factorization returned by nag_lapack_zgttrf (f07cr) and an initial solution returned by nag_lapack_zgttrs (f07cs). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07cv(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

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

Description

nag_lapack_zgtrfs (f07cv) should normally be preceded by calls to nag_lapack_zgttrf (f07cr) and nag_lapack_zgttrs (f07cs). nag_lapack_zgttrf (f07cr) 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_zgttrs (f07cs) 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_zgtrfs (f07cv) 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 $X$ .
$trans ='T'$: Solve $A^{T} X = B$ for $X$ .
$trans ='C'$: Solve $A^{H} X = B$ for $X$ .

Constraint:

trans ='N'

'T'

'C'

2: $dl (:)$ – complex 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: $d (:)$ – complex 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: $du (:)$ – complex 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: $dlf (:)$ – complex 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: $df (:)$ – complex 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: $duf (:)$ – complex 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: $du2 (:)$ – complex 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: $b (ldb :)$ – 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

matrix of right-hand sides

B

11: $x (ldx :)$ – complex 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: $x (ldx :)$ – complex 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_zgtcon (f07cu) 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

A^{H} 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 real analogue of this function is nag_lapack_dgtrfs (f07ch).

Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array})

and

B = (\begin{array}{r} 2.4 - 5.0 i & 2.7 + 6.9 i \\ 3.4 + 18.2 i & - 6.9 - 5.3 i \\ - 14.7 + 9.7 i & - 6.0 - 0.6 i \\ 31.9 - 7.7 i & - 3.9 + 9.3 i \\ - 1.0 + 1.6 i & - 3.0 + 12.2 i \end{array}) .

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

Open in the MATLAB editor: f07cv_example

function f07cv_example


fprintf('f07cv example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize
[dlf, df, duf, du2, ipiv, info] = ...
  f07cr(dl, d, du);

% RHS
b  = [ 2.4 -  5i,    2.7 +  6.9i;
       3.4 + 18.2i, -6.9 -  5.3i;
     -14.7 +  9.7i, -6   -  0.6i;
      31.9 -  7.7i, -3.9 +  9.3i;
      -1   +  1.6i, -3   + 12.2i];

% Solve for x
trans = 'No transpose';
[x, info] = f07cs( ...
                   trans, dlf, df, duf, du2, ipiv, b);

% Refine solution
[x, ferr, berr, info] = ...
  f07cv( ...
         trans, dl, d, du, dlf, df, duf, du2, 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);

f07cv example results

Solution:
   1.0000 + 1.0000i   2.0000 - 1.0000i
   3.0000 - 1.0000i   1.0000 + 2.0000i
   4.0000 + 5.0000i  -1.0000 + 1.0000i
  -1.0000 - 2.0000i   2.0000 + 1.0000i
   1.0000 - 1.0000i   2.0000 - 2.0000i

Forward  error bounds =    5.5e-14     7.7e-14
Backward error bounds =    3.6e-17     1.0e-16

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

Chapter Contents

Chapter Introduction

NAG Toolbox