SPBEQU (3lapack)

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.