?ptcon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?ptcon

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

Syntax

lapack_int LAPACKE_sptcon( lapack_int n, const float* d, const float* e, float anorm, float* rcond );

lapack_int LAPACKE_dptcon( lapack_int n, const double* d, const double* e, double anorm, double* rcond );

lapack_int LAPACKE_cptcon( lapack_int n, const float* d, const lapack_complex_float* e, float anorm, float* rcond );

lapack_int LAPACKE_zptcon( lapack_int n, const double* d, const lapack_complex_double* e, double anorm, double* rcond );

Include Files

mkl.h

Description

The routine computes the reciprocal of the condition number (in the 1-norm) of a real symmetric or complex Hermitian positive-definite tridiagonal matrix using the factorization A = L*D*L^T for real flavors and A = L*D*L^H for complex flavors or A = U^T*D*U for real flavors and A = U^H*D*U for complex flavors computed by ?pttrf :

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

The norm ||A^-1|| is computed by a direct method, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^-1||).

Before calling this routine:

compute anorm as ||A||₁ = max_jΣ_i |a_ij|
call ?pttrf to compute the factorization of A.

Input Parameters

n	The order of the matrix `A`; `n`≥ 0.
d	Arrays, dimension (`n`). The array `d` contains the `n` diagonal elements of the diagonal matrix `D` from the factorization of `A`, as computed by ?pttrf ;
e	Array, size `(n -1)`. Contains off-diagonal elements of the unit bidiagonal factor `U` or `L` from the factorization computed by ?pttrf .
anorm	The 1- 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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Developer Reference for Intel® oneAPI Math Kernel Library for C

?ptcon