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|
*> \brief \b DTRSEN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSEN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsen.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsen.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsen.f">
*> [TXT]</a>
*> \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 elements 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 April 2012
*
*> \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 computational routine (version 3.4.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. 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( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
$ WANTSP
INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
$ NN
DOUBLE PRECISION EST, RNORM, SCALE
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE
EXTERNAL LSAME, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
WANTQ = LSAME( COMPQ, 'V' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -8
ELSE
*
* Set M to the dimension of the specified invariant subspace,
* and test LWORK and LIWORK.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( T( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
*
N1 = M
N2 = N - M
NN = N1*N2
*
IF( WANTSP ) THEN
LWMIN = MAX( 1, 2*NN )
LIWMIN = MAX( 1, NN )
ELSE IF( LSAME( JOB, 'N' ) ) THEN
LWMIN = MAX( 1, N )
LIWMIN = 1
ELSE IF( LSAME( JOB, 'E' ) ) THEN
LWMIN = MAX( 1, NN )
LIWMIN = 1
END IF
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTS )
$ S = ONE
IF( WANTSP )
$ SEP = DLANGE( '1', N, N, T, LDT, WORK )
GO TO 40
END IF
*
* Collect the selected blocks at the top-left corner of T.
*
KS = 0
PAIR = .FALSE.
DO 20 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
SWAP = SELECT( K )
IF( K.LT.N ) THEN
IF( T( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
SWAP = SWAP .OR. SELECT( K+1 )
END IF
END IF
IF( SWAP ) THEN
KS = KS + 1
*
* Swap the K-th block to position KS.
*
IERR = 0
KK = K
IF( K.NE.KS )
$ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
$ IERR )
IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
* Blocks too close to swap: exit.
*
INFO = 1
IF( WANTS )
$ S = ZERO
IF( WANTSP )
$ SEP = ZERO
GO TO 40
END IF
IF( PAIR )
$ KS = KS + 1
END IF
END IF
20 CONTINUE
*
IF( WANTS ) THEN
*
* Solve Sylvester equation for R:
*
* T11*R - R*T22 = scale*T12
*
CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
$ LDT, WORK, N1, SCALE, IERR )
*
* Estimate the reciprocal of the condition number of the cluster
* of eigenvalues.
*
RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
IF( RNORM.EQ.ZERO ) THEN
S = ONE
ELSE
S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
$ SQRT( RNORM ) )
END IF
END IF
*
IF( WANTSP ) THEN
*
* Estimate sep(T11,T22).
*
EST = ZERO
KASE = 0
30 CONTINUE
CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve T11*R - R*T22 = scale*X.
*
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
ELSE
*
* Solve T11**T*R - R*T22**T = scale*X.
*
CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
END IF
GO TO 30
END IF
*
SEP = SCALE / EST
END IF
*
40 CONTINUE
*
* Store the output eigenvalues in WR and WI.
*
DO 50 K = 1, N
WR( K ) = T( K, K )
WI( K ) = ZERO
50 CONTINUE
DO 60 K = 1, N - 1
IF( T( K+1, K ).NE.ZERO ) THEN
WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
$ SQRT( ABS( T( K+1, K ) ) )
WI( K+1 ) = -WI( K )
END IF
60 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DTRSEN
*
END
|
