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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zpotrs (f07fs)

▸▿ 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_zpotrs (f07fs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by nag_lapack_zpotrf (f07fr).

Syntax

[b, info] = f07fs(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

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

Description

nag_lapack_zpotrs (f07fs) is used to solve a complex Hermitian positive definite system of linear equations

A X = B

, this function must be preceded by a call to nag_lapack_zpotrf (f07fr) which computes the Cholesky factorization of

A

. The solution

X

is computed by forward and backward substitution.

uplo ='U'

A = U^{H} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{H} Y = B

and then

U X = Y

uplo ='L'

A = L L^{H}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{H} 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.

$uplo ='U'$: $A = U^{H} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{H}$ , 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)

The Cholesky factor of

A

, as returned by nag_lapack_zpotrf (f07fr).

3: $b (ldb :)$ – 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 array a.
$n$ , 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 $uplo ='U'$ , $|E| \leq c (n) ε |U^{H}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε |L| |L^{H}|$ ,

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_zporfs (f07fv), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_lapack_zpocon (f07fu).

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_zporfs (f07fv) to refine the solution and return an error estimate.

The real analogue of this function is nag_lapack_dpotrs (f07fe).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array})

and

B = (\begin{array}{r} 3.93 - 6.14 i & 1.48 + 6.58 i \\ 6.17 + 9.42 i & 4.65 - 4.75 i \\ - 7.17 - 21.83 i & - 4.91 + 2.29 i \\ 1.99 - 14.38 i & 7.64 - 10.79 i \end{array}) .

Here

A

is Hermitian positive definite and must first be factorized by nag_lapack_zpotrf (f07fr).

Open in the MATLAB editor: f07fs_example

function f07fs_example


fprintf('f07fs example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'Lower';
a = [ 3.23 + 0i,     0    + 0i,     0    + 0i,     0    + 0i;
      1.51 + 1.92i,  3.58 + 0i,     0    + 0i,     0    + 0i;
      1.90 - 0.84i, -0.23 - 1.11i,  4.09 + 0i,     0    + 0i;
      0.42 - 2.50i, -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];

[L, info] = f07fr( ...
                   uplo, a);

% Rhs
b = [ 3.93 -  6.14i,  1.48 +  6.58i;
      6.17 +  9.42i,  4.65 -  4.75i;
     -7.17 - 21.83i, -4.91 +  2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];

% Solve AX = B
[x, info] = f07fs( ...
                   uplo, L, b);

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

f07fs example results

Solution(s)
   1.0000 - 1.0000i  -1.0000 + 2.0000i
  -0.0000 + 3.0000i   3.0000 - 4.0000i
  -4.0000 - 5.0000i  -2.0000 + 3.0000i
   2.0000 + 1.0000i   4.0000 - 5.0000i

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Chapter Contents

Chapter Introduction

NAG Toolbox