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    1:       SUBROUTINE ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, LDA, N
   10:       DOUBLE PRECISION   AMAX, SCOND
   11: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   S( * )
   14:       COMPLEX*16         A( LDA, * )
   15: *     ..
   16: *
   17: *  Purpose
   18: *  =======
   19: *
   20: *  ZPOEQU computes row and column scalings intended to equilibrate a
   21: *  Hermitian positive definite matrix A and reduce its condition number
   22: *  (with respect to the two-norm).  S contains the scale factors,
   23: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
   24: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
   25: *  choice of S puts the condition number of B within a factor N of the
   26: *  smallest possible condition number over all possible diagonal
   27: *  scalings.
   28: *
   29: *  Arguments
   30: *  =========
   31: *
   32: *  N       (input) INTEGER
   33: *          The order of the matrix A.  N >= 0.
   34: *
   35: *  A       (input) COMPLEX*16 array, dimension (LDA,N)
   36: *          The N-by-N Hermitian positive definite matrix whose scaling
   37: *          factors are to be computed.  Only the diagonal elements of A
   38: *          are referenced.
   39: *
   40: *  LDA     (input) INTEGER
   41: *          The leading dimension of the array A.  LDA >= max(1,N).
   42: *
   43: *  S       (output) DOUBLE PRECISION array, dimension (N)
   44: *          If INFO = 0, S contains the scale factors for A.
   45: *
   46: *  SCOND   (output) DOUBLE PRECISION
   47: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   48: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   49: *          large nor too small, it is not worth scaling by S.
   50: *
   51: *  AMAX    (output) DOUBLE PRECISION
   52: *          Absolute value of largest matrix element.  If AMAX is very
   53: *          close to overflow or very close to underflow, the matrix
   54: *          should be scaled.
   55: *
   56: *  INFO    (output) INTEGER
   57: *          = 0:  successful exit
   58: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   59: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   60: *
   61: *  =====================================================================
   62: *
   63: *     .. Parameters ..
   64:       DOUBLE PRECISION   ZERO, ONE
   65:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
   66: *     ..
   67: *     .. Local Scalars ..
   68:       INTEGER            I
   69:       DOUBLE PRECISION   SMIN
   70: *     ..
   71: *     .. External Subroutines ..
   72:       EXTERNAL           XERBLA
   73: *     ..
   74: *     .. Intrinsic Functions ..
   75:       INTRINSIC          DBLE, MAX, MIN, SQRT
   76: *     ..
   77: *     .. Executable Statements ..
   78: *
   79: *     Test the input parameters.
   80: *
   81:       INFO = 0
   82:       IF( N.LT.0 ) THEN
   83:          INFO = -1
   84:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
   85:          INFO = -3
   86:       END IF
   87:       IF( INFO.NE.0 ) THEN
   88:          CALL XERBLA( 'ZPOEQU', -INFO )
   89:          RETURN
   90:       END IF
   91: *
   92: *     Quick return if possible
   93: *
   94:       IF( N.EQ.0 ) THEN
   95:          SCOND = ONE
   96:          AMAX = ZERO
   97:          RETURN
   98:       END IF
   99: *
  100: *     Find the minimum and maximum diagonal elements.
  101: *
  102:       S( 1 ) = DBLE( A( 1, 1 ) )
  103:       SMIN = S( 1 )
  104:       AMAX = S( 1 )
  105:       DO 10 I = 2, N
  106:          S( I ) = DBLE( A( I, I ) )
  107:          SMIN = MIN( SMIN, S( I ) )
  108:          AMAX = MAX( AMAX, S( I ) )
  109:    10 CONTINUE
  110: *
  111:       IF( SMIN.LE.ZERO ) THEN
  112: *
  113: *        Find the first non-positive diagonal element and return.
  114: *
  115:          DO 20 I = 1, N
  116:             IF( S( I ).LE.ZERO ) THEN
  117:                INFO = I
  118:                RETURN
  119:             END IF
  120:    20    CONTINUE
  121:       ELSE
  122: *
  123: *        Set the scale factors to the reciprocals
  124: *        of the diagonal elements.
  125: *
  126:          DO 30 I = 1, N
  127:             S( I ) = ONE / SQRT( S( I ) )
  128:    30    CONTINUE
  129: *
  130: *        Compute SCOND = min(S(I)) / max(S(I))
  131: *
  132:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  133:       END IF
  134:       RETURN
  135: *
  136: *     End of ZPOEQU
  137: *
  138:       END