Return to dlatdf.f CVS log

Up to [local] / rpl / lapack / lapack

Cohérence Lapack.

    1: *> \brief \b DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DLATDF + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
   22: *                          JPIV )
   23: * 
   24: *       .. Scalar Arguments ..
   25: *       INTEGER            IJOB, LDZ, N
   26: *       DOUBLE PRECISION   RDSCAL, RDSUM
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       INTEGER            IPIV( * ), JPIV( * )
   30: *       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
   31: *       ..
   32: *  
   33: *
   34: *> \par Purpose:
   35: *  =============
   36: *>
   37: *> \verbatim
   38: *>
   39: *> DLATDF uses the LU factorization of the n-by-n matrix Z computed by
   40: *> DGETC2 and computes a contribution to the reciprocal Dif-estimate
   41: *> by solving Z * x = b for x, and choosing the r.h.s. b such that
   42: *> the norm of x is as large as possible. On entry RHS = b holds the
   43: *> contribution from earlier solved sub-systems, and on return RHS = x.
   44: *>
   45: *> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
   46: *> where P and Q are permutation matrices. L is lower triangular with
   47: *> unit diagonal elements and U is upper triangular.
   48: *> \endverbatim
   49: *
   50: *  Arguments:
   51: *  ==========
   52: *
   53: *> \param[in] IJOB
   54: *> \verbatim
   55: *>          IJOB is INTEGER
   56: *>          IJOB = 2: First compute an approximative null-vector e
   57: *>              of Z using DGECON, e is normalized and solve for
   58: *>              Zx = +-e - f with the sign giving the greater value
   59: *>              of 2-norm(x). About 5 times as expensive as Default.
   60: *>          IJOB .ne. 2: Local look ahead strategy where all entries of
   61: *>              the r.h.s. b is chosen as either +1 or -1 (Default).
   62: *> \endverbatim
   63: *>
   64: *> \param[in] N
   65: *> \verbatim
   66: *>          N is INTEGER
   67: *>          The number of columns of the matrix Z.
   68: *> \endverbatim
   69: *>
   70: *> \param[in] Z
   71: *> \verbatim
   72: *>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
   73: *>          On entry, the LU part of the factorization of the n-by-n
   74: *>          matrix Z computed by DGETC2:  Z = P * L * U * Q
   75: *> \endverbatim
   76: *>
   77: *> \param[in] LDZ
   78: *> \verbatim
   79: *>          LDZ is INTEGER
   80: *>          The leading dimension of the array Z.  LDA >= max(1, N).
   81: *> \endverbatim
   82: *>
   83: *> \param[in,out] RHS
   84: *> \verbatim
   85: *>          RHS is DOUBLE PRECISION array, dimension (N)
   86: *>          On entry, RHS contains contributions from other subsystems.
   87: *>          On exit, RHS contains the solution of the subsystem with
   88: *>          entries acoording to the value of IJOB (see above).
   89: *> \endverbatim
   90: *>
   91: *> \param[in,out] RDSUM
   92: *> \verbatim
   93: *>          RDSUM is DOUBLE PRECISION
   94: *>          On entry, the sum of squares of computed contributions to
   95: *>          the Dif-estimate under computation by DTGSYL, where the
   96: *>          scaling factor RDSCAL (see below) has been factored out.
   97: *>          On exit, the corresponding sum of squares updated with the
   98: *>          contributions from the current sub-system.
   99: *>          If TRANS = 'T' RDSUM is not touched.
  100: *>          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
  101: *> \endverbatim
  102: *>
  103: *> \param[in,out] RDSCAL
  104: *> \verbatim
  105: *>          RDSCAL is DOUBLE PRECISION
  106: *>          On entry, scaling factor used to prevent overflow in RDSUM.
  107: *>          On exit, RDSCAL is updated w.r.t. the current contributions
  108: *>          in RDSUM.
  109: *>          If TRANS = 'T', RDSCAL is not touched.
  110: *>          NOTE: RDSCAL only makes sense when DTGSY2 is called by
  111: *>                DTGSYL.
  112: *> \endverbatim
  113: *>
  114: *> \param[in] IPIV
  115: *> \verbatim
  116: *>          IPIV is INTEGER array, dimension (N).
  117: *>          The pivot indices; for 1 <= i <= N, row i of the
  118: *>          matrix has been interchanged with row IPIV(i).
  119: *> \endverbatim
  120: *>
  121: *> \param[in] JPIV
  122: *> \verbatim
  123: *>          JPIV is INTEGER array, dimension (N).
  124: *>          The pivot indices; for 1 <= j <= N, column j of the
  125: *>          matrix has been interchanged with column JPIV(j).
  126: *> \endverbatim
  127: *
  128: *  Authors:
  129: *  ========
  130: *
  131: *> \author Univ. of Tennessee 
  132: *> \author Univ. of California Berkeley 
  133: *> \author Univ. of Colorado Denver 
  134: *> \author NAG Ltd. 
  135: *
  136: *> \date June 2016
  137: *
  138: *> \ingroup doubleOTHERauxiliary
  139: *
  140: *> \par Further Details:
  141: *  =====================
  142: *>
  143: *>  This routine is a further developed implementation of algorithm
  144: *>  BSOLVE in [1] using complete pivoting in the LU factorization.
  145: *
  146: *> \par Contributors:
  147: *  ==================
  148: *>
  149: *>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
  150: *>     Umea University, S-901 87 Umea, Sweden.
  151: *
  152: *> \par References:
  153: *  ================
  154: *>
  155: *> \verbatim
  156: *>
  157: *>
  158: *>  [1] Bo Kagstrom and Lars Westin,
  159: *>      Generalized Schur Methods with Condition Estimators for
  160: *>      Solving the Generalized Sylvester Equation, IEEE Transactions
  161: *>      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
  162: *>
  163: *>  [2] Peter Poromaa,
  164: *>      On Efficient and Robust Estimators for the Separation
  165: *>      between two Regular Matrix Pairs with Applications in
  166: *>      Condition Estimation. Report IMINF-95.05, Departement of
  167: *>      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
  168: *> \endverbatim
  169: *>
  170: *  =====================================================================
  171:       SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
  172:      $                   JPIV )
  173: *
  174: *  -- LAPACK auxiliary routine (version 3.6.1) --
  175: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  176: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  177: *     June 2016
  178: *
  179: *     .. Scalar Arguments ..
  180:       INTEGER            IJOB, LDZ, N
  181:       DOUBLE PRECISION   RDSCAL, RDSUM
  182: *     ..
  183: *     .. Array Arguments ..
  184:       INTEGER            IPIV( * ), JPIV( * )
  185:       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
  186: *     ..
  187: *
  188: *  =====================================================================
  189: *
  190: *     .. Parameters ..
  191:       INTEGER            MAXDIM
  192:       PARAMETER          ( MAXDIM = 8 )
  193:       DOUBLE PRECISION   ZERO, ONE
  194:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  195: *     ..
  196: *     .. Local Scalars ..
  197:       INTEGER            I, INFO, J, K
  198:       DOUBLE PRECISION   BM, BP, PMONE, SMINU, SPLUS, TEMP
  199: *     ..
  200: *     .. Local Arrays ..
  201:       INTEGER            IWORK( MAXDIM )
  202:       DOUBLE PRECISION   WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
  203: *     ..
  204: *     .. External Subroutines ..
  205:       EXTERNAL           DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
  206:      $                   DSCAL
  207: *     ..
  208: *     .. External Functions ..
  209:       DOUBLE PRECISION   DASUM, DDOT
  210:       EXTERNAL           DASUM, DDOT
  211: *     ..
  212: *     .. Intrinsic Functions ..
  213:       INTRINSIC          ABS, SQRT
  214: *     ..
  215: *     .. Executable Statements ..
  216: *
  217:       IF( IJOB.NE.2 ) THEN
  218: *
  219: *        Apply permutations IPIV to RHS
  220: *
  221:          CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
  222: *
  223: *        Solve for L-part choosing RHS either to +1 or -1.
  224: *
  225:          PMONE = -ONE
  226: *
  227:          DO 10 J = 1, N - 1
  228:             BP = RHS( J ) + ONE
  229:             BM = RHS( J ) - ONE
  230:             SPLUS = ONE
  231: *
  232: *           Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
  233: *           SMIN computed more efficiently than in BSOLVE [1].
  234: *
  235:             SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
  236:             SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
  237:             SPLUS = SPLUS*RHS( J )
  238:             IF( SPLUS.GT.SMINU ) THEN
  239:                RHS( J ) = BP
  240:             ELSE IF( SMINU.GT.SPLUS ) THEN
  241:                RHS( J ) = BM
  242:             ELSE
  243: *
  244: *              In this case the updating sums are equal and we can
  245: *              choose RHS(J) +1 or -1. The first time this happens
  246: *              we choose -1, thereafter +1. This is a simple way to
  247: *              get good estimates of matrices like Byers well-known
  248: *              example (see [1]). (Not done in BSOLVE.)
  249: *
  250:                RHS( J ) = RHS( J ) + PMONE
  251:                PMONE = ONE
  252:             END IF
  253: *
  254: *           Compute the remaining r.h.s.
  255: *
  256:             TEMP = -RHS( J )
  257:             CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
  258: *
  259:    10    CONTINUE
  260: *
  261: *        Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
  262: *        in BSOLVE and will hopefully give us a better estimate because
  263: *        any ill-conditioning of the original matrix is transfered to U
  264: *        and not to L. U(N, N) is an approximation to sigma_min(LU).
  265: *
  266:          CALL DCOPY( N-1, RHS, 1, XP, 1 )
  267:          XP( N ) = RHS( N ) + ONE
  268:          RHS( N ) = RHS( N ) - ONE
  269:          SPLUS = ZERO
  270:          SMINU = ZERO
  271:          DO 30 I = N, 1, -1
  272:             TEMP = ONE / Z( I, I )
  273:             XP( I ) = XP( I )*TEMP
  274:             RHS( I ) = RHS( I )*TEMP
  275:             DO 20 K = I + 1, N
  276:                XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
  277:                RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
  278:    20       CONTINUE
  279:             SPLUS = SPLUS + ABS( XP( I ) )
  280:             SMINU = SMINU + ABS( RHS( I ) )
  281:    30    CONTINUE
  282:          IF( SPLUS.GT.SMINU )
  283:      $      CALL DCOPY( N, XP, 1, RHS, 1 )
  284: *
  285: *        Apply the permutations JPIV to the computed solution (RHS)
  286: *
  287:          CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
  288: *
  289: *        Compute the sum of squares
  290: *
  291:          CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  292: *
  293:       ELSE
  294: *
  295: *        IJOB = 2, Compute approximate nullvector XM of Z
  296: *
  297:          CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
  298:          CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
  299: *
  300: *        Compute RHS
  301: *
  302:          CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
  303:          TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
  304:          CALL DSCAL( N, TEMP, XM, 1 )
  305:          CALL DCOPY( N, XM, 1, XP, 1 )
  306:          CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
  307:          CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
  308:          CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
  309:          CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
  310:          IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
  311:      $      CALL DCOPY( N, XP, 1, RHS, 1 )
  312: *
  313: *        Compute the sum of squares
  314: *
  315:          CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  316: *
  317:       END IF
  318: *
  319:       RETURN
  320: *
  321: *     End of DLATDF
  322: *
  323:       END