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Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive-definite band matrix and reduce its condition number.

Syntax

lapack_int LAPACKE_spbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const float* ab, lapack_int ldab, float* s, float* scond, float* amax );

lapack_int LAPACKE_dpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const double* ab, lapack_int ldab, double* s, double* scond, double* amax );

lapack_int LAPACKE_cpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const lapack_complex_float* ab, lapack_int ldab, float* s, float* scond, float* amax );

lapack_int LAPACKE_zpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const lapack_complex_double* ab, lapack_int ldab, double* s, double* scond, double* amax );

Include Files

mkl.h

Description

The routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive definite band matrix A and reduce its condition number (with respect to the two-norm). The output array s returns scale factors such that contains

These factors are chosen so that the scaled matrix B with elements b_ij=s[i-1]*a_ij*s[j-1] has diagonal elements equal to 1. This choice of s puts the condition number of B within a factor n of the smallest possible condition number over all possible diagonal scalings.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
uplo	Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored in the array `ab`: If `uplo = 'U'`, the array `ab` stores the upper triangular part of the matrix `A`. If `uplo = 'L'`, the array `ab` stores the lower triangular part of the matrix `A`.
n	The order of matrix `A`; `n≥ 0`.
kd	The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
ab	Array, size max(1, ldab*n) . The array `ap` contains either the upper or the lower triangular part of the matrix `A` (as specified by `uplo`) in band storage (see Matrix Storage Schemes).
ldab	The leading dimension of the array `ab`; `ldab`≥`kd` +1.

Output Parameters

Array, size (n).

If info = 0, the array s contains the scale factors for A.

scond

If info = 0, scond contains the ratio of the smallest s[i] to the largest s[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, the i-th diagonal element of A is nonpositive.

Application Notes

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

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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