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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbtrf (f07hd)

## Purpose

nag_lapack_dpbtrf (f07hd) computes the Cholesky factorization of a real symmetric positive definite band matrix.

## Syntax

[ab, info] = f07hd(uplo, kd, ab, 'n', n)
[ab, info] = nag_lapack_dpbtrf(uplo, kd, ab, 'n', n)

## Description

nag_lapack_dpbtrf (f07hd) forms the Cholesky factorization of a real symmetric positive definite band matrix $A$ either as $A={U}^{\mathrm{T}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as $A$.

## References

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:     $\mathrm{uplo}$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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

If ${\mathbf{uplo}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤ck+1εUTU ,$
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision.
If ${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

The total number of floating-point operations is approximately $n{\left(k+1\right)}^{2}$, assuming $n\gg k$.
A call to nag_lapack_dpbtrf (f07hd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpbtrf (f07hr).

## Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .$
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