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: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (:)$ – complex array

The dimension of the array ap must be at least

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

The

n

n

Hermitian 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 $i \geq j$ .

Optional Input Parameters

None.

Output Parameters

1: $ap (:)$ – complex 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 $A = U^{H} U$ or $A = L L^{H}$ , in the same storage format as $A$ .
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_zhptrf (f07pr) 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.

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_zpptrf (f07gr) may be followed by calls to the functions:

nag_lapack_zpptrs (f07gs) to solve $A X = B$ ;
nag_lapack_zppcon (f07gu) to estimate the condition number of $A$ ;
nag_lapack_zpptri (f07gw) to compute the inverse of $A$ .

The real analogue of this function is nag_lapack_dpptrf (f07gd).

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}) .

using packed storage.

Open in the MATLAB editor: f07gr_example

function f07gr_example


fprintf('f07gr example results\n\n');

uplo = 'L';
n = int64(4);
ap = [3.23 + 0i    1.51 + 1.92i    1.90 - 0.84i    0.42 - 2.50i ...
                   3.58 + 0i      -0.23 - 1.11i   -1.18 - 1.37i ...
                                   4.09 + 0.00i    2.33 + 0.14i ...
                                                   4.29 + 0.00i];

[L, info] = f07gr( ...
                     uplo, n, ap);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, L, 'factor');

f07gr 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