Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 7/13/2023
Public

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?pbequ

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


Equation

These factors are chosen so that the scaled matrix B with elements bij=s[i-1]*aij*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; ldabkd +1.

Output Parameters

s

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.