?gbequ

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?gbequ

Computes row and column scaling factors intended to equilibrate a banded matrix and reduce its condition number.

Syntax

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

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

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

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

call gbequ( ab, r, c [,kl] [,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 band 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 absolute value 1.

See ?laqgb auxiliary function that uses scaling factors computed by ?gbequ.

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 sgbequ DOUBLE PRECISION for dgbequ COMPLEX for cgbequ DOUBLE COMPLEX for zgbequ. Array, size `ldab` by *. Contains the original band matrix `A` stored in rows from 1 to `kl` + `ku` + 1. The second dimension of `ab` must be at least `max(1,n)`.
ldab	INTEGER. The leading dimension of `ab`; `ldab`≥`kl`+`ku`+1.

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 of the matrix `A`. If `info = 0`, the array `c` contains the column scale factors of 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)`.
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)`.
amax	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Absolute value of the largest element of the matrix `A`.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i` and `i≤m`, the `i`-th row of `A` is exactly zero; `i>m`, the (`i`-`m`)th column of `A` is exactly zero.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gbequ interface are as follows:

ab	Holds the array `A` of size (`kl+ku+1,n`).
r	Holds the vector of length (`m`).
c	Holds the vector of length `n`.
kl	If omitted, assumed `kl = ku`.
ku	Restored as `ku = lda-kl-1`.

Application Notes

All the components of r and c 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.

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

If rowcnd≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by r.

If colcnd≥ 0.1, it is not worth scaling by c.

If amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.

Parent topic: Matrix Equilibration: LAPACK Computational Routines

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