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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dpotrs (f07fe)

▸▿ 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_dpotrs (f07fe) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by nag_lapack_dpotrf (f07fd).

Syntax

[b, info] = f07fe(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

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

Description

nag_lapack_dpotrs (f07fe) is used to solve a real symmetric positive definite system of linear equations

A X = B

, this function must be preceded by a call to nag_lapack_dpotrf (f07fd) which computes the Cholesky factorization of

A

. The solution

X

is computed by forward and backward substitution.

uplo ='U'

A = U^{T} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{T} Y = B

and then

U X = Y

uplo ='L'

A = L L^{T}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{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.

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

Constraint:

uplo ='U'

'L'

2: $a (lda :)$ – double 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_dpotrf (f07fd).

3: $b (ldb :)$ – double 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: $b (ldb :)$ – double 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 $uplo ='U'$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

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_dporfs (f07fh), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_lapack_dpocon (f07fg).

Further Comments

The total number of floating-point operations is approximately

2 n^{2} r

This function may be followed by a call to nag_lapack_dporfs (f07fh) to refine the solution and return an error estimate.

The complex analogue of this function is nag_lapack_zpotrs (f07fs).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

Here

A

is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd).

Open in the MATLAB editor: f07fe_example

function f07fe_example


fprintf('f07fe example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
     -3.12,  5.03, 0,    0;
      0.56, -0.83, 0.76, 0;
     -0.10,  1.18, 0.34, 1.18];

% Factorize
[L, info] = f07fd( ...
                   uplo, a);

% Rhs
b = [  8.70, 8.30;
     -13.35, 2.13;
       1.89, 1.61;
      -4.14, 5.00];

% Solve
[x, info] = f07fe( ...
                   uplo, L, b);

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

f07fe example results

Solution(s)
    1.0000    4.0000
   -1.0000    3.0000
    2.0000    2.0000
   -3.0000    1.0000

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Chapter Contents

Chapter Introduction

NAG Toolbox