Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?gtrfs

Refines the solution of a system of linear equations with a tridiagonal coefficient matrix and estimates its error.

Syntax

lapack_int LAPACKE_sgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int nrhs, const float* dl, const float* d, const float* du, const float* dlf, const float* df, const float* duf, const float* du2, const lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int nrhs, const double* dl, const double* d, const double* du, const double* dlf, const double* df, const double* duf, const double* du2, const lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int nrhs, const lapack_complex_float* dl, const lapack_complex_float* d, const lapack_complex_float* du, const lapack_complex_float* dlf, const lapack_complex_float* df, const lapack_complex_float* duf, const lapack_complex_float* du2, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int nrhs, const lapack_complex_double* dl, const lapack_complex_double* d, const lapack_complex_double* du, const lapack_complex_double* dlf, const lapack_complex_double* df, const lapack_complex_double* duf, const lapack_complex_double* du2, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

  • mkl.h

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X = B or AH*X = B with a tridiagonal matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward errorβ. This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:

|δaij|/|aij| β|aij|, |δbi|/|bi| β|bi| such that (A + δA)x = (b + δb).

Finally, the routine estimates the component-wise forward error in the computed solution ||x - xe||/||x|| (here xe is the exact solution).

Before calling this routine:

  • call the factorization routine ?gttrf

  • call the solver routine ?gttrs.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

trans

Must be 'N' or 'T' or 'C'.

Indicates the form of the equations:

If trans = 'N', the system has the form A*X = B.

If trans = 'T', the system has the form AT*X = B.

If trans = 'C', the system has the form AH*X = B.

n

The order of the matrix A; n 0.

nrhs

The number of right-hand sides, that is, the number of columns of the matrix B; nrhs 0.

dl

Array dl of size n -1 contains the subdiagonal elements of A.

d

Array d of size n contains the diagonal elements of A.

du

Array du of size n -1 contains the superdiagonal elements of A.

dlf

Array dlf of size n -1 contains the (n - 1) multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf.

df

Array df of size n contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

duf

Array duf of size n -1 contains the (n - 1) elements of the first superdiagonal of U.

du2

Array du2 of size n -2 contains the (n - 2) elements of the second superdiagonal of U.

b

Array b (size max(1,ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout) contains the right-hand side matrix B.

x

Array x (size max(1,ldx*nrhs) for column major layout and max(1, ldx*n) contains the solution matrix X, as computed by ?gttrs.

ldb

The leading dimension of b; ldb max(1, n) for column major layout and ldbnrhs for row major layout.

ldx

The leading dimension of x; ldx max(1, n) for column major layout and ldxnrhs for row major layout.

ipiv

Array, size at least max(1, n). The ipiv array, as returned by ?gttrf.

Output Parameters

x

The refined solution matrix X.

ferr, berr

Arrays, size at least max(1,nrhs). Contain the component-wise forward and backward errors, respectively, for each solution vector.

Return Values

This function returns a value info.

If info = 0, the execution is successful.

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