?gbequ

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?gbequ

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

Syntax

lapack_int LAPACKE_sgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const float* ab, lapack_int ldab, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_dgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const double* ab, lapack_int ldab, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

lapack_int LAPACKE_cgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_float* ab, lapack_int ldab, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_zgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_double* ab, lapack_int ldab, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

Include Files

mkl.h

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 - 1]*a_ij*c[j - 1] have absolute value 1.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
m	The number of rows of the matrix `A`; `m≥ 0`.
n	The number of columns of the matrix `A`; `n≥ 0`.
kl	The number of subdiagonals within the band of `A`; `kl≥ 0`.
ku	The number of superdiagonals within the band of `A`; `ku≥ 0`.
ab	Array, size max(1, ldabn) for column major layout and max(1, ldabm) for row major layout. Contains the original band matrix `A`.
ldab	The leading dimension of `ab`; `ldab`≥`kl`+`ku`+1.

Output Parameters

r, c	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	If `info = 0` or `info>m`, `rowcnd` contains the ratio of the smallest `r[i]` to the largest `r[i]`.
colcnd	If `info = 0`, `colcnd` contains the ratio of the smallest `c[i]` to the largest `c[i]`.
amax	Absolute value of the largest element of the matrix `A`.

Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info = -i, parameter i 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.

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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Developer Reference for Intel® oneAPI Math Kernel Library for C

?gbequ

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes

See Also