NAG Library Function Document

nag_complex_lin_eqn_mult_rhs (f04adc)

void	nag_complex_lin_eqn_mult_rhs (Integer n, Integer nrhs, Complex a[], Integer tda, const Complex b[], Integer tdb, Complex x[], Integer tdx, NagError *fail)

3 Description

Given a set of complex linear equations

A X = B

, the function first computes an

L U

factorization of

A

with partial pivoting,

P A = L U

, where

P

is a permutation matrix,

L

is lower triangular and

U

is unit upper triangular. The columns

x

of the solution

X

are found by forward and backward substitution in

L y = P b

and

U x = y

, where

b

is a column of the right-hand side matrix

B

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 1$ .
2: nrhs – IntegerInput: On entry: $r$ , the number of right-hand sides.
Constraint: $nrhs \geq 1$ .
3: a[ $n \times tda$ ] – ComplexInput/Output: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the $n$ by $n$ matrix $A$ .

On exit: $A$ is overwritten by the lower triangular matrix $L$ and the off-diagonal elements of the upper triangular matrix $U$ . The unit diagonal elements of $U$ are not stored.
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: b[ $n \times tdb$ ] – const ComplexInput: Note: the $(i, j)$ th element of the matrix $B$ is stored in $b [(i - 1) \times tdb + j - 1]$ .

On entry: the $n$ by $r$ right-hand side matrix $B$ .
6: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq nrhs$ .
7: x[ $n \times tdx$ ] – ComplexOutput: Note: the $(i, j)$ th element of the matrix $X$ is stored in $x [(i - 1) \times tdx + j - 1]$ .

On exit: the $n$ by $r$ solution matrix $X$ . See also Section 6.
8: tdx – IntegerInput: On entry: the stride separating matrix column elements in the array x.

Constraint: $tdx \geq nrhs$ .
9: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . The arguments must satisfy $tda \geq n$ .
On entry, $tdb = ⟨value⟩$ while $nrhs = ⟨value⟩$ . These arguments must satisfy $tdb \geq nrhs$ .
On entry, $tdx = ⟨value⟩$ while $nrhs = ⟨value⟩$ . These arguments must satisfy $tdx \geq nrhs$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
On entry, $nrhs = ⟨value⟩$ .
Constraint: $nrhs \geq 1$ .
NE_SINGULAR: The matrix $A$ is singular, possibly due to rounding errors.

7 Accuracy

The accuracy of the computed solution depends on the conditioning of the original matrix. For a detailed error analysis see page 106 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_complex_lin_eqn_mult_rhs (f04adc) is approximately proportional to

n^{3}

The function may be called with the same array supplied for arguments b and x, in which case the solution vectors will overwrite the right-hand sides.

10 Example

To solve the set of linear equations

A X = B

where

A = (\begin{matrix} 1 & 1 + 2 i & 2 + 10 i \\ 1 + i & 3 i & - 5 + 14 i \\ 1 + i & 5 i & - 8 + 20 i \end{matrix}) and B = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) .

NAG Library Function Documentnag_complex_lin_eqn_mult_rhs (f04adc)

+− Contents

1 Purpose

2 Specification