?gbequb

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gbequb

Computes row and column scaling factors restricted to a power of radix to equilibrate a banded matrix and reduce its condition number.

Syntax

call sgbequb( m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info )

call dgbequb( m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info )

call cgbequb( m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info )

call zgbequb( m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info )

Include Files

mkl.fi, lapack.f90

Description

The routine computes row and column scalings intended to equilibrate an m-by-n banded matrix A and reduce its condition number. The output array r returns the row scale factors and the array c - the column scale factors. These factors are chosen to try to make the largest element in each row and column of the matrix B with elements b(ij)=r(i)*a(ij)*c(j) have an absolute value of at most the radix.

r(i) and c(j) are restricted to be a power of the radix 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.

SMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to compute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:

SMLNUM = slamch ('s')
BIGNUM = 1 / SMLNUM

This routine differs from ?gbequ by restricting the scaling factors to a power of the radix. Except for over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer equal to approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Input Parameters

m	INTEGER. The number of rows of the matrix `A`; `m≥ 0`.
n	INTEGER. The number of columns of the matrix `A`; `n≥ 0`.
kl	INTEGER. The number of subdiagonals within the band of `A`; `kl≥ 0`.
ku	INTEGER. The number of superdiagonals within the band of `A`; `ku≥ 0`.
ab	REAL for sgbequb DOUBLE PRECISION for dgbequb COMPLEX for cgbequb DOUBLE COMPLEX for zgbequb. Array: size `ldab` by * Contains the original banded matrix `A` stored in rows from 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(n,j+kl)`. The second dimension of ab must be at least `max(1,n)`.
ldab	INTEGER. The leading dimension of a; `ldab≥ max(1, m)`.

Output Parameters

r, c	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays: r (size m), c (size n). If `info = 0`, or info>m, the array r contains the row scale factors for the matrix `A`. If `info = 0`, the array c contains the column scale factors for the matrix `A`.
rowcnd	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Absolute value of the largest element of the matrix `A`. If amax is very close to SMLNUM or BIGNUM, the matrix should be scaled.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, the `i`-th diagonal element of `A` is nonpositive. `i≤m`, the `i`-th row of `A` is exactly zero; `i>m`, the (`i`-m)-th column of `A` is exactly zero.

Parent topic: Matrix Equilibration: LAPACK Computational Routines

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