Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn14.pdf

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)

Specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $a (lda :)$ – double 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

symmetric positive definite 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.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

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

The upper or lower triangle of $A$ stores the Cholesky factor $U$ or $L$ as specified by uplo.
2: $info$ – 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 $_$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$ . To factorize a symmetric matrix which is not positive definite, call nag_lapack_dsytrf (f07md) instead.

Accuracy

uplo ='U'

, the computed factor

U

is the exact factor of a perturbed matrix

A + E

, where

|E| \leq c (n) ε |U^{T}| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. If

uplo ='L'

, a similar statement holds for the computed factor

L

. It follows that

|e_{i j}| \leq c (n) ε \sqrt{a_{i i} a_{j j}}

Further Comments

The total number of floating-point operations is approximately

\frac{1}{3} n^{3}

A call to nag_lapack_dpotrf (f07fd) may be followed by calls to the functions:

nag_lapack_dpotrs (f07fe) to solve $A X = B$ ;
nag_lapack_dpocon (f07fg) to estimate the condition number of $A$ ;
nag_lapack_dpotri (f07fj) to compute the inverse of $A$ .

The complex analogue of this function is nag_lapack_zpotrf (f07fr).

Example

This example computes the Cholesky factorization of the matrix

A

, where

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}) .

Open in the MATLAB editor: f07fd_example

function f07fd_example


fprintf('f07fd example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
     -3.12,  5.03, 0,    0;
      0.56, -0.83, 0.76, 0;
     -0.10,  1.18, 0.34, 1.18];

[L, info] = f07fd( ...
                   uplo, a);

[ifail] = x04ca( ...
                 uplo, 'Non-unit', L, 'Cholesky factor');

f07fd example results

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

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

Chapter Contents

Chapter Introduction

NAG Toolbox