dtrsen.f

*> \brief \b DTRSEN
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  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.
*
*> \ingroup trsen
*
*> \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 --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. 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