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    1:       SUBROUTINE DLATDF( 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:       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  DLATDF uses the LU factorization of the n-by-n matrix Z computed by
   22: *  DGETC2 and computes a contribution to the reciprocal Dif-estimate
   23: *  by solving Z * x = b for x, and choosing the r.h.s. b such that
   24: *  the norm of x is as large as possible. On entry RHS = b holds the
   25: *  contribution from earlier solved sub-systems, and on return RHS = x.
   26: *
   27: *  The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
   28: *  where P and Q are permutation matrices. L is lower triangular with
   29: *  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 DGECON, e is normalized and solve for
   37: *              Zx = +-e - f with the sign giving the greater value
   38: *              of 2-norm(x). About 5 times as expensive as Default.
   39: *          IJOB .ne. 2: Local look ahead strategy where all entries of
   40: *              the r.h.s. b is choosen as either +1 or -1 (Default).
   41: *
   42: *  N       (input) INTEGER
   43: *          The number of columns of the matrix Z.
   44: *
   45: *  Z       (input) DOUBLE PRECISION array, dimension (LDZ, N)
   46: *          On entry, the LU part of the factorization of the n-by-n
   47: *          matrix Z computed by DGETC2:  Z = P * L * U * Q
   48: *
   49: *  LDZ     (input) INTEGER
   50: *          The leading dimension of the array Z.  LDA >= max(1, N).
   51: *
   52: *  RHS     (input/output) DOUBLE PRECISION array, dimension N.
   53: *          On entry, RHS contains contributions from other subsystems.
   54: *          On exit, RHS contains the solution of the subsystem with
   55: *          entries acoording to the value of IJOB (see above).
   56: *
   57: *  RDSUM   (input/output) DOUBLE PRECISION
   58: *          On entry, the sum of squares of computed contributions to
   59: *          the Dif-estimate under computation by DTGSYL, where the
   60: *          scaling factor RDSCAL (see below) has been factored out.
   61: *          On exit, the corresponding sum of squares updated with the
   62: *          contributions from the current sub-system.
   63: *          If TRANS = 'T' RDSUM is not touched.
   64: *          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
   65: *
   66: *  RDSCAL  (input/output) DOUBLE PRECISION
   67: *          On entry, scaling factor used to prevent overflow in RDSUM.
   68: *          On exit, RDSCAL is updated w.r.t. the current contributions
   69: *          in RDSUM.
   70: *          If TRANS = 'T', RDSCAL is not touched.
   71: *          NOTE: RDSCAL only makes sense when DTGSY2 is called by
   72: *                DTGSYL.
   73: *
   74: *  IPIV    (input) INTEGER array, dimension (N).
   75: *          The pivot indices; for 1 <= i <= N, row i of the
   76: *          matrix has been interchanged with row IPIV(i).
   77: *
   78: *  JPIV    (input) INTEGER array, dimension (N).
   79: *          The pivot indices; for 1 <= j <= N, column j of the
   80: *          matrix has been interchanged with column JPIV(j).
   81: *
   82: *  Further Details
   83: *  ===============
   84: *
   85: *  Based on contributions by
   86: *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   87: *     Umea University, S-901 87 Umea, Sweden.
   88: *
   89: *  This routine is a further developed implementation of algorithm
   90: *  BSOLVE in [1] using complete pivoting in the LU factorization.
   91: *
   92: *  [1] Bo Kagstrom and Lars Westin,
   93: *      Generalized Schur Methods with Condition Estimators for
   94: *      Solving the Generalized Sylvester Equation, IEEE Transactions
   95: *      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
   96: *
   97: *  [2] Peter Poromaa,
   98: *      On Efficient and Robust Estimators for the Separation
   99: *      between two Regular Matrix Pairs with Applications in
  100: *      Condition Estimation. Report IMINF-95.05, Departement of
  101: *      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
  102: *
  103: *  =====================================================================
  104: *
  105: *     .. Parameters ..
  106:       INTEGER            MAXDIM
  107:       PARAMETER          ( MAXDIM = 8 )
  108:       DOUBLE PRECISION   ZERO, ONE
  109:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  110: *     ..
  111: *     .. Local Scalars ..
  112:       INTEGER            I, INFO, J, K
  113:       DOUBLE PRECISION   BM, BP, PMONE, SMINU, SPLUS, TEMP
  114: *     ..
