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Mise à jour globale de Lapack 3.2.2.

    1:       SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
    2:      $                   JPIV )
    3: *
    4: *  -- LAPACK auxiliary routine (version 3.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     November 2006
    8: *
    9: *     .. Scalar Arguments ..
   10:       INTEGER            IJOB, LDZ, N
   11:       DOUBLE PRECISION   RDSCAL, RDSUM
   12: *     ..
   13: *     .. Array Arguments ..
   14:       INTEGER            IPIV( * ), JPIV( * )
   15:       COMPLEX*16         RHS( * ), Z( LDZ, * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  ZLATDF computes the contribution to the reciprocal Dif-estimate
   22: *  by solving for x in Z * x = b, where b is chosen such that the norm
   23: *  of x is as large as possible. It is assumed that LU decomposition
   24: *  of Z has been computed by ZGETC2. On entry RHS = f holds the
   25: *  contribution from earlier solved sub-systems, and on return RHS = x.
   26: *
   27: *  The factorization of Z returned by ZGETC2 has the form
   28: *  Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
   29: *  triangular with unit diagonal elements and U is upper triangular.
   30: *
   31: *  Arguments
   32: *  =========
   33: *
   34: *  IJOB    (input) INTEGER
   35: *          IJOB = 2: First compute an approximative null-vector e
   36: *              of Z using ZGECON, e is normalized and solve for
   37: *              Zx = +-e - f with the sign giving the greater value of
   38: *              2-norm(x).  About 5 times as expensive as Default.
   39: *          IJOB .ne. 2: Local look ahead strategy where
   40: *              all entries of the r.h.s. b is choosen as either +1 or
   41: *              -1.  Default.
   42: *
   43: *  N       (input) INTEGER
   44: *          The number of columns of the matrix Z.
   45: *
   46: *  Z       (input) DOUBLE PRECISION array, dimension (LDZ, N)
   47: *          On entry, the LU part of the factorization of the n-by-n
   48: *          matrix Z computed by ZGETC2:  Z = P * L * U * Q
   49: *
   50: *  LDZ     (input) INTEGER
   51: *          The leading dimension of the array Z.  LDA >= max(1, N).
   52: *
   53: *  RHS     (input/output) DOUBLE PRECISION array, dimension (N).
   54: *          On entry, RHS contains contributions from other subsystems.
   55: *          On exit, RHS contains the solution of the subsystem with
   56: *          entries according to the value of IJOB (see above).
   57: *
   58: *  RDSUM   (input/output) DOUBLE PRECISION
   59: *          On entry, the sum of squares of computed contributions to
   60: *          the Dif-estimate under computation by ZTGSYL, where the
   61: *          scaling factor RDSCAL (see below) has been factored out.
   62: *          On exit, the corresponding sum of squares updated with the
   63: *          contributions from the current sub-system.
   64: *          If TRANS = 'T' RDSUM is not touched.
   65: *          NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
   66: *
   67: *  RDSCAL  (input/output) DOUBLE PRECISION
   68: *          On entry, scaling factor used to prevent overflow in RDSUM.
   69: *          On exit, RDSCAL is updated w.r.t. the current contributions
   70: *          in RDSUM.
   71: *          If TRANS = 'T', RDSCAL is not touched.
   72: *          NOTE: RDSCAL only makes sense when ZTGSY2 is called by
   73: *          ZTGSYL.
   74: *
   75: *  IPIV    (input) INTEGER array, dimension (N).
   76: *          The pivot indices; for 1 <= i <= N, row i of the
   77: *          matrix has been interchanged with row IPIV(i).
   78: *
   79: *  JPIV    (input) INTEGER array, dimension (N).
   80: *          The pivot indices; for 1 <= j <= N, column j of the
   81: *          matrix has been interchanged with column JPIV(j).
   82: *
   83: *  Further Details
   84: *  ===============
   85: *
   86: *  Based on contributions by
   87: *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   88: *     Umea University, S-901 87 Umea, Sweden.
   89: *
   90: *  This routine is a further developed implementation of algorithm
   91: *  BSOLVE in [1] using complete pivoting in the LU factorization.
   92: *
   93: *   [1]   Bo Kagstrom and Lars Westin,
   94: *         Generalized Schur Methods with Condition Estimators for
   95: *         Solving the Generalized Sylvester Equation, IEEE Transactions
   96: *         on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
   97: *
   98: *   [2]   Peter Poromaa,
   99: *         On Efficient and Robust Estimators for the Separation
  100: *         between two Regular Matrix Pairs with Applications in
  101: *         Condition Estimation. Report UMINF-95.05, Department of
  102: *         Computing Science, Umea University, S-901 87 Umea, Sweden,
  103: *         1995.
  104: *
  105: *  =====================================================================
  106: *
  107: *     .. Parameters ..
  108:       INTEGER            MAXDIM
  109:       PARAMETER          ( MAXDIM = 2 )
  110:       DOUBLE PRECISION   ZERO, ONE
  111:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  112:       COMPLEX*16         CONE
  113:       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
  114: *     ..
