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spbequ

 NAME
      SPBEQU - compute row and column scalings intended to equili-
      brate 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 equili-
      brate a symmetric positive definite band matrix A and reduce
      its condition number (with respect to the two-norm).  S con-
      tains 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 diago-
      nal 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 under-
              flow, 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 entry is nonpo-
              sitive.