dtrsen.f

          SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
         $                   M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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( * )
    *     ..
    *
    *  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*Z = ( T11 T12 ) n1
    *                         (  0  T22 ) n2
    *                            n1  n2
    *
    *  where N = n1+n2 and Z' 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', then the reordered real Schur factorization of A is given
    *  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', 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 ..
          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'*R - R*T22' = 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