?gbcon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?gbcon

Estimates the reciprocal of the condition number of a band matrix in the 1-norm or the infinity-norm.

Syntax

lapack_int LAPACKE_sgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl, lapack_int ku, const float* ab, lapack_int ldab, const lapack_int* ipiv, float anorm, float* rcond );

lapack_int LAPACKE_dgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl, lapack_int ku, const double* ab, lapack_int ldab, const lapack_int* ipiv, double anorm, double* rcond );

lapack_int LAPACKE_cgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_float* ab, lapack_int ldab, const lapack_int* ipiv, float anorm, float* rcond );

lapack_int LAPACKE_zgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_double* ab, lapack_int ldab, const lapack_int* ipiv, double anorm, double* rcond );

Include Files

mkl.h

Description

The routine estimates the reciprocal of the condition number of a general band matrix A in the 1-norm or infinity-norm:

κ₁(A) = ||A||₁||A^-1||₁ = κ_∞(A^T) = κ_∞(A^H)

κ_∞(A) = ||A||_∞||A^-1||_∞ = κ₁(A^T) = κ₁(A^H).

An estimate is obtained for ||A^-1||, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^-1||).

Before calling this routine:

compute anorm (either ||A||₁ = max_jΣ_i |a_ij| or ||A||_∞ = max_iΣ_j |a_ij|)
call ?gbtrf to compute the LU factorization of A.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
norm	Must be '1' or 'O' or 'I'. If `norm` = '1' or 'O', then the routine estimates the condition number of matrix `A` in 1-norm. If `norm = 'I'`, then the routine estimates the condition number of matrix `A` in infinity-norm.
n	The order 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.
ldab	The leading dimension of the array `ab`. (`ldab`≥ 2*`kl` + `ku` +1).
ipiv	Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?gbtrf.
ab	The array `ab`of size max(1, ldab*n) contains the factored band matrix `A`, as returned by ?gbtrf.
anorm	The norm of the original matrix `A`(see Description).

Output Parameters

rcond

An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, parameter i had an illegal value.

Application Notes

The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations A*x = b or A^H*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n(ku + 2kl) floating-point operations for real flavors and 8n(ku + 2kl) for complex flavors.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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