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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpotrf (f07fr)

## Purpose

nag_lapack_zpotrf (f07fr) computes the Cholesky factorization of a complex Hermitian positive definite matrix.

## Syntax

[a, info] = f07fr(uplo, a, 'n', n)
[a, info] = nag_lapack_zpotrf(uplo, a, 'n', n)

## Description

nag_lapack_zpotrf (f07fr) forms the Cholesky factorization of a complex Hermitian positive definite matrix A$A$ either as A = UHU$A={U}^{\mathrm{H}}U$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = LLH$A=L{L}^{\mathrm{H}}$ 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 UHU${U}^{\mathrm{H}}U$, 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 LLH$L{L}^{\mathrm{H}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     a(lda, : $:$) – complex 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$n$ Hermitian 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, : $:$) – complex 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 Hermitian matrix which is not positive definite, call nag_lapack_zhetrf (f07mr) 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|UH||U| , $|E|≤c(n)ε|UH||U| ,$
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 real floating point operations is approximately (4/3)n3$\frac{4}{3}{n}^{3}$.
A call to nag_lapack_zpotrf (f07fr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpotrf (f07fd).

## Example

```function nag_lapack_zpotrf_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
[aOut, info] = nag_lapack_zpotrf(uplo, a)
```
```

aOut =

1.7972 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.8402 + 1.0683i   1.3164 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
1.0572 - 0.4674i  -0.4702 + 0.3131i   1.5604 + 0.0000i   0.0000 + 0.0000i
0.2337 - 1.3910i   0.0834 + 0.0368i   0.9360 + 0.9900i   0.6603 + 0.0000i

info =

0

```
```function f07fr_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
[aOut, info] = f07fr(uplo, a)
```
```

aOut =

1.7972 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.8402 + 1.0683i   1.3164 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
1.0572 - 0.4674i  -0.4702 + 0.3131i   1.5604 + 0.0000i   0.0000 + 0.0000i
0.2337 - 1.3910i   0.0834 + 0.0368i   0.9360 + 0.9900i   0.6603 + 0.0000i

info =

0

```