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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpotrf (f07fd)

## Purpose

nag_lapack_dpotrf (f07fd) computes the Cholesky factorization of a real symmetric positive definite matrix.

## Syntax

[a, info] = f07fd(uplo, a, 'n', n)
[a, info] = nag_lapack_dpotrf(uplo, a, 'n', n)

## Description

nag_lapack_dpotrf (f07fd) forms the Cholesky factorization of a real symmetric positive definite 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$U$ is an upper triangular matrix and L$L$ is lower triangular.

## 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:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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 positive definite matrix A$A$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of a$a$ must be stored and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of a$a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

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

lda

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of A$A$ stores the Cholesky factor U$U$ or L$L$ as specified by uplo.
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: a, 4: lda, 5: 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$A$. To factorize a symmetric matrix which is not positive definite, call nag_lapack_dsytrf (f07md) instead.

## 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(n)ε|UT||U| , $|E|≤c(n)ε|UT||U| ,$
c(n)$c\left(n\right)$ is a modest linear function of n$n$, 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(n)ε×sqrt(aiiajj)$|{e}_{ij}|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

The total number of floating point operations is approximately (1/3)n3$\frac{1}{3}{n}^{3}$.
A call to nag_lapack_dpotrf (f07fd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpotrf (f07fr).

## Example

```function nag_lapack_dpotrf_example
uplo = 'L';
a = [4.16, 0, 0, 0;
-3.12, 5.03, 0, 0;
0.56, -0.83, 0.76, 0;
-0.1, 1.18, 0.34, 1.18];
[aOut, info] = nag_lapack_dpotrf(uplo, a)
```
```

aOut =

2.0396         0         0         0
-1.5297    1.6401         0         0
0.2746   -0.2500    0.7887         0
-0.0490    0.6737    0.6617    0.5347

info =

0

```
```function f07fd_example
uplo = 'L';
a = [4.16, 0, 0, 0;
-3.12, 5.03, 0, 0;
0.56, -0.83, 0.76, 0;
-0.1, 1.18, 0.34, 1.18];
[aOut, info] = f07fd(uplo, a)
```
```

aOut =

2.0396         0         0         0
-1.5297    1.6401         0         0
0.2746   -0.2500    0.7887         0
-0.0490    0.6737    0.6617    0.5347

info =

0

```