Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 6/24/2024
Public

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?getrs

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

Syntax

call sgetrs( trans, n, nrhs, a, lda, ipiv, b, ldb, info )

call dgetrs( trans, n, nrhs, a, lda, ipiv, b, ldb, info )

call cgetrs( trans, n, nrhs, a, lda, ipiv, b, ldb, info )

call zgetrs( trans, n, nrhs, a, lda, ipiv, b, ldb, info )

call getrs( a, ipiv, b [, trans] [,info] )

Include Files

  • mkl.fi, lapack.f90

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

trans

CHARACTER*1. 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

INTEGER. The order of A; the number of rows in B(n 0).

nrhs

INTEGER. The number of right-hand sides; nrhs 0.

a, b

REAL for sgetrs

DOUBLE PRECISION for dgetrs

COMPLEX for cgetrs

DOUBLE COMPLEX for zgetrs.

Arrays: a (size lda by *), b (size ldb by *).

The array a contains LU factorization of matrix A resulting from the call of ?getrf.

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

The second dimension of a must be at least max(1,n) and the second dimension of b at least max(1,nrhs).

lda

INTEGER. The leading dimension of a; lda max(1, n).

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ipiv

INTEGER.

Array, size at least max(1, n). The ipiv array, as returned by ?getrf.

Output Parameters

b

Overwritten by the solution matrix X.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine getrs interface are as follows:

a

Holds the matrix A of size (n, n).

b

Holds the matrix B of size (n, nrhs).

ipiv

Holds the vector of length n.

trans

Must be 'N', 'C', or 'T'. The default value is 'N'.

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.