--- rpl/lapack/lapack/dtrsen.f 2011/07/22 07:38:13 1.8
+++ rpl/lapack/lapack/dtrsen.f 2011/11/21 20:43:06 1.9
@@ -1,10 +1,322 @@
+*> \brief \b DTRSEN
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DTRSEN + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
+* M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER COMPQ, JOB
+* INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
+* DOUBLE PRECISION S, SEP
+* ..
+* .. Array Arguments ..
+* LOGICAL SELECT( * )
+* INTEGER IWORK( * )
+* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
+* $ WR( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DTRSEN reorders the real Schur factorization of a real matrix
+*> A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
+*> the leading diagonal blocks of the upper quasi-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.
+*>
+*> T must be in Schur canonical form (as returned by DHSEQR), that is,
+*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
+*> 2-by-2 diagonal block has its diagonal elemnts equal and its
+*> off-diagonal elements of opposite sign.
+*> \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 a real eigenvalue w(j), SELECT(j) must be set to
+*> .TRUE.. To select a complex conjugate pair of eigenvalues
+*> w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
+*> either SELECT(j) or SELECT(j+1) or both must be set to
+*> .TRUE.; a complex conjugate pair of eigenvalues must be
+*> either both included in the cluster or both excluded.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix T. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] T
+*> \verbatim
+*> T is DOUBLE PRECISION array, dimension (LDT,N)
+*> On entry, the upper quasi-triangular matrix T, in Schur
+*> canonical form.
+*> On exit, T is overwritten by the reordered matrix T, again in
+*> Schur canonical form, with the selected eigenvalues in the
+*> leading diagonal blocks.
+*> \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 DOUBLE PRECISION 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
+*> orthogonal 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] WR
+*> \verbatim
+*> WR is DOUBLE PRECISION array, dimension (N)
+*> \endverbatim
+*> \param[out] WI
+*> \verbatim
+*> WI is DOUBLE PRECISION array, dimension (N)
+*>
+*> The real and imaginary parts, respectively, of the reordered
+*> eigenvalues of T. The eigenvalues are stored in the same
+*> order as on the diagonal of T, with WR(i) = T(i,i) and, if
+*> T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
+*> WI(i+1) = -WI(i). Note that if a complex eigenvalue is
+*> sufficiently ill-conditioned, then its value may differ
+*> significantly from its value before reordering.
+*> \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 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.
+*> If JOB = 'N', LWORK >= max(1,N);
+*> 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] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
+*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
+*> \endverbatim
+*>
+*> \param[in] LIWORK
+*> \verbatim
+*> LIWORK is INTEGER
+*> The dimension of the array IWORK.
+*> If JOB = 'N' or 'E', LIWORK >= 1;
+*> if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
+*>
+*> If LIWORK = -1, then a workspace query is assumed; the
+*> routine only calculates the optimal size of the IWORK array,
+*> returns this value as the first entry of the IWORK array, and
+*> no error message related to LIWORK 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: reordering of T failed because some eigenvalues are too
+*> close to separate (the problem is very ill-conditioned);
+*> T may have been partially reordered, and WR and WI
+*> contain the eigenvalues in the same order as in T; S and
+*> SEP (if requested) are set to zero.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup doubleOTHERcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> DTRSEN first collects the selected eigenvalues by computing an
+*> orthogonal 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**T * T * Z = ( T11 T12 ) n1
+*> ( 0 T22 ) n2
+*> n1 n2
+*>
+*> where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
+*> of Z span the specified invariant subspace of T.
+*>
+*> If T has been obtained from the real Schur factorization of a matrix
+*> A = Q*T*Q**T, then the reordered real Schur factorization of A is given
+*> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
$ M, S, SEP, WORK, LWORK, IWORK, LIWORK, 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 --
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPQ, JOB
@@ -18,208 +330,6 @@
$ WR( * )
* ..
*
-* Purpose
-* =======
-*
-* DTRSEN reorders the real Schur factorization of a real matrix
-* A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
-* the leading diagonal blocks of the upper quasi-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.
-*
-* T must be in Schur canonical form (as returned by DHSEQR), that is,
-* block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
-* 2-by-2 diagonal block has its diagonal elemnts equal and its
-* off-diagonal elements of opposite sign.
-*
-* 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 a real eigenvalue w(j), SELECT(j) must be set to
-* .TRUE.. To select a complex conjugate pair of eigenvalues
-* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
-* either SELECT(j) or SELECT(j+1) or both must be set to
-* .TRUE.; a complex conjugate pair of eigenvalues must be
-* either both included in the cluster or both excluded.
-*
-* N (input) INTEGER
-* The order of the matrix T. N >= 0.
-*
-* T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
-* On entry, the upper quasi-triangular matrix T, in Schur
-* canonical form.
-* On exit, T is overwritten by the reordered matrix T, again in
-* Schur canonical form, with the selected eigenvalues in the
-* leading diagonal blocks.
-*
-* LDT (input) INTEGER
-* The leading dimension of the array T. LDT >= max(1,N).
-*
-* Q (input/output) DOUBLE PRECISION 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
-* orthogonal 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.
-*
-* WR (output) DOUBLE PRECISION array, dimension (N)
-* WI (output) DOUBLE PRECISION array, dimension (N)
-* The real and imaginary parts, respectively, of the reordered
-* eigenvalues of T. The eigenvalues are stored in the same
-* order as on the diagonal of T, with WR(i) = T(i,i) and, if
-* T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
-* WI(i+1) = -WI(i). Note that if a complex eigenvalue is
-* sufficiently ill-conditioned, then its value may differ
-* significantly from its value before reordering.
-*
-* 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) 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.
-* If JOB = 'N', LWORK >= max(1,N);
-* 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.
-*
-* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
-* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
-*
-* LIWORK (input) INTEGER
-* The dimension of the array IWORK.
-* If JOB = 'N' or 'E', LIWORK >= 1;
-* if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
-*
-* If LIWORK = -1, then a workspace query is assumed; the
-* routine only calculates the optimal size of the IWORK array,
-* returns this value as the first entry of the IWORK array, and
-* no error message related to LIWORK is issued by XERBLA.
-*
-* INFO (output) INTEGER
-* = 0: successful exit
-* < 0: if INFO = -i, the i-th argument had an illegal value
-* = 1: reordering of T failed because some eigenvalues are too
-* close to separate (the problem is very ill-conditioned);
-* T may have been partially reordered, and WR and WI
-* contain the eigenvalues in the same order as in T; S and
-* SEP (if requested) are set to zero.
-*
-* Further Details
-* ===============
-*
-* DTRSEN first collects the selected eigenvalues by computing an
-* orthogonal 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**T * T * Z = ( T11 T12 ) n1
-* ( 0 T22 ) n2
-* n1 n2
-*
-* where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
-* of Z span the specified invariant subspace of T.
-*
-* If T has been obtained from the real Schur factorization of a matrix
-* A = Q*T*Q**T, then the reordered real Schur factorization of A is given
-* by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 ..