PPEQU - 5.2 English - 68552

PPEQU computes row and column scalings.

Purpose:

  PPEQU computes row and column scalings intended to equilibrate a
  symmetric positive definite matrix A in packed storage 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.

Parameters:

• UPLO – [in]

  UPLO is CHARACTER*1

  = ‘U’: Upper triangle of A is stored;

  = ‘L’: Lower triangle of A is stored.

• N – [in]

  N is INTEGER

  The order of the matrix A. N >= 0.

• AP – [in]

  AP is REAL array, dimension (N*(N+1)/2)

  The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

  if UPLO = ‘U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

  if UPLO = ‘L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

• S – [out]

  S is REAL array, dimension (N)

  If INFO = 0, S contains the scale factors for A.

• SCOND – [out]

  SCOND is REAL

  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 – [out]

  AMAX is REAL

  Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.

• INFO – [out]

  INFO is 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.