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: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

3: $ab (ldab :)$ – complex array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The

n

n

Hermitian positive definite band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

Optional Input Parameters

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

Constraint: $n \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – complex array: The first dimension of the array ab will be $kd + 1$ .
The second dimension of the array ab will be $\max (1, n)$ .

The upper or lower triangle of $A$ stores the Cholesky factor $U$ or $L$ as specified by uplo, using the same storage format as described above.
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 $A$ . There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_zgbtrf (f07br) or as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

Accuracy

uplo ='U'

, the computed factor

U

is the exact factor of a perturbed matrix

A + E

, where

|E| \leq c (k + 1) ε |U^{H}| |U|,

c (k + 1)

is a modest linear function of

k + 1

, 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 (k + 1) ε \sqrt{a_{i i} a_{j j}}

Further Comments

The total number of real floating-point operations is approximately

4 n {(k + 1)}^{2}

, assuming

n ≫ k

A call to nag_lapack_zpbtrf (f07hr) may be followed by calls to the functions:

nag_lapack_zpbtrs (f07hs) to solve $A X = B$ ;
nag_lapack_zpbcon (f07hu) to estimate the condition number of $A$ .

The real analogue of this function is nag_lapack_dpbtrf (f07hd).

Example

This example computes the Cholesky factorization of the matrix

A

, where

A = (\begin{array}{r} 9.39 + 0.00 i & 1.08 - 1.73 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.08 + 1.73 i & 1.69 + 0.00 i & - 0.04 + 0.29 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 0.04 - 0.29 i & 2.65 + 0.00 i & - 0.33 + 2.24 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.33 - 2.24 i & 2.17 + 0.00 i \end{array}) .

Open in the MATLAB editor: f07hr_example

function f07hr_example


fprintf('f07hr example results\n\n');

uplo = 'L';
kd = int64(1);
n  = int64(4);
ab = [ 9.39 + 0i,     1.69 + 0i,      2.65 + 0i,      2.17 + 0i;
       1.08 - 1.73i, -0.04 + 0.29i,  -0.33 + 2.24i    0    + 0i];

% Factorize
[abf, info] = f07hr( ...
                     uplo, kd, ab);

ku = int64(0);
[ifail] = x04de( ...
                 n, n, kd, ku, abf, 'Cholesky factor');

f07hr example results

 Cholesky factor
             1          2          3          4
 1      3.0643
        0.0000

 2      0.3524     1.1167
       -0.5646     0.0000

 3                -0.0358     1.6066
                   0.2597     0.0000

 4                           -0.2054     0.4289
                              1.3942     0.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox