NAG Library Function Document

nag_complex_lu_solve_mult_rhs (f04akc)

void	nag_complex_lu_solve_mult_rhs (Integer n, Integer nrhs, const Complex a[], Integer tda, const Integer pivot[], Complex b[], Integer tdb, NagError *fail)

3 Description

To solve a set of complex linear equations

A X = B

, the function must be preceded by a call to nag_complex_lu (f03ahc) which 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.

4 References

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[ $n \times tda$ ] – const ComplexInput: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: details of the $L U$ factorization, as returned by nag_complex_lu (f03ahc).
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: pivot[n] – const IntegerInput: On entry: details of the row interchanges as returned by nag_complex_lu (f03ahc).
6: b[ $n \times tdb$ ] – ComplexInput/Output: 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 $B$ .

On exit: $B$ is overwritten by the solution matrix $X$ .
7: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq nrhs$ .
8: 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 $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
On entry, $tdb = ⟨value⟩$ while $nrhs = ⟨value⟩$ . These arguments must satisfy $tdb \geq nrhs$ .
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
On entry, $nrhs = ⟨value⟩$ .
Constraint: $nrhs \geq 1$ .

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_lu_solve_mult_rhs (f04akc) is approximately proportional to

n^{2} r

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_lu_solve_mult_rhs (f04akc)

+− Contents

1 Purpose

2 Specification