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    1: *> \brief \b DLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DLA_GBRPVGRW + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_gbrpvgrw.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_gbrpvgrw.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gbrpvgrw.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       DOUBLE PRECISION FUNCTION DLA_GBRPVGRW( N, KL, KU, NCOLS, AB,
   22: *                                               LDAB, AFB, LDAFB )
   23: *
   24: *       .. Scalar Arguments ..
   25: *       INTEGER            N, KL, KU, NCOLS, LDAB, LDAFB
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * )
   29: *       ..
   30: *
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *> DLA_GBRPVGRW computes the reciprocal pivot growth factor
   38: *> norm(A)/norm(U). The "max absolute element" norm is used. If this is
   39: *> much less than 1, the stability of the LU factorization of the
   40: *> (equilibrated) matrix A could be poor. This also means that the
   41: *> solution X, estimated condition numbers, and error bounds could be
   42: *> unreliable.
   43: *> \endverbatim
   44: *
   45: *  Arguments:
   46: *  ==========
   47: *
   48: *> \param[in] N
   49: *> \verbatim
   50: *>          N is INTEGER
   51: *>     The number of linear equations, i.e., the order of the
   52: *>     matrix A.  N >= 0.
   53: *> \endverbatim
   54: *>
   55: *> \param[in] KL
   56: *> \verbatim
   57: *>          KL is INTEGER
   58: *>     The number of subdiagonals within the band of A.  KL >= 0.
   59: *> \endverbatim
   60: *>
   61: *> \param[in] KU
   62: *> \verbatim
   63: *>          KU is INTEGER
   64: *>     The number of superdiagonals within the band of A.  KU >= 0.
   65: *> \endverbatim
   66: *>
   67: *> \param[in] NCOLS
   68: *> \verbatim
   69: *>          NCOLS is INTEGER
   70: *>     The number of columns of the matrix A.  NCOLS >= 0.
   71: *> \endverbatim
   72: *>
   73: *> \param[in] AB
   74: *> \verbatim
   75: *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
   76: *>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
   77: *>     The j-th column of A is stored in the j-th column of the
   78: *>     array AB as follows:
   79: *>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
   80: *> \endverbatim
   81: *>
   82: *> \param[in] LDAB
   83: *> \verbatim
   84: *>          LDAB is INTEGER
   85: *>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
   86: *> \endverbatim
   87: *>
   88: *> \param[in] AFB
   89: *> \verbatim
   90: *>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
   91: *>     Details of the LU factorization of the band matrix A, as
   92: *>     computed by DGBTRF.  U is stored as an upper triangular
   93: *>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
   94: *>     and the multipliers used during the factorization are stored
   95: *>     in rows KL+KU+2 to 2*KL+KU+1.
   96: *> \endverbatim
   97: *>
   98: *> \param[in] LDAFB
   99: *> \verbatim
  100: *>          LDAFB is INTEGER
  101: *>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
  102: *> \endverbatim
  103: *
  104: *  Authors:
  105: *  ========
  106: *
  107: *> \author Univ. of Tennessee
  108: *> \author Univ. of California Berkeley
  109: *> \author Univ. of Colorado Denver
  110: *> \author NAG Ltd.
  111: *
  112: *> \ingroup doubleGBcomputational
  113: *
  114: *  =====================================================================
  115:       DOUBLE PRECISION FUNCTION DLA_GBRPVGRW( N, KL, KU, NCOLS, AB,
  116:      $                                        LDAB, AFB, LDAFB )
  117: *
  118: *  -- LAPACK computational routine --
  119: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  120: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  121: *
  122: *     .. Scalar Arguments ..
  123:       INTEGER            N, KL, KU, NCOLS, LDAB, LDAFB
  124: *     ..
  125: *     .. Array Arguments ..
  126:       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * )
  127: *     ..
  128: *
  129: *  =====================================================================
  130: *
  131: *     .. Local Scalars ..
  132:       INTEGER            I, J, KD
  133:       DOUBLE PRECISION   AMAX, UMAX, RPVGRW
  134: *     ..
  135: *     .. Intrinsic Functions ..
  136:       INTRINSIC          ABS, MAX, MIN
  137: *     ..
  138: *     .. Executable Statements ..
  139: *
  140:       RPVGRW = 1.0D+0
  141: 
  142:       KD = KU + 1
  143:       DO J = 1, NCOLS
  144:          AMAX = 0.0D+0
  145:          UMAX = 0.0D+0
  146:          DO I = MAX( J-KU, 1 ), MIN( J+KL, N )
  147:             AMAX = MAX( ABS( AB( KD+I-J, J)), AMAX )
  148:          END DO
  149:          DO I = MAX( J-KU, 1 ), J
  150:             UMAX = MAX( ABS( AFB( KD+I-J, J ) ), UMAX )
  151:          END DO
  152:          IF ( UMAX /= 0.0D+0 ) THEN
  153:             RPVGRW = MIN( AMAX / UMAX, RPVGRW )
  154:          END IF
  155:       END DO
  156:       DLA_GBRPVGRW = RPVGRW
  157: *
  158: *     End of DLA_GBRPVGRW
  159: *
  160:       END