  115: *     .. Local Scalars ..
  116:       INTEGER            I, INFO, J, K
  117:       DOUBLE PRECISION   RTEMP, SCALE, SMINU, SPLUS
  118:       COMPLEX*16         BM, BP, PMONE, TEMP
  119: *     ..
  120: *     .. Local Arrays ..
  121:       DOUBLE PRECISION   RWORK( MAXDIM )
  122:       COMPLEX*16         WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
  123: *     ..
  124: *     .. External Subroutines ..
  125:       EXTERNAL           ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
  126:      $                   ZSCAL
  127: *     ..
  128: *     .. External Functions ..
  129:       DOUBLE PRECISION   DZASUM
  130:       COMPLEX*16         ZDOTC
  131:       EXTERNAL           DZASUM, ZDOTC
  132: *     ..
  133: *     .. Intrinsic Functions ..
  134:       INTRINSIC          ABS, DBLE, SQRT
  135: *     ..
  136: *     .. Executable Statements ..
  137: *
  138:       IF( IJOB.NE.2 ) THEN
  139: *
  140: *        Apply permutations IPIV to RHS
  141: *
  142:          CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
  143: *
  144: *        Solve for L-part choosing RHS either to +1 or -1.
  145: *
  146:          PMONE = -CONE
  147:          DO 10 J = 1, N - 1
  148:             BP = RHS( J ) + CONE
  149:             BM = RHS( J ) - CONE
  150:             SPLUS = ONE
  151: *
  152: *           Lockahead for L- part RHS(1:N-1) = +-1
  153: *           SPLUS and SMIN computed more efficiently than in BSOLVE[1].
  154: *
  155:             SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
  156:      $              J ), 1 ) )
  157:             SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
  158:             SPLUS = SPLUS*DBLE( RHS( J ) )
  159:             IF( SPLUS.GT.SMINU ) THEN
  160:                RHS( J ) = BP
  161:             ELSE IF( SMINU.GT.SPLUS ) THEN
  162:                RHS( J ) = BM
  163:             ELSE
  164: *
  165: *              In this case the updating sums are equal and we can
  166: *              choose RHS(J) +1 or -1. The first time this happens we
  167: *              choose -1, thereafter +1. This is a simple way to get
  168: *              good estimates of matrices like Byers well-known example
  169: *              (see [1]). (Not done in BSOLVE.)
  170: *
  171:                RHS( J ) = RHS( J ) + PMONE
  172:                PMONE = CONE
  173:             END IF
  174: *
  175: *           Compute the remaining r.h.s.
  176: *
  177:             TEMP = -RHS( J )
  178:             CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
  179:    10    CONTINUE
  180: *
  181: *        Solve for U- part, lockahead for RHS(N) = +-1. This is not done
  182: *        In BSOLVE and will hopefully give us a better estimate because
  183: *        any ill-conditioning of the original matrix is transfered to U
  184: *        and not to L. U(N, N) is an approximation to sigma_min(LU).
  185: *
  186:          CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
  187:          WORK( N ) = RHS( N ) + CONE
  188:          RHS( N ) = RHS( N ) - CONE
  189:          SPLUS = ZERO
  190:          SMINU = ZERO
  191:          DO 30 I = N, 1, -1
  192:             TEMP = CONE / Z( I, I )
  193:             WORK( I ) = WORK( I )*TEMP
  194:             RHS( I ) = RHS( I )*TEMP
  195:             DO 20 K = I + 1, N
  196:                WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
  197:                RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
  198:    20       CONTINUE
  199:             SPLUS = SPLUS + ABS( WORK( I ) )
  200:             SMINU = SMINU + ABS( RHS( I ) )
  201:    30    CONTINUE
  202:          IF( SPLUS.GT.SMINU )
  203:      $      CALL ZCOPY( N, WORK, 1, RHS, 1 )
  204: *
  205: *        Apply the permutations JPIV to the computed solution (RHS)
  206: *
  207:          CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
  208: *
  209: *        Compute the sum of squares
  210: *
  211:          CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  212:          RETURN
  213:       END IF
  214: *
  215: *     ENTRY IJOB = 2
  216: *
  217: *     Compute approximate nullvector XM of Z
  218: *
  219:       CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
  220:       CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
  221: *
  222: *     Compute RHS
  223: *
  224:       CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
  225:       TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
  226:       CALL ZSCAL( N, TEMP, XM, 1 )
  227:       CALL ZCOPY( N, XM, 1, XP, 1 )
  228:       CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
  229:       CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
  230:       CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
  231:       CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
  232:       IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
  233:      $   CALL ZCOPY( N, XP, 1, RHS, 1 )
  234: *
  235: *     Compute the sum of squares
  236: *
  237:       CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  238:       RETURN
  239: *
  240: *     End of ZLATDF
  241: *
  242:       END