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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgesv (f07aa)

## Purpose

nag_lapack_dgesv (f07aa) computes the solution to a real system of linear equations
 AX = B , $AX=B ,$
where A$A$ is an n$n$ by n$n$ matrix and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[a, ipiv, b, info] = f07aa(a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, ipiv, b, info] = nag_lapack_dgesv(a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgesv (f07aa) uses the LU$LU$ decomposition with partial pivoting and row interchanges to factor A$A$ as
 A = PLU , $A=PLU ,$
where P$P$ is a permutation matrix, L$L$ is unit lower triangular, and U$U$ is upper triangular. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.

## References

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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ coefficient matrix A$A$.
2:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
n$n$, the number of linear equations, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

lda ldb

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factors L$L$ and U$U$ from the factorization A = PLU$A=PLU$; the unit diagonal elements of L$L$ are not stored.
2:     ipiv(n) – int64int32nag_int array
If no constraints are violated, the pivot indices that define the permutation matrix P$P$; at the i$i$th step row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
3:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: nrhs_p, 3: a, 4: lda, 5: ipiv, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, uii${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor U$U$ is exactly singular, so the solution could not be computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies the equation of the form
 (A + E) x̂ = b , $(A+E) x^=b ,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) $‖ x^ - x ‖1 ‖ x ‖1 ≤ κ(A) ‖ E ‖1 ‖ A ‖1$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A $A$ 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 nag_lapack_dgesv (f07aa), nag_lapack_dgecon (f07ag) can be used to estimate the condition number of A $A$ and nag_lapack_dgerfs (f07ah) can be used to obtain approximate error bounds. Alternatives to nag_lapack_dgesv (f07aa), which return condition and error estimates directly are nag_linsys_real_square_solve (f04ba) and nag_lapack_dgesvx (f07ab).

The total number of floating point operations is approximately (2/3) n3 + 2n2 r $\frac{2}{3}{n}^{3}+2{n}^{2}r$, where r $r$ is the number of right-hand sides.
The complex analogue of this function is nag_lapack_zgesv (f07an).

## Example

```function nag_lapack_dgesv_example
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52;
24.35;
0.77;
-6.22];
[aOut, ipiv, bOut, info] = nag_lapack_dgesv(a, b)
```
```

aOut =

5.2500   -2.9500   -0.9500   -3.8000
0.3429    3.8914    2.3757    0.4129
0.3010   -0.4631   -1.5139    0.2948
-0.2114   -0.3299    0.0047    0.1314

ipiv =

2
2
3
4

bOut =

1.0000
-1.0000
3.0000
-5.0000

info =

0

```
```function f07aa_example
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52;
24.35;
0.77;
-6.22];
[aOut, ipiv, bOut, info] = f07aa(a, b)
```
```

aOut =

5.2500   -2.9500   -0.9500   -3.8000
0.3429    3.8914    2.3757    0.4129
0.3010   -0.4631   -1.5139    0.2948
-0.2114   -0.3299    0.0047    0.1314

ipiv =

2
2
3
4

bOut =

1.0000
-1.0000
3.0000
-5.0000

info =

0

```