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',

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 ldbnrhs 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:


Equation

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.

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