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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zgbtrs (f07bs)

▸▿ 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_zgbtrs (f07bs) solves a complex band system of linear equations with multiple right-hand sides,

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

where

A

has been factorized by nag_lapack_zgbtrf (f07br).

Syntax

[b, info] = f07bs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

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

Description

nag_lapack_zgbtrs (f07bs) is used to solve a complex band system of linear equations

A X = B

A^{T} X = B

A^{H} X = B

, the function must be preceded by a call to nag_lapack_zgbtrf (f07br) 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'

, the solution is computed by solving

U^{T} Y = B

and then

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

trans ='C'

, the solution is computed by solving

U^{H} 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: $trans$ – string (length ≥ 1)

Indicates the form of the equations.

$trans ='N'$: $A X = B$ is solved for $X$ .
$trans ='T'$: $A^{T} X = B$ is solved for $X$ .
$trans ='C'$: $A^{H} X = B$ 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: $ab (ldab :)$ – complex 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_zgbtrf (f07br).

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_zgbtrf (f07br).

6: $b (ldb :)$ – 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 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: $b (ldb :)$ – 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

|E| \leq c (k) ε |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^{H})

(which is the same as

cond (A^{T})

) can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_zgbrfs (f07bv), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_zgbcon (f07bu) with

norm_p ='I'

Further Comments

The total number of real floating-point operations is approximately

8 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_zgbrfs (f07bv) to refine the solution and return an error estimate.

The real analogue of this function is nag_lapack_dgbtrs (f07be).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0.00 + 0.00 i \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0.00 + 0.00 i & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array})

and

B = (\begin{array}{r} - 1.06 + 21.50 i & 12.85 + 2.84 i \\ - 22.72 - 53.90 i & - 70.22 + 21.57 i \\ 28.24 - 38.60 i & - 20.7 - 31.23 i \\ - 34.56 + 16.73 i & 26.01 + 31.97 i \end{array}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_zgbtrf (f07br).

Open in the MATLAB editor: f07bs_example

function f07bs_example


fprintf('f07bs example results\n\n');

m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0    + 0i,      0    +  0i,     0    + 0i,     0    + 0i;
       0    + 0i,      0    +  0i,     0.97 - 2.84i,  0.59 - 0.48i;
       0    + 0i,     -2.05 -  0.85i, -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i,  -1.48 -  1.75i, -1.06 + 1.94i, -0.46 - 1.72i;
       0    + 6.3i,   -0.77 +  2.83i,  4.48 - 1.09i,  0    + 0i];

b = [ -1.06 + 21.5i,  12.85 +  2.84i;
     -22.72 - 53.9i, -70.22 + 21.57i;
      28.24 - 38.6i, -20.73 -  1.23i;
     -34.56 + 16.73i, 26.01 + 31.97i];

% Factorize
[abf, ipiv, info] = f07br( ...
                           m, kl, ku, ab);

%Solve
trans = 'N';
[x, info] = f07bs( ...
                   trans, kl, ku, abf, ipiv, b);

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

f07bs example results

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox