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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

uplo ='U'

, the upper triangle of

A

is stored.

uplo ='L'

, the lower triangle of

A

is stored.

Constraint:

uplo ='U'

'L'

2: $a (lda :)$ – complex array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

Hermitian matrix

A

If $uplo ='U'$ , the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

3: $b (ldb :)$ – 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)

Note: to solve the equations

A x = b

, where

b

is a single right-hand side, b may be supplied as a one-dimensional array with length

ldb = \max (1, n)

The

n

r

right-hand side matrix

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, b and the second dimension of the array a.
$n$ , the number of linear equations, i.e., 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, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

If $info = 0$ , the factor $U$ or $L$ from the Cholesky factorization $A = U^{H} U$ or $A = L L^{H}$ .
2: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

Note: to solve the equations $A x = b$ , where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $ldb = \max (1, n)$ .

If $info = 0$ , the $n$ by $r$ solution matrix $X$ .
3: $info$ – 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.

$info > 0$: The leading minor of order $_$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

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‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

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

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, 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.

nag_lapack_zposvx (f07fp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_solve (f04cd) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_solve (f04cd) calls nag_lapack_zposv (f07fn) to solve the equations.

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3} + 8 n^{2} r

, where

r

is the number of right-hand sides.

The real analogue of this function is nag_lapack_dposv (f07fa).

Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite matrix

A = (\begin{array}{r} 3.23 & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 \end{array})

and

b = (\begin{array}{r} 3.93 - 6.14 i \\ 6.17 + 9.42 i \\ - 7.17 - 21.83 i \\ 1.99 - 14.38 i \end{array}) .

Details of the Cholesky factorization of

A

are also output.

Open in the MATLAB editor: f07fn_example

function f07fn_example


fprintf('f07fn example results\n\n');

% Upper triangular part of Hermitian matrix A
uplo = 'Upper';
a = [ 3.23  + 0i,  1.51 - 1.92i,  1.90 + 0.84i,  0.42 + 2.50i;
      0     + 0i,  3.58 + 0i,    -0.23 + 1.11i, -1.18 + 1.37i;
      0     + 0i,  0    + 0i,     4.09 + 0i,     2.33 - 0.14i;
      0     + 0i,  0    + 0i,     0    + 0i,     4.29 + 0i];

% RHS
b = [ 3.93 -  6.14i;
      6.17 +  9.42i;
     -7.17 - 21.83i;
      1.99 - 14.38i];

% Solve
[af, x, info] = f07fn( ...
                       uplo, a, b);

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

[ifail] = x04da( ...
                 uplo, 'Non-unit', af, 'Cholesky factor U');

f07fn example results

Solution
   1.0000 - 1.0000i
  -0.0000 + 3.0000i
  -4.0000 - 5.0000i
   2.0000 + 1.0000i

 Cholesky factor U
             1          2          3          4
 1      1.7972     0.8402     1.0572     0.2337
        0.0000    -1.0683     0.4674     1.3910

 2                 1.3164    -0.4702     0.0834
                   0.0000    -0.3131    -0.0368

 3                            1.5604     0.9360
                              0.0000    -0.9900

 4                                       0.6603
                                         0.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox