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    1:       SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
    2: *
    3: *     -- LAPACK routine (version 3.3.1)                                 --
    4: *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
    5: *     -- Jason Riedy of Univ. of California Berkeley.                 --
    6: *     -- November 2008                                                --
    7: *
    8: *     -- LAPACK is a software package provided by Univ. of Tennessee, --
    9: *     -- Univ. of California Berkeley and NAG Ltd.                    --
   10: *
   11:       IMPLICIT NONE
   12: *     ..
   13: *     .. Scalar Arguments ..
   14:       INTEGER            INFO, LDA, N
   15:       DOUBLE PRECISION   AMAX, SCOND
   16: *     ..
   17: *     .. Array Arguments ..
   18:       COMPLEX*16         A( LDA, * )
   19:       DOUBLE PRECISION   S( * )
   20: *     ..
   21: *
   22: *  Purpose
   23: *  =======
   24: *
   25: *  ZPOEQUB computes row and column scalings intended to equilibrate a
   26: *  symmetric positive definite matrix A and reduce its condition number
   27: *  (with respect to the two-norm).  S contains the scale factors,
   28: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
   29: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
   30: *  choice of S puts the condition number of B within a factor N of the
   31: *  smallest possible condition number over all possible diagonal
   32: *  scalings.
   33: *
   34: *  Arguments
   35: *  =========
   36: *
   37: *  N       (input) INTEGER
   38: *          The order of the matrix A.  N >= 0.
   39: *
   40: *  A       (input) COMPLEX*16 array, dimension (LDA,N)
   41: *          The N-by-N symmetric positive definite matrix whose scaling
   42: *          factors are to be computed.  Only the diagonal elements of A
   43: *          are referenced.
   44: *
   45: *  LDA     (input) INTEGER
   46: *          The leading dimension of the array A.  LDA >= max(1,N).
   47: *
   48: *  S       (output) DOUBLE PRECISION array, dimension (N)
   49: *          If INFO = 0, S contains the scale factors for A.
   50: *
   51: *  SCOND   (output) DOUBLE PRECISION
   52: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   53: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   54: *          large nor too small, it is not worth scaling by S.
   55: *
   56: *  AMAX    (output) DOUBLE PRECISION
   57: *          Absolute value of largest matrix element.  If AMAX is very
   58: *          close to overflow or very close to underflow, the matrix
   59: *          should be scaled.
   60: *
   61: *  INFO    (output) INTEGER
   62: *          = 0:  successful exit
   63: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   64: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   65: *
   66: *  =====================================================================
   67: *
   68: *     .. Parameters ..
   69:       DOUBLE PRECISION   ZERO, ONE
   70:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
   71: *     ..
   72: *     .. Local Scalars ..
   73:       INTEGER            I
   74:       DOUBLE PRECISION   SMIN, BASE, TMP
   75:       COMPLEX*16         ZDUM
   76: *     ..
   77: *     .. External Functions ..
   78:       DOUBLE PRECISION   DLAMCH
   79:       EXTERNAL           DLAMCH
   80: *     ..
   81: *     .. External Subroutines ..
   82:       EXTERNAL           XERBLA
   83: *     ..
   84: *     .. Intrinsic Functions ..
   85:       INTRINSIC          MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
   86: *     ..
   87: *     .. Executable Statements ..
   88: *
   89: *     Test the input parameters.
   90: *
   91: *     Positive definite only performs 1 pass of equilibration.
   92: *
   93:       INFO = 0
   94:       IF( N.LT.0 ) THEN
   95:          INFO = -1
   96:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
   97:          INFO = -3
   98:       END IF
   99:       IF( INFO.NE.0 ) THEN
  100:          CALL XERBLA( 'ZPOEQUB', -INFO )
  101:          RETURN
  102:       END IF
  103: *
  104: *     Quick return if possible.
  105: *
  106:       IF( N.EQ.0 ) THEN
  107:          SCOND = ONE
  108:          AMAX = ZERO
  109:          RETURN
  110:       END IF
  111: 
  112:       BASE = DLAMCH( 'B' )
  113:       TMP = -0.5D+0 / LOG ( BASE )
  114: *
  115: *     Find the minimum and maximum diagonal elements.
  116: *
  117:       S( 1 ) = A( 1, 1 )
  118:       SMIN = S( 1 )
  119:       AMAX = S( 1 )
  120:       DO 10 I = 2, N
  121:          S( I ) = A( I, I )
  122:          SMIN = MIN( SMIN, S( I ) )
  123:          AMAX = MAX( AMAX, S( I ) )
  124:    10 CONTINUE
  125: *
  126:       IF( SMIN.LE.ZERO ) THEN
  127: *
  128: *        Find the first non-positive diagonal element and return.
  129: *
  130:          DO 20 I = 1, N
  131:             IF( S( I ).LE.ZERO ) THEN
  132:                INFO = I
  133:                RETURN
  134:             END IF
  135:    20    CONTINUE
  136:       ELSE
  137: *
  138: *        Set the scale factors to the reciprocals
  139: *        of the diagonal elements.
  140: *
  141:          DO 30 I = 1, N
  142:             S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
  143:    30    CONTINUE
  144: *
  145: *        Compute SCOND = min(S(I)) / max(S(I)).
  146: *
  147:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  148:       END IF
  149: *
  150:       RETURN
  151: *
  152: *     End of ZPOEQUB
  153: *
  154:       END