?pbcon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?pbcon

Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix.

Syntax

lapack_int LAPACKE_spbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const float* ab, lapack_int ldab, float anorm, float* rcond );

lapack_int LAPACKE_dpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const double* ab, lapack_int ldab, double anorm, double* rcond );

lapack_int LAPACKE_cpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const lapack_complex_float* ab, lapack_int ldab, float anorm, float* rcond );

lapack_int LAPACKE_zpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd, const lapack_complex_double* ab, lapack_int ldab, double anorm, double* rcond );

Include Files

mkl.h

Description

The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix A:

κ₁(A) = ||A||₁ ||A^-1||₁ (since A is symmetric or Hermitian, κ_∞(A) = κ₁(A)).

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 ?pbtrf to compute the Cholesky factorization of A.

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 how the input matrix `A` has been factored: If `uplo = 'U'`, `A` is factored as `A = U^TU` for real flavors or `A = U^HU` for complex flavors, and `U` is stored. If `uplo = 'L'`, `A` is factored as `A = LL^T` for real flavors or `A = LL^H` for complex flavors, and `L` is stored.
n	The order of the matrix `A`; `n`≥ 0.
kd	The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
ldab	The leading dimension of the array `ab`. (`ldab`≥`kd` +1).
ab	The array `ab` of size max(1, ldab*n) contains the factored matrix `A` in band form, as returned by ?pbtrf.
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; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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?pbcon

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