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

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of linear equations, i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs$ – Integer Input: On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs \geq 0$ .
3: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $n \times n$ coefficient matrix $A$ .

On exit: the factors $L$ and $U$ from the factorization $A = P L U$ ; the unit diagonal elements of $L$ are not stored.
4: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f07aaf is called.

Constraint: $lda \geq \max (1, n)$ .
5: $ipiv (n)$ – 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 $ipiv (i)$ . $ipiv (i) = i$ indicates a row interchange was not required.
6: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .

On entry: the $n \times r$ right-hand side matrix $B$ .

On exit: if $info = 0$ , the $n \times r$ solution matrix $X$ .
7: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f07aaf is called.

Constraint: $ldb \geq \max (1, n)$ .
8: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: Element $⟨ value ⟩$ 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.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies the equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{1} = O (ε) {‖ A ‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{1}}{{‖ x ‖}_{1}} \leq κ (A) \frac{{‖ E ‖}_{1}}{{‖ A ‖}_{1}}

where

κ (A) = {‖ 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.

8 Parallelism 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.

9 Further Comments

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.

10 Example

This example solves the equations

A x = b,

where

A

is the general matrix

A = (\begin{matrix} 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 \end{matrix}) and b = (\begin{matrix} 9.52 \\ 24.35 \\ 0.77 \\ - 6.22 \end{matrix}) .

Details of the

L U

factorization of

A

are also output.

f07aa: FL CL CPP AD

NAG FL Interfacef07aaf (dgesv)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f07aaf (dgesv)