NAG Library Routine Document

F07AAF (DGESV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07AAF (DGESV) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

matrix and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07AAF (

N, NRHS, A, LDA, IPIV, B, LDB, INFO)

INTEGER	N, NRHS, LDA, IPIV(N), LDB, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,)

The routine may be called by its LAPACK name dgesv.

3 Description

F07AAF (DGESV) uses the

L U

decomposition with partial pivoting and row interchanges to factor

A

A = P L U,

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

A X = B

4 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

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of linear equations, i.e., the order of the matrix $A$ .
Constraint: $N \geq 0$ .
2: NRHS – INTEGERInput: 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) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
On entry: the $n$ by $n$ 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 – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F07AAF (DGESV) is called.
Constraint: $LDA \geq \max (1, N)$ .
5: IPIV(N) – INTEGER arrayOutput: 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) arrayInput/Output: Note: the second dimension of the array B must be at least $\max (1, NRHS)$ .
On entry: the $n$ by $r$ right-hand side matrix $B$ .

On exit: if $INFO = 0$ , the $n$ by $r$ solution matrix $X$ .
7: LDB – INTEGERInput: On entry: the first dimension of the array B as declared in the (sub)program from which F07AAF (DGESV) is called.
Constraint: $LDB \geq \max (1, N)$ .
8: INFO – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

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

$INFO > 0$: If $INFO = i$ , $u_{i i}$ 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 (DGESV), F07AGF (DGECON) can be used to estimate the condition number of

A

and F07AHF (DGERFS) can be used to obtain approximate error bounds. Alternatives to F07AAF (DGESV), which return condition and error estimates directly are F04BAF and F07ABF (DGESVX).

8 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 (ZGESV).

9 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.

NAG Library Routine DocumentF07AAF (DGESV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07AAF (DGESV)