spoequ.f.html

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      SUBROUTINE <a name="SPOEQU.1"></a><a href="spoequ.f.html#SPOEQU.1">SPOEQU</a>( N, A, LDA, S, SCOND, AMAX, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            INFO, LDA, N
      REAL               AMAX, SCOND
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      REAL               A( LDA, * ), S( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="SPOEQU.18"></a><a href="spoequ.f.html#SPOEQU.1">SPOEQU</a> computes row and column scalings intended to equilibrate a
</span><span class="comment">*</span><span class="comment">  symmetric positive definite matrix A and reduce its condition number
</span><span class="comment">*</span><span class="comment">  (with respect to the two-norm).  S contains the scale factors,
</span><span class="comment">*</span><span class="comment">  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
</span><span class="comment">*</span><span class="comment">  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
</span><span class="comment">*</span><span class="comment">  choice of S puts the condition number of B within a factor N of the
</span><span class="comment">*</span><span class="comment">  smallest possible condition number over all possible diagonal
</span><span class="comment">*</span><span class="comment">  scalings.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix A.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input) REAL array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          The N-by-N symmetric positive definite matrix whose scaling
</span><span class="comment">*</span><span class="comment">          factors are to be computed.  Only the diagonal elements of A
</span><span class="comment">*</span><span class="comment">          are referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A.  LDA &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  S       (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          If INFO = 0, S contains the scale factors for A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SCOND   (output) REAL
</span><span class="comment">*</span><span class="comment">          If INFO = 0, S contains the ratio of the smallest S(i) to
</span><span class="comment">*</span><span class="comment">          the largest S(i).  If SCOND &gt;= 0.1 and AMAX is neither too
</span><span class="comment">*</span><span class="comment">          large nor too small, it is not worth scaling by S.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  AMAX    (output) REAL
</span><span class="comment">*</span><span class="comment">          Absolute value of largest matrix element.  If AMAX is very
</span><span class="comment">*</span><span class="comment">          close to overflow or very close to underflow, the matrix
</span><span class="comment">*</span><span class="comment">          should be scaled.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">          &gt; 0:  if INFO = i, the i-th diagonal element is nonpositive.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I
      REAL               SMIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="XERBLA.70"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          MAX, MIN, SQRT
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span>      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 <a name="XERBLA.86"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="SPOEQU.86"></a><a href="spoequ.f.html#SPOEQU.1">SPOEQU</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Find the minimum and maximum diagonal elements.
</span><span class="comment">*</span><span class="comment">
</span>      S( 1 ) = A( 1, 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
      DO 10 I = 2, N
         S( I ) = A( I, I )
         SMIN = MIN( SMIN, S( I ) )
         AMAX = MAX( AMAX, S( I ) )
   10 CONTINUE
<span class="comment">*</span><span class="comment">
</span>      IF( SMIN.LE.ZERO ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Find the first non-positive diagonal element and return.
</span><span class="comment">*</span><span class="comment">
</span>         DO 20 I = 1, N
            IF( S( I ).LE.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   20    CONTINUE
      ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Set the scale factors to the reciprocals
</span><span class="comment">*</span><span class="comment">        of the diagonal elements.
</span><span class="comment">*</span><span class="comment">
</span>         DO 30 I = 1, N
            S( I ) = ONE / SQRT( S( I ) )
   30    CONTINUE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Compute SCOND = min(S(I)) / max(S(I))
</span><span class="comment">*</span><span class="comment">
</span>         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="SPOEQU.134"></a><a href="spoequ.f.html#SPOEQU.1">SPOEQU</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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