Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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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 = maxjΣi |aij| or ||A|| = maxiΣj |aij|)

  • 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.