ZGBEQU (3lapack)

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

SYNOPSIS

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

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

      DOUBLE         PRECISION AMAX, COLCND, ROWCND

      DOUBLE         PRECISION C( * ), R( * )

      COMPLEX*16     AB( LDAB, * )

PURPOSE

  ZGBEQU 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) COMPLEX*16 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) DOUBLE PRECISION array, dimension (M)
          If INFO = 0, or INFO > M, R contains the row scale factors for A.

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

  ROWCND  (output) DOUBLE PRECISION
          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) DOUBLE PRECISION
          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) DOUBLE PRECISION
          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