SGBEQU (3lapack)

compute row and column scalings intended to equilibrate an M-by-N band matrix A and reduce its condition number

SYNOPSIS

  SUBROUTINE SGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO
                     )

      INTEGER        INFO, KL, KU, LDAB, M, N

      REAL           AMAX, COLCND, ROWCND

      REAL           AB( LDAB, * ), C( * ), R( * )

PURPOSE

  SGBEQU computes row and column scalings intended to equilibrate an M-by-N
  band matrix A and reduce its condition number.  R returns the row scale
  factors and C the column scale factors, chosen to try to make the largest
  element in each row and column of the matrix B with elements
  B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

  R(i) and C(j) are restricted to be between SMLNUM = smallest safe number
  and BIGNUM = largest safe number.  Use of these scaling factors is not
  guaranteed to reduce the condition number of A but works well in practice.

ARGUMENTS

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

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

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.

  AB      (input) REAL array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th column of
          A is stored in the j-th column of the array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.

  R       (output) REAL array, dimension (M)
          If INFO = 0, or INFO > M, R contains the row scale factors for A.

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

  ROWCND  (output) REAL
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the smallest
          R(i) to the largest R(i).  If ROWCND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by R.

  COLCND  (output) REAL
          If INFO = 0, COLCND contains the ratio of the smallest C(i) to the
          largest C(i).  If COLCND >= 0.1, it is not worth scaling by C.

  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, and i is
          <= M:  the i-th row of A is exactly zero
          >  M:  the (i-M)-th column of A is exactly zero