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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$A$ either as A = UTU$A={U}^{\mathrm{T}}U$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = LLT$A=L{L}^{\mathrm{T}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ (or L$L$) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as A$A$.

## References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
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$A$ is stored and how A$A$ is to be factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored and A$A$ is factorized as UTU${U}^{\mathrm{T}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored and A$A$ is factorized as LLT$L{L}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ symmetric band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

ldab

### Output Parameters

1:     ab(ldab, : $:$) – double array
The first dimension of the array ab will be kd + 1${\mathbf{kd}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldabkd + 1$\mathit{ldab}\ge {\mathbf{kd}}+1$.
The upper or lower triangle of A$A$ stores the Cholesky factor U$U$ or L$L$ as specified by uplo, using the same storage format as described above.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: kd, 4: ab, 5: ldab, 6: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the leading minor of order i$i$ is not positive definite and the factorization could not be completed. Hence A$A$ itself is not positive definite. This may indicate an error in forming the matrix A$A$. 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 uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the computed factor U$U$ is the exact factor of a perturbed matrix A + E$A+E$, where
 |E| ≤ c(k + 1)ε|UT||U| , $|E|≤c(k+1)ε|UT||U| ,$
c(k + 1)$c\left(k+1\right)$ is a modest linear function of k + 1$k+1$, and ε$\epsilon$ is the machine precision.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factor L$L$. It follows that |eij|c(k + 1)ε×sqrt(aiiajj)$|{e}_{ij}|\le c\left(k+1\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

The total number of floating point operations is approximately n(k + 1)2$n{\left(k+1\right)}^{2}$, assuming nk$n\gg k$.
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

```function nag_lapack_dpbtrf_example
uplo = 'L';
kd = int64(1);
ab = [5.49, 5.63, 2.6, 5.17;
2.68, -2.39, -2.22, 5.495816452771857e+222];
[abOut, info] = nag_lapack_dpbtrf(uplo, kd, ab)
```
```

1.0e+222 *

0.0000    0.0000    0.0000    0.0000
0.0000   -0.0000   -0.0000    5.4958

info =

0

```
```function f07hd_example
uplo = 'L';
kd = int64(1);
ab = [5.49, 5.63, 2.6, 5.17;
2.68, -2.39, -2.22, 5.495816452771857e+222];
[abOut, info] = f07hd(uplo, kd, ab)
```
```

1.0e+222 *

0.0000    0.0000    0.0000    0.0000
0.0000   -0.0000   -0.0000    5.4958

info =

0

```