?gtrfs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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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 A^T*X = B or A^H*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:

|δa_ij|/|a_ij| ≤β|a_ij|, |δb_i|/|b_i| ≤β|b_i| such that (A + δA)x = (b + δb).

Finally, the routine estimates the component-wise forward error in the computed solution ||x - x_e||_∞/||x||_∞ (here x_e 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 `AX = B`. If `trans = 'T'`, the system has the form `A`^T`X` = `B`. If `trans = 'C'`, the system has the form `A`^H`*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 `ldb`≥nrhs for row major layout.
ldx	The leading dimension of `x`; `ldx`≥ max(1, n) for column major layout and `ldx`≥nrhs 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.

Parent topic: Refining the Solution and Estimating Its Error: LAPACK Computational Routines

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