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 );

• mkl.h

## Description

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

 A*X = B if trans='N', AT*X = B if trans='T', AH*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 A*X = B is solved for X. If trans = 'T', then AT*X = B is solved for X. If trans = 'C', then AH*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 AT and AH 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.