shseqr.f

          SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
         $                   LDZ, WORK, LWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
          CHARACTER          COMPZ, JOB
    *     ..
    *     .. Array Arguments ..
          REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
         $                   Z( LDZ, * )
    *     ..
    *     Purpose
    *     =======
    *
    *     SHSEQR computes the eigenvalues of a Hessenberg matrix H
    *     and, optionally, the matrices T and Z from the Schur decomposition
    *     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
    *     Schur form), and Z is the orthogonal matrix of Schur vectors.
    *
    *     Optionally Z may be postmultiplied into an input orthogonal
    *     matrix Q so that this routine can give the Schur factorization
    *     of a matrix A which has been reduced to the Hessenberg form H
    *     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
    *
    *     Arguments
    *     =========
    *
    *     JOB   (input) CHARACTER*1
    *           = 'E':  compute eigenvalues only;
    *           = 'S':  compute eigenvalues and the Schur form T.
    *
    *     COMPZ (input) CHARACTER*1
    *           = 'N':  no Schur vectors are computed;
    *           = 'I':  Z is initialized to the unit matrix and the matrix Z
    *                   of Schur vectors of H is returned;
    *           = 'V':  Z must contain an orthogonal matrix Q on entry, and
    *                   the product Q*Z is returned.
    *
    *     N     (input) INTEGER
    *           The order of the matrix H.  N .GE. 0.
    *
    *     ILO   (input) INTEGER
    *     IHI   (input) INTEGER
    *           It is assumed that H is already upper triangular in rows
    *           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
    *           set by a previous call to SGEBAL, and then passed to SGEHRD
    *           when the matrix output by SGEBAL is reduced to Hessenberg
    *           form. Otherwise ILO and IHI should be set to 1 and N
    *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
    *           If N = 0, then ILO = 1 and IHI = 0.
    *
    *     H     (input/output) REAL array, dimension (LDH,N)
    *           On entry, the upper Hessenberg matrix H.
    *           On exit, if INFO = 0 and JOB = 'S', then H contains the
    *           upper quasi-triangular matrix T from the Schur decomposition
    *           (the Schur form); 2-by-2 diagonal blocks (corresponding to
    *           complex conjugate pairs of eigenvalues) are returned in
    *           standard form, with H(i,i) = H(i+1,i+1) and
    *           H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
    *           contents of H are unspecified on exit.  (The output value of
    *           H when INFO.GT.0 is given under the description of INFO
    *           below.)
    *
    *           Unlike earlier versions of SHSEQR, this subroutine may
    *           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
    *           or j = IHI+1, IHI+2, ... N.
    *
    *     LDH   (input) INTEGER
    *           The leading dimension of the array H. LDH .GE. max(1,N).
    *
    *     WR    (output) REAL array, dimension (N)
    *     WI    (output) REAL array, dimension (N)
    *           The real and imaginary parts, respectively, of the computed
    *           eigenvalues. If two eigenvalues are computed as a complex
    *           conjugate pair, they are stored in consecutive elements of
    *           WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
    *           WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in
    *           the same order as on the diagonal of the Schur form returned
    *           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
    *           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
    *           WI(i+1) = -WI(i).
    *
    *     Z     (input/output) REAL array, dimension (LDZ,N)
    *           If COMPZ = 'N', Z is not referenced.
    *           If COMPZ = 'I', on entry Z need not be set and on exit,
    *           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
    *           vectors of H.  If COMPZ = 'V', on entry Z must contain an
    *           N-by-N matrix Q, which is assumed to be equal to the unit
    *           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
    *           if INFO = 0, Z contains Q*Z.
    *           Normally Q is the orthogonal matrix generated by SORGHR
    *           after the call to SGEHRD which formed the Hessenberg matrix
    *           H. (The output value of Z when INFO.GT.0 is given under
    *           the description of INFO below.)
    *
    *     LDZ   (input) INTEGER
    *           The leading dimension of the array Z.  if COMPZ = 'I' or
    *           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.
    *
    *     WORK  (workspace/output) REAL array, dimension (LWORK)
    *           On exit, if INFO = 0, WORK(1) returns an estimate of
    *           the optimal value for LWORK.
    *
    *     LWORK (input) INTEGER
    *           The dimension of the array WORK.  LWORK .GE. max(1,N)
    *           is sufficient, but LWORK typically as large as 6*N may
    *           be required for optimal performance.  A workspace query
    *           to determine the optimal workspace size is recommended.
    *
    *           If LWORK = -1, then SHSEQR does a workspace query.
    *           In this case, SHSEQR checks the input parameters and
    *           estimates the optimal workspace size for the given
    *           values of N, ILO and IHI.  The estimate is returned
    *           in WORK(1).  No error message related to LWORK is
    *           issued by XERBLA.  Neither H nor Z are accessed.
    *
    *
    *     INFO  (output) INTEGER
    *             =  0:  successful exit
    *           .LT. 0:  if INFO = -i, the i-th argument had an illegal
    *                    value
    *           .GT. 0:  if INFO = i, SHSEQR failed to compute all of
    *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
    *                and WI contain those eigenvalues which have been
    *                successfully computed.  (Failures are rare.)
    *
    *                If INFO .GT. 0 and JOB = 'E', then on exit, the
    *                remaining unconverged eigenvalues are the eigen-
    *                values of the upper Hessenberg matrix rows and
    *                columns ILO through INFO of the final, output
    *                value of H.
    *
    *                If INFO .GT. 0 and JOB   = 'S', then on exit
    *
    *           (*)  (initial value of H)*U  = U*(final value of H)
    *
    *                where U is an orthogonal matrix.  The final
    *                value of H is upper Hessenberg and quasi-triangular
    *                in rows and columns INFO+1 through IHI.
    *
    *                If INFO .GT. 0 and COMPZ = 'V', then on exit
    *
    *                  (final value of Z)  =  (initial value of Z)*U
    *
    *                where U is the orthogonal matrix in (*) (regard-
    *                less of the value of JOB.)
    *
    *                If INFO .GT. 0 and COMPZ = 'I', then on exit
    *                      (final value of Z)  = U
    *                where U is the orthogonal matrix in (*) (regard-
    *                less of the value of JOB.)
    *
    *                If INFO .GT. 0 and COMPZ = 'N', then Z is not
    *                accessed.
    *
    *     ================================================================
    *             Default values supplied by
    *             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
    *             It is suggested that these defaults be adjusted in order
    *             to attain best performance in each particular
    *             computational environment.
    *
    *            ISPEC=1:  The SLAHQR vs SLAQR0 crossover point.
    *                      Default: 75. (Must be at least 11.)
    *
    *            ISPEC=2:  Recommended deflation window size.
    *                      This depends on ILO, IHI and NS.  NS is the
    *                      number of simultaneous shifts returned
    *                      by ILAENV(ISPEC=4).  (See ISPEC=4 below.)
    *                      The default for (IHI-ILO+1).LE.500 is NS.
    *                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
    *
    *            ISPEC=3:  Nibble crossover point. (See ILAENV for
    *                      details.)  Default: 14% of deflation window
    *                      size.
    *
    *            ISPEC=4:  Number of simultaneous shifts, NS, in
    *                      a multi-shift QR iteration.
    *
    *                      If IHI-ILO+1 is ...
    *
    *                      greater than      ...but less    ... the
    *                      or equal to ...      than        default is
    *
    *                           1               30          NS -   2(+)
    *                          30               60          NS -   4(+)
    *                          60              150          NS =  10(+)
    *                         150              590          NS =  **
    *                         590             3000          NS =  64
    *                        3000             6000          NS = 128
    *                        6000             infinity      NS = 256
    *
    *                  (+)  By default some or all matrices of this order
    *                       are passed to the implicit double shift routine
    *                       SLAHQR and NS is ignored.  See ISPEC=1 above
    *                       and comments in IPARM for details.
    *
    *                       The asterisks (**) indicate an ad-hoc
    *                       function of N increasing from 10 to 64.
    *
    *            ISPEC=5:  Select structured matrix multiply.
    *                      (See ILAENV for details.) Default: 3.
    *
    *     ================================================================
    *     Based on contributions by
    *        Karen Braman and Ralph Byers, Department of Mathematics,
    *        University of Kansas, USA
    *
    *     ================================================================
    *     References:
    *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
    *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
    *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
    *       929--947, 2002.
    *
    *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
    *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
    *       of Matrix Analysis, volume 23, pages 948--973, 2002.
    *
    *     ================================================================
    *     .. Parameters ..
    *
    *     ==== Matrices of order NTINY or smaller must be processed by
    *     .    SLAHQR because of insufficient subdiagonal scratch space.
