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: $ap (:)$ – double array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

3: $b (ldb :)$ – 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

right-hand side matrix

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b.
$n$ , 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: $ap (:)$ – double array: The dimension of the array ap will be $\max (1, n \times (n + 1) / 2)$

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

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_dppsvx (f07gb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_packed_solve (f04be) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_real_posdef_packed_solve (f04be) calls nag_lapack_dppsv (f07ga) to solve the equations.

Further Comments

The total number of floating-point operations is approximately

\frac{1}{3} n^{3} + 2 n^{2} r

, where

r

is the number of right-hand sides.

The complex analogue of this function is nag_lapack_zppsv (f07gn).

Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite matrix

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and b = (\begin{matrix} 8.70 \\ - 13.35 \\ 1.89 \\ - 4.14 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

Open in the MATLAB editor: f07ga_example

function f07ga_example


fprintf('f07ga example results\n\n');

% Symmetric matrix A, upper triangular part packed in ap
uplo = 'U';
ap = [4.16  ...
     -3.12  5.03  ...
      0.56 -0.83  0.76  ...
     -0.10  1.18  0.34  1.18];

% Rhs
b = [  8.7;
     -13.35;
      1.89;
     -4.14];
n = int64(size(b,1));

% Solve
[apf, x, info] = f07ga( ...
                        uplo, ap, b);

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

[ifail] = x04cc( ...
                 uplo, 'N', n, apf, 'Cholesky factor');

f07ga example results

Solution
    1.0000   -1.0000    2.0000   -3.0000

 Cholesky factor
             1          2          3          4
 1      2.0396    -1.5297     0.2746    -0.0490
 2                 1.6401    -0.2500     0.6737
 3                            0.7887     0.6617
 4                                       0.5347

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

Chapter Contents

Chapter Introduction

NAG Toolbox