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    1: *> \brief \b DPPEQU
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DPPEQU + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppequ.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppequ.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppequ.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       CHARACTER          UPLO
   25: *       INTEGER            INFO, N
   26: *       DOUBLE PRECISION   AMAX, SCOND
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       DOUBLE PRECISION   AP( * ), S( * )
   30: *       ..
   31: *
   32: *
   33: *> \par Purpose:
   34: *  =============
   35: *>
   36: *> \verbatim
   37: *>
   38: *> DPPEQU computes row and column scalings intended to equilibrate a
   39: *> symmetric positive definite matrix A in packed storage and reduce
   40: *> its condition number (with respect to the two-norm).  S contains the
   41: *> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
   42: *> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
   43: *> This choice of S puts the condition number of B within a factor N of
   44: *> the smallest possible condition number over all possible diagonal
   45: *> scalings.
   46: *> \endverbatim
   47: *
   48: *  Arguments:
   49: *  ==========
   50: *
   51: *> \param[in] UPLO
   52: *> \verbatim
   53: *>          UPLO is CHARACTER*1
   54: *>          = 'U':  Upper triangle of A is stored;
   55: *>          = 'L':  Lower triangle of A is stored.
   56: *> \endverbatim
   57: *>
   58: *> \param[in] N
   59: *> \verbatim
   60: *>          N is INTEGER
   61: *>          The order of the matrix A.  N >= 0.
   62: *> \endverbatim
   63: *>
   64: *> \param[in] AP
   65: *> \verbatim
   66: *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
   67: *>          The upper or lower triangle of the symmetric matrix A, packed
   68: *>          columnwise in a linear array.  The j-th column of A is stored
   69: *>          in the array AP as follows:
   70: *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   71: *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
   72: *> \endverbatim
   73: *>
   74: *> \param[out] S
   75: *> \verbatim
   76: *>          S is DOUBLE PRECISION array, dimension (N)
   77: *>          If INFO = 0, S contains the scale factors for A.
   78: *> \endverbatim
   79: *>
   80: *> \param[out] SCOND
   81: *> \verbatim
   82: *>          SCOND is DOUBLE PRECISION
   83: *>          If INFO = 0, S contains the ratio of the smallest S(i) to
   84: *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   85: *>          large nor too small, it is not worth scaling by S.
   86: *> \endverbatim
   87: *>
   88: *> \param[out] AMAX
   89: *> \verbatim
   90: *>          AMAX is DOUBLE PRECISION
   91: *>          Absolute value of largest matrix element.  If AMAX is very
   92: *>          close to overflow or very close to underflow, the matrix
   93: *>          should be scaled.
   94: *> \endverbatim
   95: *>
   96: *> \param[out] INFO
   97: *> \verbatim
   98: *>          INFO is INTEGER
   99: *>          = 0:  successful exit
  100: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  101: *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
  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: *> \date December 2016
  113: *
  114: *> \ingroup doubleOTHERcomputational
  115: *
  116: *  =====================================================================
  117:       SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
  118: *
  119: *  -- LAPACK computational routine (version 3.7.0) --
  120: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  121: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  122: *     December 2016
  123: *
  124: *     .. Scalar Arguments ..
  125:       CHARACTER          UPLO
  126:       INTEGER            INFO, N
  127:       DOUBLE PRECISION   AMAX, SCOND
  128: *     ..
  129: *     .. Array Arguments ..
  130:       DOUBLE PRECISION   AP( * ), S( * )
  131: *     ..
  132: *
  133: *  =====================================================================
  134: *
  135: *     .. Parameters ..
  136:       DOUBLE PRECISION   ONE, ZERO
  137:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
  138: *     ..
  139: *     .. Local Scalars ..
  140:       LOGICAL            UPPER
  141:       INTEGER            I, JJ
  142:       DOUBLE PRECISION   SMIN
  143: *     ..
  144: *     .. External Functions ..
  145:       LOGICAL            LSAME
  146:       EXTERNAL           LSAME
  147: *     ..
  148: *     .. External Subroutines ..
  149:       EXTERNAL           XERBLA
  150: *     ..
  151: *     .. Intrinsic Functions ..
  152:       INTRINSIC          MAX, MIN, SQRT
  153: *     ..
  154: *     .. Executable Statements ..
  155: *
  156: *     Test the input parameters.
  157: *
  158:       INFO = 0
  159:       UPPER = LSAME( UPLO, 'U' )
  160:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  161:          INFO = -1
  162:       ELSE IF( N.LT.0 ) THEN
  163:          INFO = -2
  164:       END IF
  165:       IF( INFO.NE.0 ) THEN
  166:          CALL XERBLA( 'DPPEQU', -INFO )
  167:          RETURN
  168:       END IF
  169: *
  170: *     Quick return if possible
  171: *
  172:       IF( N.EQ.0 ) THEN
  173:          SCOND = ONE
  174:          AMAX = ZERO
  175:          RETURN
  176:       END IF
  177: *
  178: *     Initialize SMIN and AMAX.
  179: *
  180:       S( 1 ) = AP( 1 )
  181:       SMIN = S( 1 )
  182:       AMAX = S( 1 )
  183: *
  184:       IF( UPPER ) THEN
  185: *
  186: *        UPLO = 'U':  Upper triangle of A is stored.
  187: *        Find the minimum and maximum diagonal elements.
  188: *
  189:          JJ = 1
  190:          DO 10 I = 2, N
  191:             JJ = JJ + I
  192:             S( I ) = AP( JJ )
  193:             SMIN = MIN( SMIN, S( I ) )
  194:             AMAX = MAX( AMAX, S( I ) )
  195:    10    CONTINUE
  196: *
  197:       ELSE
  198: *
  199: *        UPLO = 'L':  Lower triangle of A is stored.
  200: *        Find the minimum and maximum diagonal elements.
  201: *
  202:          JJ = 1
  203:          DO 20 I = 2, N
  204:             JJ = JJ + N - I + 2
  205:             S( I ) = AP( JJ )
  206:             SMIN = MIN( SMIN, S( I ) )
  207:             AMAX = MAX( AMAX, S( I ) )
  208:    20    CONTINUE
  209:       END IF
  210: *
  211:       IF( SMIN.LE.ZERO ) THEN
  212: *
  213: *        Find the first non-positive diagonal element and return.
  214: *
  215:          DO 30 I = 1, N
  216:             IF( S( I ).LE.ZERO ) THEN
  217:                INFO = I
  218:                RETURN
  219:             END IF
  220:    30    CONTINUE
  221:       ELSE
  222: *
  223: *        Set the scale factors to the reciprocals
  224: *        of the diagonal elements.
  225: *
  226:          DO 40 I = 1, N
  227:             S( I ) = ONE / SQRT( S( I ) )
  228:    40    CONTINUE
  229: *
  230: *        Compute SCOND = min(S(I)) / max(S(I))
  231: *
  232:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  233:       END IF
  234:       RETURN
  235: *
  236: *     End of DPPEQU
  237: *
  238:       END