  115: *     .. Local Arrays ..
  116:       INTEGER            IWORK( MAXDIM )
  117:       DOUBLE PRECISION   WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
  118: *     ..
  119: *     .. External Subroutines ..
  120:       EXTERNAL           DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
  121:      $                   DSCAL
  122: *     ..
  123: *     .. External Functions ..
  124:       DOUBLE PRECISION   DASUM, DDOT
  125:       EXTERNAL           DASUM, DDOT
  126: *     ..
  127: *     .. Intrinsic Functions ..
  128:       INTRINSIC          ABS, SQRT
  129: *     ..
  130: *     .. Executable Statements ..
  131: *
  132:       IF( IJOB.NE.2 ) THEN
  133: *
  134: *        Apply permutations IPIV to RHS
  135: *
  136:          CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
  137: *
  138: *        Solve for L-part choosing RHS either to +1 or -1.
  139: *
  140:          PMONE = -ONE
  141: *
  142:          DO 10 J = 1, N - 1
  143:             BP = RHS( J ) + ONE
  144:             BM = RHS( J ) - ONE
  145:             SPLUS = ONE
  146: *
  147: *           Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
  148: *           SMIN computed more efficiently than in BSOLVE [1].
  149: *
  150:             SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
  151:             SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
  152:             SPLUS = SPLUS*RHS( J )
  153:             IF( SPLUS.GT.SMINU ) THEN
  154:                RHS( J ) = BP
  155:             ELSE IF( SMINU.GT.SPLUS ) THEN
  156:                RHS( J ) = BM
  157:             ELSE
  158: *
  159: *              In this case the updating sums are equal and we can
  160: *              choose RHS(J) +1 or -1. The first time this happens
  161: *              we choose -1, thereafter +1. This is a simple way to
  162: *              get good estimates of matrices like Byers well-known
  163: *              example (see [1]). (Not done in BSOLVE.)
  164: *
  165:                RHS( J ) = RHS( J ) + PMONE
  166:                PMONE = ONE
  167:             END IF
  168: *
  169: *           Compute the remaining r.h.s.
  170: *
  171:             TEMP = -RHS( J )
  172:             CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
  173: *
  174:    10    CONTINUE
  175: *
  176: *        Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
  177: *        in BSOLVE and will hopefully give us a better estimate because
  178: *        any ill-conditioning of the original matrix is transfered to U
  179: *        and not to L. U(N, N) is an approximation to sigma_min(LU).
  180: *
  181:          CALL DCOPY( N-1, RHS, 1, XP, 1 )
  182:          XP( N ) = RHS( N ) + ONE
  183:          RHS( N ) = RHS( N ) - ONE
  184:          SPLUS = ZERO
  185:          SMINU = ZERO
  186:          DO 30 I = N, 1, -1
  187:             TEMP = ONE / Z( I, I )
  188:             XP( I ) = XP( I )*TEMP
  189:             RHS( I ) = RHS( I )*TEMP
  190:             DO 20 K = I + 1, N
  191:                XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
  192:                RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
  193:    20       CONTINUE
  194:             SPLUS = SPLUS + ABS( XP( I ) )
  195:             SMINU = SMINU + ABS( RHS( I ) )
  196:    30    CONTINUE
  197:          IF( SPLUS.GT.SMINU )
  198:      $      CALL DCOPY( N, XP, 1, RHS, 1 )
  199: *
  200: *        Apply the permutations JPIV to the computed solution (RHS)
  201: *
  202:          CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
  203: *
  204: *        Compute the sum of squares
  205: *
  206:          CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  207: *
  208:       ELSE
  209: *
  210: *        IJOB = 2, Compute approximate nullvector XM of Z
  211: *
  212:          CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
  213:          CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
  214: *
  215: *        Compute RHS
  216: *
  217:          CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
  218:          TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
  219:          CALL DSCAL( N, TEMP, XM, 1 )
  220:          CALL DCOPY( N, XM, 1, XP, 1 )
  221:          CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
  222:          CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
  223:          CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
  224:          CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
  225:          IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
  226:      $      CALL DCOPY( N, XP, 1, RHS, 1 )
  227: *
  228: *        Compute the sum of squares
  229: *
  230:          CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
  231: *
  232:       END IF
  233: *
  234:       RETURN
  235: *
  236: *     End of DLATDF
  237: *
  238:       END