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 $U^{H} U$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{H}$ , where $L$ is lower triangular.

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

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 Hermitian matrix which is not positive definite, call nag_lapack_zhetrf (f07mr) 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^{H}| |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 real floating-point operations is approximately

\frac{4}{3} n^{3}

A call to nag_lapack_zpotrf (f07fr) may be followed by calls to the functions:

nag_lapack_zpotrs (f07fs) to solve $A X = B$ ;
nag_lapack_zpocon (f07fu) to estimate the condition number of $A$ ;
nag_lapack_zpotri (f07fw) to compute the inverse of $A$ .

The real analogue of this function is nag_lapack_dpotrf (f07fd).

Example

This example computes the Cholesky factorization of the matrix

A

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}) .

Open in the MATLAB editor: f07fr_example

function f07fr_example


fprintf('f07fr example results\n\n');

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

[L, info] = f07fr( ...
                   uplo, a);

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

f07fr example results

 factor
             1          2          3          4
 1      1.7972
        0.0000

 2      0.8402     1.3164
        1.0683     0.0000

 3      1.0572    -0.4702     1.5604
       -0.4674     0.3131     0.0000

 4      0.2337     0.0834     0.9360     0.6603
       -1.3910     0.0368     0.9900     0.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox