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    1: *> \brief \b DPOEQU
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DPOEQU + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequ.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequ.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequ.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, LDA, N
   25: *       DOUBLE PRECISION   AMAX, SCOND
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   A( LDA, * ), S( * )
   29: *       ..
   30: *
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *> DPOEQU computes row and column scalings intended to equilibrate a
   38: *> symmetric positive definite matrix A and reduce its condition number
   39: *> (with respect to the two-norm).  S contains the scale factors,
   40: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
   41: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
   42: *> choice of S puts the condition number of B within a factor N of the
   43: *> smallest possible condition number over all possible diagonal
   44: *> scalings.
   45: *> \endverbatim
   46: *
   47: *  Arguments:
   48: *  ==========
   49: *
   50: *> \param[in] N
   51: *> \verbatim
   52: *>          N is INTEGER
   53: *>          The order of the matrix A.  N >= 0.
   54: *> \endverbatim
   55: *>
   56: *> \param[in] A
   57: *> \verbatim
   58: *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   59: *>          The N-by-N symmetric positive definite matrix whose scaling
   60: *>          factors are to be computed.  Only the diagonal elements of A
   61: *>          are referenced.
   62: *> \endverbatim
   63: *>
   64: *> \param[in] LDA
   65: *> \verbatim
   66: *>          LDA is INTEGER
   67: *>          The leading dimension of the array A.  LDA >= max(1,N).
   68: *> \endverbatim
   69: *>
   70: *> \param[out] S
   71: *> \verbatim
   72: *>          S is DOUBLE PRECISION array, dimension (N)
   73: *>          If INFO = 0, S contains the scale factors for A.
   74: *> \endverbatim
   75: *>
   76: *> \param[out] SCOND
   77: *> \verbatim
   78: *>          SCOND is DOUBLE PRECISION
   79: *>          If INFO = 0, S contains the ratio of the smallest S(i) to
   80: *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   81: *>          large nor too small, it is not worth scaling by S.
   82: *> \endverbatim
   83: *>
   84: *> \param[out] AMAX
   85: *> \verbatim
   86: *>          AMAX is DOUBLE PRECISION
   87: *>          Absolute value of largest matrix element.  If AMAX is very
   88: *>          close to overflow or very close to underflow, the matrix
   89: *>          should be scaled.
   90: *> \endverbatim
   91: *>
   92: *> \param[out] INFO
   93: *> \verbatim
   94: *>          INFO is INTEGER
   95: *>          = 0:  successful exit
   96: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   97: *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   98: *> \endverbatim
   99: *
  100: *  Authors:
  101: *  ========
  102: *
  103: *> \author Univ. of Tennessee
  104: *> \author Univ. of California Berkeley
  105: *> \author Univ. of Colorado Denver
  106: *> \author NAG Ltd.
  107: *
  108: *> \ingroup doublePOcomputational
  109: *
  110: *  =====================================================================
  111:       SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
  112: *
  113: *  -- LAPACK computational routine --
  114: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  115: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  116: *
  117: *     .. Scalar Arguments ..
  118:       INTEGER            INFO, LDA, N
  119:       DOUBLE PRECISION   AMAX, SCOND
  120: *     ..
  121: *     .. Array Arguments ..
  122:       DOUBLE PRECISION   A( LDA, * ), S( * )
  123: *     ..
  124: *
  125: *  =====================================================================
  126: *
  127: *     .. Parameters ..
  128:       DOUBLE PRECISION   ZERO, ONE
  129:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  130: *     ..
  131: *     .. Local Scalars ..
  132:       INTEGER            I
  133:       DOUBLE PRECISION   SMIN
  134: *     ..
  135: *     .. External Subroutines ..
  136:       EXTERNAL           XERBLA
  137: *     ..
  138: *     .. Intrinsic Functions ..
  139:       INTRINSIC          MAX, MIN, SQRT
  140: *     ..
  141: *     .. Executable Statements ..
  142: *
  143: *     Test the input parameters.
  144: *
  145:       INFO = 0
  146:       IF( N.LT.0 ) THEN
  147:          INFO = -1
  148:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  149:          INFO = -3
  150:       END IF
  151:       IF( INFO.NE.0 ) THEN
  152:          CALL XERBLA( 'DPOEQU', -INFO )
  153:          RETURN
  154:       END IF
  155: *
  156: *     Quick return if possible
  157: *
  158:       IF( N.EQ.0 ) THEN
  159:          SCOND = ONE
  160:          AMAX = ZERO
  161:          RETURN
  162:       END IF
  163: *
  164: *     Find the minimum and maximum diagonal elements.
  165: *
  166:       S( 1 ) = A( 1, 1 )
  167:       SMIN = S( 1 )
  168:       AMAX = S( 1 )
  169:       DO 10 I = 2, N
  170:          S( I ) = A( I, I )
  171:          SMIN = MIN( SMIN, S( I ) )
  172:          AMAX = MAX( AMAX, S( I ) )
  173:    10 CONTINUE
  174: *
  175:       IF( SMIN.LE.ZERO ) THEN
  176: *
  177: *        Find the first non-positive diagonal element and return.
  178: *
  179:          DO 20 I = 1, N
  180:             IF( S( I ).LE.ZERO ) THEN
  181:                INFO = I
  182:                RETURN
  183:             END IF
  184:    20    CONTINUE
  185:       ELSE
  186: *
  187: *        Set the scale factors to the reciprocals
  188: *        of the diagonal elements.
  189: *
  190:          DO 30 I = 1, N
  191:             S( I ) = ONE / SQRT( S( I ) )
  192:    30    CONTINUE
  193: *
  194: *        Compute SCOND = min(S(I)) / max(S(I))
  195: *
  196:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  197:       END IF
  198:       RETURN
  199: *
  200: *     End of DPOEQU
  201: *
  202:       END