zpoequ.f

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      SUBROUTINE zpoequ( N, A, LDA, S, SCOND, AMAX, INFO )
*
*  -- LAPACK routine (version 3.2) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      INTEGER            info, lda, n
      DOUBLE PRECISION   amax, scond
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   s( * )
      COMPLEX*16         a( lda, * )
*     ..
*
*  Purpose
*  =======
*
*  ZPOEQU computes row and column scalings intended to equilibrate a
*  Hermitian positive definite matrix A and reduce its condition number
*  (with respect to the two-norm).  S contains the scale factors,
*  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*  choice of S puts the condition number of B within a factor N of the
*  smallest possible condition number over all possible diagonal
*  scalings.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX*16 array, dimension (LDA,N)
*          The N-by-N Hermitian positive definite matrix whose scaling
*          factors are to be computed.  Only the diagonal elements of A
*          are referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  S       (output) DOUBLE PRECISION array, dimension (N)
*          If INFO = 0, S contains the scale factors for A.
*
*  SCOND   (output) DOUBLE PRECISION
*          If INFO = 0, S contains the ratio of the smallest S(i) to
*          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*          large nor too small, it is not worth scaling by S.
*
*  AMAX    (output) DOUBLE PRECISION
*          Absolute value of largest matrix element.  If AMAX is very
*          close to overflow or very close to underflow, the matrix
*          should be scaled.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   zero, one
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            i
      DOUBLE PRECISION   smin
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -3
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZPOEQU', -info )
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 ) THEN
         scond = one
         amax = zero
         return
      END IF
*
*     Find the minimum and maximum diagonal elements.
*
      s( 1 ) = dble( a( 1, 1 ) )
      smin = s( 1 )
      amax = s( 1 )
      DO 10 i = 2, n
         s( i ) = dble( a( i, i ) )
         smin = min( smin, s( i ) )
         amax = max( amax, s( i ) )
   10 continue
*
      IF( smin.LE.zero ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 20 i = 1, n
            IF( s( i ).LE.zero ) THEN
               info = i
               return
            END IF
   20    continue
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 30 i = 1, n
            s( i ) = one / sqrt( s( i ) )
   30    continue
*
*        Compute SCOND = min(S(I)) / max(S(I))
*
         scond = sqrt( smin ) / sqrt( amax )
      END IF
      return
*
*     End of ZPOEQU
*
      END