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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zhetrs (f07ms)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zhetrs (f07ms) solves a complex Hermitian indefinite system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by nag_lapack_zhetrf (f07mr).

Syntax

[b, info] = f07ms(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

[b, info] = nag_lapack_zhetrs(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zhetrs (f07ms) is used to solve a complex Hermitian indefinite system of linear equations

A X = B

, this function must be preceded by a call to nag_lapack_zhetrf (f07mr) which computes the Bunch–Kaufman factorization of

A

uplo ='U'

A = P U D U^{H} P^{T}

, where

P

is a permutation matrix,

U

is an upper triangular matrix and

D

is an Hermitian block diagonal matrix with

1

1

and

2

2

blocks; the solution

X

is computed by solving

P U D Y = B

and then

U^{H} P^{T} X = Y

uplo ='L'

A = P L D L^{H} P^{T}

, where

L

is a lower triangular matrix; the solution

X

is computed by solving

P L D Y = B

and then

L^{H} P^{T} X = Y

References

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

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Specifies how

A

has been factorized.

$uplo ='U'$: $A = P U D U^{H} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{H} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $a (lda :)$ – complex array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

Details of the factorization of

A

, as returned by nag_lapack_zhetrf (f07mr).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

Details of the interchanges and the block structure of

D

, as returned by nag_lapack_zhetrf (f07mr).

4: $b (ldb :)$ – complex array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, nrhs_p)

The

n

r

right-hand side matrix

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, b and the second dimension of the arrays a, ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides.

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

The $n$ by $r$ solution matrix $X$ .
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

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.

Accuracy

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $uplo ='U'$ , $|E| \leq c (n) ε P |U| |D| |U^{H}| P^{T}$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε P |L| |D| |L^{H}| P^{T}$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (n) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

Note that

cond (A, x)

can be much smaller than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_zherfs (f07mv), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_lapack_zhecon (f07mu).

Further Comments

The total number of real floating-point operations is approximately

8 n^{2} r

This function may be followed by a call to nag_lapack_zherfs (f07mv) to refine the solution and return an error estimate.

The real analogue of this function is nag_lapack_dsytrs (f07me).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 1.36 + 0.00 i & 1.58 + 0.90 i & 2.21 - 0.21 i & 3.91 + 1.50 i \\ 1.58 - 0.90 i & - 8.87 + 0.00 i & - 1.84 - 0.03 i & - 1.78 + 1.18 i \\ 2.21 + 0.21 i & - 1.84 + 0.03 i & - 4.63 + 0.00 i & 0.11 + 0.11 i \\ 3.91 - 1.50 i & - 1.78 - 1.18 i & 0.11 - 0.11 i & - 1.84 + 0.00 i \end{array})

and

B = (\begin{array}{r} 7.79 + 5.48 i & - 35.39 + 18.01 i \\ - 0.77 - 16.05 i & 4.23 - 70.02 i \\ - 9.58 + 3.88 i & - 24.79 - 8.40 i \\ 2.98 - 10.18 i & 28.68 - 39.89 i \end{array}) .

Here

A

is Hermitian indefinite and must first be factorized by nag_lapack_zhetrf (f07mr).

Open in the MATLAB editor: f07ms_example

function f07ms_example


fprintf('f07ms example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored)
uplo = 'L';
a = [-1.36 + 0i,     0    + 0i,     0    + 0i,      0    + 0i;
      1.58 - 0.90i, -8.87 + 0i,     0    + 0i,      0    + 0i;
      2.21 + 0.21i, -1.84 + 0.03i, -4.63 + 0i,      0    + 0i;
      3.91 - 1.50i, -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];

% Factorize
[af, ipiv, info] = f07mr( ...
                          uplo, a);

% RHS
b = [ 7.79 +  5.48i, -35.39 + 18.01i;
     -0.77 - 16.05i,   4.23 - 70.02i;
     -9.58 +  3.88i, -24.79 -  8.40i;
      2.98 - 10.18i,  28.68 - 39.89i];

% Solve
[x, info] = f07ms( ...
                   uplo, af, ipiv, b);

disp('Solution(s)');
disp(x);

f07ms example results

Solution(s)
   1.0000 - 1.0000i   3.0000 - 4.0000i
  -1.0000 + 2.0000i  -1.0000 + 5.0000i
   3.0000 - 2.0000i   7.0000 - 2.0000i
   2.0000 + 1.0000i  -8.0000 + 6.0000i

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Chapter Contents

Chapter Introduction

NAG Toolbox