# ?getrs

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

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

## Description

This 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(lda,*)`, `b(ldb,*)`.

  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)`, the second dimension of b at least `max(1,nrhs)`.

lda

  INTEGER. The first dimension of a; `lda ≥ max(1, n)`.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

ipiv

  INTEGER.

  Array, DIMENSION 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.

## Fortran 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 Fortran 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 `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.

Parent topic: Routines for Solving Systems of Linear Equations