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Mise à jour de lapack vers la version 3.3.0.

    1:       SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
    2:      $                   WORK, LWORK, INFO )
    3: *
    4: *  -- LAPACK computational routine (version 3.2.2) --
    5: *     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
    6: *     June 2010
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
   10:       CHARACTER          COMPZ, JOB
   11: *     ..
   12: *     .. Array Arguments ..
   13:       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
   14: *     ..
   15: *     Purpose
   16: *     =======
   17: *
   18: *     ZHSEQR computes the eigenvalues of a Hessenberg matrix H
   19: *     and, optionally, the matrices T and Z from the Schur decomposition
   20: *     H = Z T Z**H, where T is an upper triangular matrix (the
   21: *     Schur form), and Z is the unitary matrix of Schur vectors.
   22: *
   23: *     Optionally Z may be postmultiplied into an input unitary
   24: *     matrix Q so that this routine can give the Schur factorization
   25: *     of a matrix A which has been reduced to the Hessenberg form H
   26: *     by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
   27: *
   28: *     Arguments
   29: *     =========
   30: *
   31: *     JOB   (input) CHARACTER*1
   32: *           = 'E':  compute eigenvalues only;
   33: *           = 'S':  compute eigenvalues and the Schur form T.
   34: *
   35: *     COMPZ (input) CHARACTER*1
   36: *           = 'N':  no Schur vectors are computed;
   37: *           = 'I':  Z is initialized to the unit matrix and the matrix Z
   38: *                   of Schur vectors of H is returned;
   39: *           = 'V':  Z must contain an unitary matrix Q on entry, and
   40: *                   the product Q*Z is returned.
   41: *
   42: *     N     (input) INTEGER
   43: *           The order of the matrix H.  N .GE. 0.
   44: *
   45: *     ILO   (input) INTEGER
   46: *     IHI   (input) INTEGER
   47: *           It is assumed that H is already upper triangular in rows
   48: *           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
   49: *           set by a previous call to ZGEBAL, and then passed to ZGEHRD
   50: *           when the matrix output by ZGEBAL is reduced to Hessenberg
   51: *           form. Otherwise ILO and IHI should be set to 1 and N
   52: *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
   53: *           If N = 0, then ILO = 1 and IHI = 0.
   54: *
   55: *     H     (input/output) COMPLEX*16 array, dimension (LDH,N)
   56: *           On entry, the upper Hessenberg matrix H.
   57: *           On exit, if INFO = 0 and JOB = 'S', H contains the upper
   58: *           triangular matrix T from the Schur decomposition (the
   59: *           Schur form). If INFO = 0 and JOB = 'E', the contents of
   60: *           H are unspecified on exit.  (The output value of H when
   61: *           INFO.GT.0 is given under the description of INFO below.)
   62: *
   63: *           Unlike earlier versions of ZHSEQR, this subroutine may
   64: *           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
   65: *           or j = IHI+1, IHI+2, ... N.
   66: *
   67: *     LDH   (input) INTEGER
   68: *           The leading dimension of the array H. LDH .GE. max(1,N).
   69: *
   70: *     W        (output) COMPLEX*16 array, dimension (N)
   71: *           The computed eigenvalues. If JOB = 'S', the eigenvalues are
   72: *           stored in the same order as on the diagonal of the Schur
   73: *           form returned in H, with W(i) = H(i,i).
   74: *
   75: *     Z     (input/output) COMPLEX*16 array, dimension (LDZ,N)
   76: *           If COMPZ = 'N', Z is not referenced.
   77: *           If COMPZ = 'I', on entry Z need not be set and on exit,
   78: *           if INFO = 0, Z contains the unitary matrix Z of the Schur
   79: *           vectors of H.  If COMPZ = 'V', on entry Z must contain an
   80: *           N-by-N matrix Q, which is assumed to be equal to the unit
   81: *           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
   82: *           if INFO = 0, Z contains Q*Z.
   83: *           Normally Q is the unitary matrix generated by ZUNGHR
   84: *           after the call to ZGEHRD which formed the Hessenberg matrix
   85: *           H. (The output value of Z when INFO.GT.0 is given under
   86: *           the description of INFO below.)
