spbequ(3F)

NAME
     SPBEQU - compute row and column scalings intended to equilibrate a
     symmetric positive definite band matrix A and reduce its condition number
     (with respect to the two-norm)

SYNOPSIS
     SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, KD, LDAB, N

         REAL           AMAX, SCOND

         REAL           AB( LDAB, * ), S( * )

PURPOSE
     SPBEQU computes row and column scalings intended to equilibrate a
     symmetric positive definite band 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
     UPLO    (input) CHARACTER*1
             = 'U':  Upper triangular of A is stored;
             = 'L':  Lower triangular of A is stored.

     N       (input) INTEGER
             The order of the matrix A.  N >= 0.

     KD      (input) INTEGER
             The number of superdiagonals of the matrix A if UPLO = 'U', or
             the number of subdiagonals if UPLO = 'L'.  KD >= 0.

     AB      (input) REAL array, dimension (LDAB,N)
             The upper or lower triangle of the symmetric band matrix A,
             stored in the first KD+1 rows of the array.  The j-th column of A
             is stored in the j-th column of the array AB as follows:  if UPLO
             = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO =
             'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

     LDAB     (input) INTEGER
              The leading dimension of the array A.  LDAB >= KD+1.

     S       (output) REAL array, dimension (N)
             If INFO = 0, S contains the scale factors for A.
     SCOND   (output) 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    (output) 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    (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.