NAG Toolbox: nag_lapack_dpbtrf (f07hd)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)

Specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

$\mathbf{uplo}=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $U^{\mathrm{T}}U$, where $U$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathrm{kd}$ – int64int32nag_int scalar

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

Constraint: $\mathbf{kd}\ge 0$.

3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array

The first dimension of the array ab must be at least $\mathbf{kd}+1$.

The second dimension of the array ab must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ symmetric positive definite band matrix $A$.

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the second dimension of the array ab.

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array

The first dimension of the array ab will be $\mathbf{kd}+1$.

The second dimension of the array ab will be $\max \left(1,\mathbf{n}\right)$.

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:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

   $\mathbf{info}>0$

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 symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_dgbtrf (f07bd) or as a full symmetric matrix, by calling nag_lapack_dsytrf (f07md).

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07hd_example

function f07hd_example


fprintf('f07hd example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form 
uplo = 'Lower';
kd = int64(1);
n  = int64(4);
ab = [5.49,  5.63,  2.60, 5.17;
      2.68, -2.39, -2.22, 0.00];

% Factorize
[abf, info] = f07hd( ...
                     uplo, kd, ab);

ku = int64(0);
[ifail] = x04ce( ...
                 n, n, kd, ku, abf, 'Cholesky factor');

f07hd example results

 Cholesky factor
             1          2          3          4
 1      2.3431
 2      1.1438     2.0789
 3                -1.1497     1.1306
 4                           -1.9635     1.1465