    *     .    (This is a hard limit.) ====
    *
    *     ==== NL allocates some local workspace to help small matrices
    *     .    through a rare SLAHQR failure.  NL .GT. NTINY = 11 is
    *     .    required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom-
    *     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
    *     .    allows up to six simultaneous shifts and a 16-by-16
    *     .    deflation window.  ====
    *
          INTEGER            NTINY
          PARAMETER          ( NTINY = 11 )
          INTEGER            NL
          PARAMETER          ( NL = 49 )
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
    *     ..
    *     .. Local Arrays ..
          REAL               HL( NL, NL ), WORKL( NL )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, KBOT, NMIN
          LOGICAL            INITZ, LQUERY, WANTT, WANTZ
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          LOGICAL            LSAME
          EXTERNAL           ILAENV, LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN, REAL
    *     ..
    *     .. Executable Statements ..
    *
    *     ==== Decode and check the input parameters. ====
    *
          WANTT = LSAME( JOB, 'S' )
          INITZ = LSAME( COMPZ, 'I' )
          WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
          WORK( 1 ) = REAL( MAX( 1, N ) )
          LQUERY = LWORK.EQ.-1
    *
          INFO = 0
          IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
             INFO = -1
          ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
             INFO = -4
          ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
             INFO = -5
          ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
             INFO = -11
          ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -13
          END IF
    *
          IF( INFO.NE.0 ) THEN
    *
    *        ==== Quick return in case of invalid argument. ====
    *
             CALL XERBLA( 'SHSEQR', -INFO )
             RETURN
    *
          ELSE IF( N.EQ.0 ) THEN
    *
    *        ==== Quick return in case N = 0; nothing to do. ====
    *
             RETURN
    *
          ELSE IF( LQUERY ) THEN
    *
    *        ==== Quick return in case of a workspace query ====
    *
             CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
         $                IHI, Z, LDZ, WORK, LWORK, INFO )
    *        ==== Ensure reported workspace size is backward-compatible with
    *        .    previous LAPACK versions. ====
             WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
             RETURN
    *
          ELSE
    *
    *        ==== copy eigenvalues isolated by SGEBAL ====
    *
             DO 10 I = 1, ILO - 1
                WR( I ) = H( I, I )
                WI( I ) = ZERO
       10    CONTINUE
             DO 20 I = IHI + 1, N
                WR( I ) = H( I, I )
                WI( I ) = ZERO
       20    CONTINUE
    *
    *        ==== Initialize Z, if requested ====
    *
             IF( INITZ )
         $      CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
    *
    *        ==== Quick return if possible ====
    *
             IF( ILO.EQ.IHI ) THEN
                WR( ILO ) = H( ILO, ILO )
                WI( ILO ) = ZERO
                RETURN
             END IF
    *
    *        ==== SLAHQR/SLAQR0 crossover point ====
    *
             NMIN = ILAENV( 1, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, ILO,
         $          IHI, LWORK )
             NMIN = MAX( NTINY, NMIN )
    *
    *        ==== SLAQR0 for big matrices; SLAHQR for small ones ====
    *
             IF( N.GT.NMIN ) THEN
                CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
         $                   IHI, Z, LDZ, WORK, LWORK, INFO )
             ELSE
    *
    *           ==== Small matrix ====
    *
                CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
         $                   IHI, Z, LDZ, INFO )
    *
                IF( INFO.GT.0 ) THEN
    *
    *              ==== A rare SLAHQR failure!  SLAQR0 sometimes succeeds
    *              .    when SLAHQR fails. ====
    *
                   KBOT = INFO
    *
                   IF( N.GE.NL ) THEN
    *
    *                 ==== Larger matrices have enough subdiagonal scratch
    *                 .    space to call SLAQR0 directly. ====
    *
                      CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
         $                         WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
    *
                   ELSE
    *
    *                 ==== Tiny matrices don't have enough subdiagonal
    *                 .    scratch space to benefit from SLAQR0.  Hence,
    *                 .    tiny matrices must be copied into a larger
    *                 .    array before calling SLAQR0. ====
    *
                      CALL SLACPY( 'A', N, N, H, LDH, HL, NL )
                      HL( N+1, N ) = ZERO
                      CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
         $                         NL )
                      CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
         $                         WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
                      IF( WANTT .OR. INFO.NE.0 )
         $               CALL SLACPY( 'A', N, N, HL, NL, H, LDH )
                   END IF
                END IF
             END IF
    *
    *        ==== Clear out the trash, if necessary. ====
    *
             IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
         $      CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
    *
    *        ==== Ensure reported workspace size is backward-compatible with
    *        .    previous LAPACK versions. ====
    *
             WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
          END IF
    *
    *     ==== End of SHSEQR ====
    *
          END