   87: *
   88: *     LDZ   (input) INTEGER
   89: *           The leading dimension of the array Z.  if COMPZ = 'I' or
   90: *           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.
   91: *
   92: *     WORK  (workspace/output) COMPLEX*16 array, dimension (LWORK)
   93: *           On exit, if INFO = 0, WORK(1) returns an estimate of
   94: *           the optimal value for LWORK.
   95: *
   96: *     LWORK (input) INTEGER
   97: *           The dimension of the array WORK.  LWORK .GE. max(1,N)
   98: *           is sufficient and delivers very good and sometimes
   99: *           optimal performance.  However, LWORK as large as 11*N
  100: *           may be required for optimal performance.  A workspace
  101: *           query is recommended to determine the optimal workspace
  102: *           size.
  103: *
  104: *           If LWORK = -1, then ZHSEQR does a workspace query.
  105: *           In this case, ZHSEQR checks the input parameters and
  106: *           estimates the optimal workspace size for the given
  107: *           values of N, ILO and IHI.  The estimate is returned
  108: *           in WORK(1).  No error message related to LWORK is
  109: *           issued by XERBLA.  Neither H nor Z are accessed.
  110: *
  111: *
  112: *     INFO  (output) INTEGER
  113: *             =  0:  successful exit
  114: *           .LT. 0:  if INFO = -i, the i-th argument had an illegal
  115: *                    value
  116: *           .GT. 0:  if INFO = i, ZHSEQR failed to compute all of
  117: *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
  118: *                and WI contain those eigenvalues which have been
  119: *                successfully computed.  (Failures are rare.)
  120: *
  121: *                If INFO .GT. 0 and JOB = 'E', then on exit, the
  122: *                remaining unconverged eigenvalues are the eigen-
  123: *                values of the upper Hessenberg matrix rows and
  124: *                columns ILO through INFO of the final, output
  125: *                value of H.
  126: *
  127: *                If INFO .GT. 0 and JOB   = 'S', then on exit
  128: *
  129: *           (*)  (initial value of H)*U  = U*(final value of H)
  130: *
  131: *                where U is a unitary matrix.  The final
  132: *                value of  H is upper Hessenberg and triangular in
  133: *                rows and columns INFO+1 through IHI.
  134: *
  135: *                If INFO .GT. 0 and COMPZ = 'V', then on exit
  136: *
  137: *                  (final value of Z)  =  (initial value of Z)*U
  138: *
  139: *                where U is the unitary matrix in (*) (regard-
  140: *                less of the value of JOB.)
  141: *
  142: *                If INFO .GT. 0 and COMPZ = 'I', then on exit
  143: *                      (final value of Z)  = U
  144: *                where U is the unitary matrix in (*) (regard-
  145: *                less of the value of JOB.)
  146: *
  147: *                If INFO .GT. 0 and COMPZ = 'N', then Z is not
  148: *                accessed.
  149: *
  150: *     ================================================================
  151: *             Default values supplied by
  152: *             ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
  153: *             It is suggested that these defaults be adjusted in order
  154: *             to attain best performance in each particular
  155: *             computational environment.
  156: *
  157: *            ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
  158: *                      Default: 75. (Must be at least 11.)
  159: *
  160: *            ISPEC=13: Recommended deflation window size.
  161: *                      This depends on ILO, IHI and NS.  NS is the
  162: *                      number of simultaneous shifts returned
  163: *                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
  164: *                      The default for (IHI-ILO+1).LE.500 is NS.
  165: *                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
  166: *
  167: *            ISPEC=14: Nibble crossover point. (See IPARMQ for
  168: *                      details.)  Default: 14% of deflation window
  169: *                      size.
  170: *
  171: *            ISPEC=15: Number of simultaneous shifts in a multishift
  172: *                      QR iteration.
  173: *
  174: *                      If IHI-ILO+1 is ...
