?ptrfs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?ptrfs

Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix and estimates its error.

Syntax

lapack_int LAPACKE_sptrfs( int matrix_layout, lapack_int n, lapack_int nrhs, const float* d, const float* e, const float* df, const float* ef, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dptrfs( int matrix_layout, lapack_int n, lapack_int nrhs, const double* d, const double* e, const double* df, const double* ef, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cptrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const float* d, const lapack_complex_float* e, const float* df, const lapack_complex_float* ef, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zptrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs, const double* d, const lapack_complex_double* e, const double* df, const lapack_complex_double* ef, 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 with a symmetric (Hermitian) positive definite 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| ≤β|a_ij|, |δb_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 ?pttrf
call the solver routine ?pttrs.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
uplo	Used for complex flavors only. Must be 'U' or 'L'. Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix `A` is stored and how `A` is factored: If `uplo = 'U'`, the array `e` stores the superdiagonal of `A`, and `A` is factored as `U^HDU`. If `uplo = 'L'`, the array `e` stores the subdiagonal of `A`, and `A` is factored as `LDL^H`.
n	The order of the matrix `A`; `n`≥ 0.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
d	The array `d` (size `n`) contains the `n` diagonal elements of the tridiagonal matrix `A`.
df	The array `df` (size `n`) contains the `n` diagonal elements of the diagonal matrix `D` from the factorization of `A` as computed by ?pttrf.
e,ef,b,x	The array `e` (size `n` -1) contains the (`n` - 1) off-diagonal elements of the tridiagonal matrix `A` (see `uplo`). The array `ef` (size `n` -1) contains the (`n` - 1) off-diagonal elements of the unit bidiagonal factor `U` or `L` from the factorization computed by ?pttrf (see `uplo`). The array `b` of size max(1, ldbnrhs) for column major layout and max(1, ldbn) for row major layout contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The array `x` of size max(1, ldxnrhs) for column major layout and max(1, ldxn) for row major layout contains the solution matrix `X` as computed by ?pttrs.
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.

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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Developer Reference for Intel® oneAPI Math Kernel Library for C

?ptrfs

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

See Also