?getrs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?getrs

Solves a system of linear equations with an LU-factored square coefficient matrix, with multiple right-hand sides.

Syntax

lapack_int LAPACKE_sgetrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const float * a , lapack_int lda , const lapack_int * ipiv , float * b , lapack_int ldb );

lapack_int LAPACKE_dgetrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const double * a , lapack_int lda , const lapack_int * ipiv , double * b , lapack_int ldb );

lapack_int LAPACKE_cgetrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zgetrs (int matrix_layout , char trans , lapack_int n , lapack_int nrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );

Include Files

mkl.h

Description

The routine solves for X the following systems of linear equations:

`A*X = B`	if `trans`='N',
`A^T*X = B`	if `trans`='T',
`A^H*X = B`	if `trans`='C' (for complex matrices only).

Before calling this routine, you must call ?getrf to compute the LU factorization of A.

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'`, then `AX = B` is solved for `X`. If `trans = 'T'`, then `A^TX = B` is solved for `X`. If `trans = 'C'`, then `A^H*X = B` is solved for `X`.
n	The order of `A`; the number of rows in `B(n≥ 0)`.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
a	Array of size max(1, `lda`*n). The array `a` contains `LU` factorization of matrix `A` resulting from the call of ?getrf.
b	Array of size max(1,`ldb`nrhs) for column major layout, and max(1,`ldb`n) for row major layout. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations.
lda	The leading dimension of `a`; `lda≥ max(1, n)`.
ldb	The leading dimension of `b`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.
ipiv	Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?getrf.

Output Parameters

b	Overwritten by the solution matrix `X`.

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

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A + E)x = b, where

|E| ≤ c(n)ε P|L||U|

c(n) is a modest linear function of n, and ε is the machine precision.

If x₀ is the true solution, the computed solution x satisfies this error bound:

where cond(A,x)= || |A^-1||A| |x| ||_∞ / ||x||_∞≤ ||A^-1||_∞ ||A||_∞ = κ_∞(A).

Note that cond(A,x) can be much smaller than κ_∞(A); the condition number of A^T and A^H might or might not be equal to κ_∞(A).

The approximate number of floating-point operations for one right-hand side vector b is 2n² for real flavors and 8n² for complex flavors.

To estimate the condition number κ_∞(A), call ?gecon.

To refine the solution and estimate the error, call ?gerfs.

Parent topic: Solving Systems of Linear Equations: LAPACK Computational Routines

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