?getrs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

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

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',
`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

trans	CHARACTER1. 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^HX = 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

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:

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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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?getrs

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