
-* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
-* the two-norm of M. Then S is computed as the lower bound
-*
-* (1 + F-norm(R)**2)**(-1/2)
-*
-* on the reciprocal of 2-norm(P), the true reciprocal condition number.
-* S cannot underestimate 1 / 2-norm(P) by more than a factor of
-* sqrt(N).
-*
-* An approximate error bound for the computed average of the
-* eigenvalues of T11 is
-*
-* EPS * norm(T) / S
-*
-* where EPS is the machine precision.
-*
-* The reciprocal condition number of the right invariant subspace
-* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
-* SEP is defined as the separation of T11 and T22:
-*
-* sep( T11, T22 ) = sigma-min( C )
-*
-* where sigma-min(C) is the smallest singular value of the
-* n1*n2-by-n1*n2 matrix
-*
-* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
-*
-* I(m) is an m by m identity matrix, and kprod denotes the Kronecker
-* product. We estimate sigma-min(C) by the reciprocal of an estimate of
-* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
-* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
-*
-* When SEP is small, small changes in T can cause large changes in
-* the invariant subspace. An approximate bound on the maximum angular
-* error in the computed right invariant subspace is
-*
-* EPS * norm(T) / SEP
-*
* =====================================================================
*
* .. Parameters ..