  175: *
  176: *                      greater than      ...but less    ... the
  177: *                      or equal to ...      than        default is
  178: *
  179: *                           1               30          NS =   2(+)
  180: *                          30               60          NS =   4(+)
  181: *                          60              150          NS =  10(+)
  182: *                         150              590          NS =  **
  183: *                         590             3000          NS =  64
  184: *                        3000             6000          NS = 128
  185: *                        6000             infinity      NS = 256
  186: *
  187: *                  (+)  By default some or all matrices of this order
  188: *                       are passed to the implicit double shift routine
  189: *                       ZLAHQR and this parameter is ignored.  See
  190: *                       ISPEC=12 above and comments in IPARMQ for
  191: *                       details.
  192: *
  193: *                 (**)  The asterisks (**) indicate an ad-hoc
  194: *                       function of N increasing from 10 to 64.
  195: *
  196: *            ISPEC=16: Select structured matrix multiply.
  197: *                      If the number of simultaneous shifts (specified
  198: *                      by ISPEC=15) is less than 14, then the default
  199: *                      for ISPEC=16 is 0.  Otherwise the default for
  200: *                      ISPEC=16 is 2.
  201: *
  202: *     ================================================================
  203: *     Based on contributions by
  204: *        Karen Braman and Ralph Byers, Department of Mathematics,
  205: *        University of Kansas, USA
  206: *
  207: *     ================================================================
  208: *     References:
  209: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  210: *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  211: *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  212: *       929--947, 2002.
  213: *
  214: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  215: *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
  216: *       of Matrix Analysis, volume 23, pages 948--973, 2002.
  217: *
  218: *     ================================================================
  219: *     .. Parameters ..
  220: *
  221: *     ==== Matrices of order NTINY or smaller must be processed by
  222: *     .    ZLAHQR because of insufficient subdiagonal scratch space.
  223: *     .    (This is a hard limit.) ====
  224:       INTEGER            NTINY
  225:       PARAMETER          ( NTINY = 11 )
  226: *
  227: *     ==== NL allocates some local workspace to help small matrices
  228: *     .    through a rare ZLAHQR failure.  NL .GT. NTINY = 11 is
  229: *     .    required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
  230: *     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
  231: *     .    allows up to six simultaneous shifts and a 16-by-16
  232: *     .    deflation window.  ====
  233:       INTEGER            NL
  234:       PARAMETER          ( NL = 49 )
  235:       COMPLEX*16         ZERO, ONE
  236:       PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
  237:      $                   ONE = ( 1.0d0, 0.0d0 ) )
  238:       DOUBLE PRECISION   RZERO
  239:       PARAMETER          ( RZERO = 0.0d0 )
  240: *     ..
  241: *     .. Local Arrays ..
  242:       COMPLEX*16         HL( NL, NL ), WORKL( NL )
  243: *     ..
  244: *     .. Local Scalars ..
  245:       INTEGER            KBOT, NMIN
  246:       LOGICAL            INITZ, LQUERY, WANTT, WANTZ
  247: *     ..
  248: *     .. External Functions ..
  249:       INTEGER            ILAENV
  250:       LOGICAL            LSAME
  251:       EXTERNAL           ILAENV, LSAME
  252: *     ..
  253: *     .. External Subroutines ..
  254:       EXTERNAL           XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET
  255: *     ..
  256: *     .. Intrinsic Functions ..
  257:       INTRINSIC          DBLE, DCMPLX, MAX, MIN
  258: *     ..
  259: *     .. Executable Statements ..
  260: *
  261: *     ==== Decode and check the input parameters. ====
  262: *
  263:       WANTT = LSAME( JOB, 'S' )
  264:       INITZ = LSAME( COMPZ, 'I' )
  265:       WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
  266:       WORK( 1 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO )
  267:       LQUERY = LWORK.EQ.-1
  268: *
  269:       INFO = 0
  270:       IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
  271:          INFO = -1
  272:       ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
  273:          INFO = -2
  274:       ELSE IF( N.LT.0 ) THEN
  275:          INFO = -3
  276:       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
  277:          INFO = -4
  278:       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
  279:          INFO = -5
  280:       ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
  281:          INFO = -7
  282:       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
  283:          INFO = -10
  284:       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
  285:          INFO = -12
  286:       END IF
  287: *
  288:       IF( INFO.NE.0 ) THEN
  289: *
  290: *        ==== Quick return in case of invalid argument. ====
  291: *
  292:          CALL XERBLA( 'ZHSEQR', -INFO )
  293:          RETURN
  294: *
  295:       ELSE IF( N.EQ.0 ) THEN
  296: *
  297: *        ==== Quick return in case N = 0; nothing to do. ====
  298: *
  299:          RETURN
  300: *
  301:       ELSE IF( LQUERY ) THEN
  302: *
  303: *        ==== Quick return in case of a workspace query ====
  304: *
  305:          CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z,
  306:      $                LDZ, WORK, LWORK, INFO )
  307: *        ==== Ensure reported workspace size is backward-compatible with
  308: *        .    previous LAPACK versions. ====
  309:          WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1,
  310:      $               N ) ) ), RZERO )
  311:          RETURN
  312: *
  313:       ELSE
  314: *
  315: *        ==== copy eigenvalues isolated by ZGEBAL ====
  316: *
  317:          IF( ILO.GT.1 )
  318:      $      CALL ZCOPY( ILO-1, H, LDH+1, W, 1 )
  319:          IF( IHI.LT.N )
  320:      $      CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 )
  321: *
  322: *        ==== Initialize Z, if requested ====
  323: *
  324:          IF( INITZ )
  325:      $      CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
  326: *
  327: *        ==== Quick return if possible ====
  328: *
  329:          IF( ILO.EQ.IHI ) THEN
  330:             W( ILO ) = H( ILO, ILO )
  331:             RETURN
  332:          END IF
  333: *
  334: *        ==== ZLAHQR/ZLAQR0 crossover point ====
  335: *
  336:          NMIN = ILAENV( 12, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
  337:      $          ILO, IHI, LWORK )
  338:          NMIN = MAX( NTINY, NMIN )
  339: *
  340: *        ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
  341: *
  342:          IF( N.GT.NMIN ) THEN
  343:             CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
  344:      $                   Z, LDZ, WORK, LWORK, INFO )
  345:          ELSE
  346: *
  347: *           ==== Small matrix ====
  348: *
  349:             CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
  350:      $                   Z, LDZ, INFO )
  351: *
  352:             IF( INFO.GT.0 ) THEN
  353: *
  354: *              ==== A rare ZLAHQR failure!  ZLAQR0 sometimes succeeds
  355: *              .    when ZLAHQR fails. ====
  356: *
  357:                KBOT = INFO
  358: *
  359:                IF( N.GE.NL ) THEN
  360: *
  361: *                 ==== Larger matrices have enough subdiagonal scratch
  362: *                 .    space to call ZLAQR0 directly. ====
  363: *
  364:                   CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W,
  365:      $                         ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
  366: *
  367:                ELSE
  368: *
  369: *                 ==== Tiny matrices don't have enough subdiagonal
  370: *                 .    scratch space to benefit from ZLAQR0.  Hence,
  371: *                 .    tiny matrices must be copied into a larger
  372: *                 .    array before calling ZLAQR0. ====
  373: *
  374:                   CALL ZLACPY( 'A', N, N, H, LDH, HL, NL )
  375:                   HL( N+1, N ) = ZERO
  376:                   CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
  377:      $                         NL )
  378:                   CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W,
  379:      $                         ILO, IHI, Z, LDZ, WORKL, NL, INFO )
  380:                   IF( WANTT .OR. INFO.NE.0 )
  381:      $               CALL ZLACPY( 'A', N, N, HL, NL, H, LDH )
  382:                END IF
  383:             END IF
  384:          END IF
  385: *
  386: *        ==== Clear out the trash, if necessary. ====
  387: *
  388:          IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
  389:      $      CALL ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
  390: *
  391: *        ==== Ensure reported workspace size is backward-compatible with
  392: *        .    previous LAPACK versions. ====
  393: *
  394:          WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ),
  395:      $               DBLE( WORK( 1 ) ) ), RZERO )
  396:       END IF
  397: *
  398: *     ==== End of ZHSEQR ====
  399: *
  400:       END