matrices, using the modified Cholesky factorization returned by nag_lapack_zpttrf (f07jr) and an initial solution returned by nag_lapack_zpttrs (f07js). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07jv(uplo, d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)

[x, ferr, berr, info] = nag_lapack_zptrfs(uplo, d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zptrfs (f07jv) should normally be preceded by calls to nag_lapack_zpttrf (f07jr) and nag_lapack_zpttrs (f07js). nag_lapack_zpttrf (f07jr) computes a modified Cholesky factorization of the matrix

A

A = L D L^{H},

where

L

is a unit lower bidiagonal matrix and

D

is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) 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_zptrfs (f07jv) 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

Note that the modified Cholesky factorization of

A

can also be expressed as

A = U^{H} D U,

where

U

is unit upper bidiagonal.

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: $uplo$ – string (length ≥ 1)

Specifies the form of the factorization as follows:

$uplo ='U'$: $A = U^{H} D U$ .
$uplo ='L'$: $A = L D L^{H}$ .

Constraint:

uplo ='U'

'L'

2: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

Must contain the

n

diagonal elements of the matrix of

A

3: $e (:)$ – complex array

The dimension of the array e must be at least

\max (1, n - 1)

uplo ='U'

, e must contain the

(n - 1)

superdiagonal elements of the matrix

A

uplo ='L'

, e must contain the

(n - 1)

subdiagonal elements of the matrix

A

4: $df (:)$ – double array

The dimension of the array df must be at least

\max (1, n)

Must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

5: $ef (:)$ – complex array

The dimension of the array ef must be at least

\max (1, n - 1)

uplo ='U'

, ef must contain the

(n - 1)

superdiagonal elements of the unit upper bidiagonal matrix

U

from the

U^{H} D U

factorization of

A

uplo ='L'

, ef must contain the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

from the

L D L^{H}

factorization of

A

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

matrix of right-hand sides

B

7: $x (ldx :)$ – 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.
$n$ , 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_zptcon (f07ju) can be used to compute the condition number of

A

Further Comments

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

A 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_dptrfs (f07jh).

Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array})

and

B = (\begin{array}{r} 64.0 + 16.0 i & - 16.0 - 32.0 i \\ 93.0 + 62.0 i & 61.0 - 66.0 i \\ 78.0 - 80.0 i & 71.0 - 74.0 i \\ 14.0 - 27.0 i & 35.0 + 15.0 i \end{array}) .

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

Open in the MATLAB editor: f07jv_example

function f07jv_example


fprintf('f07jv example results\n\n');

% Hermitian tridiagonal A stored as two diagonals
d = [ 16            41          46            21];
e = [ 16 + 16i      18 - 9i      1 - 4i         ];

% Factorize
[df, ef, info] = f07jr( ...
                        d, e);

% RHS
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,   61 - 66i;
      78 - 80i,   71 - 74i;
      14 - 27i,   35 + 15i];

% Solve
uplo = 'L';
[x, info] = f07js( ...
                   uplo, df, ef, b);


% Refine
[x, ferr, berr, info] = f07jv( ...
                               uplo, d, e, df, ef, b, x);

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

f07jv example results

Solution(s)
   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i

Forward  error bounds =    9.0e-12     6.1e-12
Backward error bounds =    0.0e+00     0.0e+00

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

Chapter Contents

Chapter Introduction

NAG Toolbox