?gtcon

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?gtcon

Estimates the reciprocal of the condition number of a tridiagonal matrix.

Syntax

lapack_int LAPACKE_sgtcon( char norm, lapack_int n, const float* dl, const float* d, const float* du, const float* du2, const lapack_int* ipiv, float anorm, float* rcond );

lapack_int LAPACKE_dgtcon( char norm, lapack_int n, const double* dl, const double* d, const double* du, const double* du2, const lapack_int* ipiv, double anorm, double* rcond );

lapack_int LAPACKE_cgtcon( char norm, lapack_int n, const lapack_complex_float* dl, const lapack_complex_float* d, const lapack_complex_float* du, const lapack_complex_float* du2, const lapack_int* ipiv, float anorm, float* rcond );

lapack_int LAPACKE_zgtcon( char norm, lapack_int n, const lapack_complex_double* dl, const lapack_complex_double* d, const lapack_complex_double* du, const lapack_complex_double* du2, const lapack_int* ipiv, double anorm, double* rcond );

Include Files

mkl.h

Description

The routine estimates the reciprocal of the condition number of a real or complex tridiagonal matrix A in the 1-norm or infinity-norm:

κ₁(A) = ||A||₁||A^-1||₁

κ_∞(A) = ||A||_∞||A^-1||_∞

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 ?gttrf to compute the LU factorization of A.

Input Parameters

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.
dl,d,du,du2	Arrays: `dl(n -1)`, `d(n)`, `du(n -1)`, `du2(n -2)`. The array `dl` contains the `(n - 1)` multipliers that define the matrix `L` from the `LU` factorization of `A` as computed by ?gttrf. The array `d` contains the `n` diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`. The array `du` contains the (`n` - 1) elements of the first superdiagonal of `U`. The array `du2` contains the (`n` - 2) elements of the second superdiagonal of `U`.
ipiv	Array, size (`n`). The array of pivot indices, as returned by ?gttrf.
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 2n² floating-point operations for real flavors and 8n² 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

?gtcon