# NAG FL Interfacef07aaf (dgesv)

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## 1Purpose

f07aaf computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ matrix and $X$ and $B$ are $n×r$ matrices.

## 2Specification

Fortran Interface
 Subroutine f07aaf ( n, nrhs, a, lda, ipiv, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, lda, ldb Integer, Intent (Out) :: ipiv(n), info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*)
#include <nag.h>
 void f07aaf_ (const Integer *n, const Integer *nrhs, double a[], const Integer *lda, Integer ipiv[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07aaf, nagf_lapacklin_dgesv or its LAPACK name dgesv.

## 3Description

f07aaf uses the $LU$ decomposition with partial pivoting and row interchanges to factor $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular, and $U$ is upper triangular. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ coefficient matrix $A$.
On exit: the factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07aaf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{ipiv}\left({\mathbf{n}}\right)$Integer array Output
On exit: if no constraints are violated, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n×r$ solution matrix $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07aaf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies the equation of the form
 $(A+E) x^=b ,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of f07aaf, f07agf can be used to estimate the condition number of $A$ and f07ahf can be used to obtain approximate error bounds. Alternatives to f07aaf, which return condition and error estimates directly are f04baf and f07abf.

## 8Parallelism and Performance

f07aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{3}+2{n}^{2}r$, where $r$ is the number of right-hand sides.
The complex analogue of this routine is f07anf.

## 10Example

This example solves the equations
 $Ax = b ,$
where $A$ is the general matrix
 $A = ( 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 ) and b = ( 9.52 24.35 0.77 -6.22 ) .$
Details of the $LU$ factorization of $A$ are also output.

### 10.1Program Text

Program Text (f07aafe.f90)

### 10.2Program Data

Program Data (f07aafe.d)

### 10.3Program Results

Program Results (f07aafe.r)