NAG Library Function Document

nag_hermitian_lin_eqn_mult_rhs (f04awc)

+− 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

1 Purpose

nag_hermitian_lin_eqn_mult_rhs (f04awc) calculates the approximate solution of a set of complex Hermitian positive definite linear equations with multiple right-hand sides,

A X = B

, where

A

has been factorized by nag_complex_cholesky (f01bnc).

2 Specification

#include <nag.h>

#include <nagf04.h>

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

3 Description

To solve a set of complex linear equations,

A X = B

, where

A

is positive definite Hermitian, this function must be preceded by a call to nag_complex_cholesky (f01bnc) which computes a Cholesky factorization

A = U^{H} U

, where

U

is an upper triangular matrix with real diagonal elements. The columns

x

of the solution

X

are found in two steps

U^{H} y = b

and

U x = y

, where

b

is a column of the right-hand side matrix

B

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

On entry: the off-diagonal elements of the upper triangular matrix $U$ as returned by nag_complex_cholesky (f01bnc). The lower triangle of the array is not used.
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: p[n] – doubleInput: On entry: the reciprocals of the diagonal elements of $U$ , as returned by nag_complex_cholesky (f01bnc).
6: 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 $B$ . See also Section 9.
7: tdb – IntegerInput: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq nrhs$ .
8: 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 9.
9: tdx – IntegerInput: On entry: the stride separating matrix column elements in the array x.

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

7 Accuracy

The solutions should be the best possible for the precision of computation having regard to the condition of

A

. The computed solution

x

, corresponding to the right-hand side

b

, satisfies the bound

\frac{‖{x - A}^{- 1} b‖}{‖A^{- 1} b‖} \leq c ε k .

Here

c

is a modest function of

n

ε

is the machine precision, and

k

is the condition number defined by

k = ‖A‖ ‖A^{- 1}‖ .

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_hermitian_lin_eqn_mult_rhs (f04awc) is approximately proportional to

n^{2} r

The function may be called with the same actual 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

is the positive definite Hermitian matrix:

(\begin{matrix} 15 & 1 - 2 i & 2 & - 4 + 3 i \\ 1 + 2 i & 20 & - 2 + i & 3 - 3 i \\ 2 & - 2 - i & 18 & - 1 + 2 i \\ - 4 - 3 i & 3 + 3 i & - 1 - 2 i & 26 \end{matrix})

and

B

is the single column vector:

(\begin{matrix} 25 + 34 i \\ 21 + 19 i \\ 12 - 21 i \\ 21 - 27 i \end{matrix}) .

NAG Library Function Documentnag_hermitian_lin_eqn_mult_rhs (f04awc)

+− 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 Library Function Document

nag_hermitian_lin_eqn_mult_rhs (f04awc)