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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dpbtrs (f07he)

▸▿ 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_dpbtrs (f07he) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by nag_lapack_dpbtrf (f07hd).

Syntax

[b, info] = f07he(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

[b, info] = nag_lapack_dpbtrs(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpbtrs (f07he) is used to solve a real symmetric positive definite band system of linear equations

A X = B

, the function must be preceded by a call to nag_lapack_dpbtrf (f07hd) 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: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

3: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The Cholesky factor of

A

, as returned by nag_lapack_dpbtrf (f07hd).

4: $b (ldb :)$ – 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 second dimension of the array ab.
$n$ , 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 $uplo ='U'$ , $|E| \leq c (k + 1) ε |U^{T}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (k + 1) ε |L| |L^{T}|$ ,

c (k + 1)

is a modest linear function of

k + 1

, 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 (k + 1) 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_dpbrfs (f07hh), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_lapack_dpbcon (f07hg).

Further Comments

The total number of floating-point operations is approximately

4 n k r

, assuming

n ≫ k

This function may be followed by a call to nag_lapack_dpbrfs (f07hh) to refine the solution and return an error estimate.

The complex analogue of this function is nag_lapack_zpbtrs (f07hs).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd).

Open in the MATLAB editor: f07he_example

function f07he_example


fprintf('f07he example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form 
uplo = 'Lower';
kd = int64(1);
n  = int64(4);
ab = [5.49,  5.63,  2.60, 5.17;
      2.68, -2.39, -2.22, 0.00];

% Factorize
[abf, info] = f07hd( ...
                     uplo, kd, ab);

% RHS
b  = [22.09,   5.1;
       9.31,  30.81;
      -5.24, -25.82;
      11.83,  22.9];

% Solve Ax = B
[x, info] = f07he( ...
                   uplo, kd, abf, b);

disp('Solution:');
disp(x);

f07he example results

Solution:
    5.0000   -2.0000
   -2.0000    6.0000
   -3.0000   -1.0000
    1.0000    4.0000

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Chapter Contents

Chapter Introduction

NAG Toolbox