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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dgbtrs (f07be)

▸▿ 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_dgbtrs (f07be) solves a real band system of linear equations with multiple right-hand sides,

A X = B or A^{T} X = B,

where

A

has been factorized by nag_lapack_dgbtrf (f07bd).

Syntax

[b, info] = f07be(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

[b, info] = nag_lapack_dgbtrs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgbtrs (f07be) is used to solve a real band system of linear equations

A X = B

A^{T} X = B

, the function must be preceded by a call to nag_lapack_dgbtrf (f07bd) which computes the

L U

factorization of

A

A = P L U

. The solution is computed by forward and backward substitution.

trans ='N'

, the solution is computed by solving

P L Y = B

and then

U X = Y

trans ='T'

'C'

, the solution is computed by solving

U^{T} Y = B

and then

L^{T} 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: $trans$ – string (length ≥ 1)

Indicates the form of the equations.

$trans ='N'$: $A X = B$ is solved for $X$ .
$trans ='T'$ or $'C'$: $A^{T} X = B$ is solved for $X$ .

Constraint:

trans ='N'

'T'

'C'

2: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

3: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

4: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

2 \times kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

L U

factorization of

A

, as returned by nag_lapack_dgbtrf (f07bd).

5: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).

6: $b (ldb :)$ – 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

|E| \leq c (k) ε P |L| |U|,

c (k)

is a modest linear function of

k = k_{l} + k_{u} + 1

, and

ε

is the machine precision. This assumes

k ≪ n

\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) 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)

, and

cond (A^{T})

can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_dgbrfs (f07bh), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_dgbcon (f07bg) with

norm_p ='I'

Further Comments

The total number of floating-point operations is approximately

2 n (2 k_{l} + k_{u}) r

, assuming

n ≫ k_{l}

and

n ≫ k_{u}

This function may be followed by a call to nag_lapack_dgbrfs (f07bh) to refine the solution and return an error estimate.

The complex analogue of this function is nag_lapack_zgbtrs (f07bs).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_dgbtrf (f07bd).

Open in the MATLAB editor: f07be_example

function f07be_example


fprintf('f07be example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0,     0,     0,     0;
       0,     0,    -3.66, -2.13;
       0,     2.54, -2.73,  4.07;
      -0.23,  2.46,  2.46, -3.82;
      -6.98,  2.56, -4.78,  0];
b  = [ 4.42, -36.01;
      27.13, -31.67;
      -6.14, -1.16;
      10.50, -25.82];
% Factorize A
[abf, ipiv, info] = f07bd( ...
                           m, kl, ku, ab);

% Compute Solution
trans = 'N';
[x, info] = f07be( ...
                      trans, kl, ku, abf, ipiv, b);

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

f07be example results

Solution(s)
   -2.0000    1.0000
    3.0000   -4.0000
    1.0000    7.0000
   -4.0000   -2.0000

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Chapter Contents

Chapter Introduction

NAG Toolbox