factorization returned by nag_lapack_dgttrf (f07cd) and an initial solution returned by nag_lapack_dgttrs (f07ce). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07ch(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

[x, ferr, berr, info] = nag_lapack_dgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgtrfs (f07ch) should normally be preceded by calls to nag_lapack_dgttrf (f07cd) and nag_lapack_dgttrs (f07ce). nag_lapack_dgttrf (f07cd) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element in each column, and

U

is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgttrs (f07ce) then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, nag_lapack_dgtrfs (f07ch) computes a component-wise backward error,

β

, the smallest relative perturbation in each element of

A

and

b

such that

\hat{x}

is the exact solution of a perturbed system

(A + E) \hat{x} = b + f, with |e_{i j}| \leq β |a_{i j}|, and |f_{j}| \leq β |b_{j}| .

The function also estimates a bound for the component-wise forward error in the computed solution defined by

\max |x_{i} - \hat{x_{i}}| / \max |\hat{x_{i}}|

, where

x

is the corresponding column of the exact solution,

X

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1)

Specifies the equations to be solved as follows:

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

Constraint:

trans ='N'

'T'

'C'

2: $dl (:)$ – double array

The dimension of the array dl must be at least

\max (1, n - 1)

Must contain the

(n - 1)

subdiagonal elements of the matrix

A

3: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

Must contain the

n

diagonal elements of the matrix

A

4: $du (:)$ – double array

The dimension of the array du must be at least

\max (1, n - 1)

Must contain the

(n - 1)

superdiagonal elements of the matrix

A

5: $dlf (:)$ – double array

The dimension of the array dlf must be at least

\max (1, n - 1)

Must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

6: $df (:)$ – double array

The dimension of the array df must be at least

\max (1, n)

Must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

7: $duf (:)$ – double array

The dimension of the array duf must be at least

\max (1, n - 1)

Must contain the

(n - 1)

elements of the first superdiagonal of

U

8: $du2 (:)$ – double array

The dimension of the array du2 must be at least

\max (1, n - 2)

Must contain the

(n - 2)

elements of the second superdiagonal of

U

9: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

Must contain the

n

pivot indices that define the permutation matrix

P

. At the

i

th step, row

i

of the matrix was interchanged with row

ipiv (i)

, and

ipiv (i)

must always be either

i

(i + 1)

ipiv (i) = i

indicating that a row interchange was not performed.

10: $b (ldb :)$ – 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

matrix of right-hand sides

B

11: $x (ldx :)$ – double array

The first dimension of the array x must be at least

\max (1, n)

The second dimension of the array x must be at least

\max (1, nrhs_p)

The

n

r

initial solution matrix

X

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays b, x and the dimension of the arrays d, df, ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the arrays b, x.
$r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $x (ldx :)$ – double array: The first dimension of the array x will be $\max (1, n)$ .
The second dimension of the array x will be $\max (1, nrhs_p)$ .

The $n$ by $r$ refined solution matrix $X$ .
2: $ferr (nrhs_p)$ – double array: Estimate of the forward error bound for each computed solution vector, such that ${‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖{\hat{x}}_{j}‖}_{\infty} \leq ferr (j)$ , where ${\hat{x}}_{j}$ is the $j$ th column of the computed solution returned in the array x and $x_{j}$ is the corresponding column of the exact solution $X$ . The estimate is almost always a slight overestimate of the true error.
3: $berr (nrhs_p)$ – double array: Estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
4: $info$ – 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

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{\infty}}{{‖x‖}_{\infty}} \leq κ (A) \frac{{‖E‖}_{\infty}}{{‖A‖}_{\infty}},

where

κ (A) = {‖A^{- 1}‖}_{\infty} {‖A‖}_{\infty}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

Function nag_lapack_dgtcon (f07cg) can be used to estimate the condition number of

A

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

A^{T} X = B

is proportional to

n r

. At most five steps of iterative refinement are performed, but usually only one or two steps are required.

The complex analogue of this function is nag_lapack_zgtrfs (f07cv).

Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and B = (\begin{matrix} 2.7 & 6.6 \\ - 0.5 & 10.8 \\ 2.6 & - 3.2 \\ 0.6 & - 11.2 \\ 2.7 & 19.1 \end{matrix}) .

Estimates for the backward errors and forward errors are also output.

Open in the MATLAB editor: f07ch_example

function f07ch_example


fprintf('f07ch example results\n\n');

% Tridiagonal matrix A stored as diagonals:
du = [        2.1    -1.0      1.9     8.0];
d  = [3.0     2.3    -5.0     -0.9     7.1];
dl = [3.4     3.6     7.0     -6.0        ];
n  = numel(d);

% Factorize A.
[dlf, df, duf, du2f, ipiv, info] = ...
  f07cd(dl, d, du);

% RHS
b = [ 2.7   6.6;
     -0.5  10.8;
      2.6  -3.2;
      0.6 -11.2;
      2.7  19.1];

% Solve Ax = b
  
trans = 'No transpose';
[x, info] = f07ce( ...
                   trans, dlf, df, duf, du2f, ipiv, b);

% Refine solution
[x, ferr, berr, info] = ...
  f07ch( ...
         trans, dl, d, du, dlf, df, duf, du2f, ipiv, b, x);

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

fprintf('Forward  error bounds = %10.1e  %10.1e\n',ferr); 
fprintf('Backward error bounds = %10.1e  %10.1e\n',berr);

f07ch example results

Solution:
   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000

Forward  error bounds =    9.4e-15     1.4e-14
Backward error bounds =    7.2e-17     5.9e-17

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

Chapter Contents

Chapter Introduction

NAG Toolbox