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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpftrf (f07wd)

## Purpose

nag_lapack_dpftrf (f07wd) computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

## Syntax

[a, info] = f07wd(transr, uplo, n, a)
[a, info] = nag_lapack_dpftrf(transr, uplo, n, a)

## Description

nag_lapack_dpftrf (f07wd) 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 a lower triangular, stored in RFP format.

## References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the RFP representation of A$A$ is normal or transposed.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'T'${\mathbf{transr}}=\text{'T'}$
The matrix A$A$ is stored in transposed RFP format.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'T'$\text{'T'}$.
2:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of A$A$ is stored.
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'$\text{'L'}$.
3:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
4:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double array
The n$n$ by n$n$ symmetric matrix A$A$, stored in RFP format, as described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

None.

None.

### Output Parameters

1:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$, the factor U$U$ or L$L$ from the Cholesky factorization A = UTU$A={U}^{\mathrm{T}}U$ or A = LLT$A=L{L}^{\mathrm{T}}$, in the same storage format as A$A$.
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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W INFO > 0${\mathbf{INFO}}>0$
The leading minor of order _$_$ 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(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)n2$\frac{1}{3}{n}^{2}$.
A call to nag_lapack_dpftrf (f07wd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpftrf (f07wr).

## Example

```function nag_lapack_dpftrf_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = nag_lapack_dpftrf(transr, uplo, n, a);

if info == 0
% Convert factor to full array form, and print it
[f, info] = nag_matop_dtfttr(transr, uplo, n, aOut);
fprintf('\n');
[ifail] = nag_file_print_matrix_real_gen(uplo, 'n', f, 'Factor');
else
fprintf('\na is not positive definite.\n');
end
```
```

Factor
1          2          3          4
1      2.0396
2     -1.5297     1.6401
3      0.2746    -0.2500     0.7887
4     -0.0490     0.6737     0.6617     0.5347

```
```function f07wd_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = f07wd(transr, uplo, n, a);

if info == 0
% Convert factor to full array form, and print it
[f, info] = f01vg(transr, uplo, n, aOut);
fprintf('\n');
[ifail] = x04ca(uplo, 'n', f, 'Factor');
else
fprintf('\na is not positive definite.\n');
end
```
```

Factor
1          2          3          4
1      2.0396
2     -1.5297     1.6401
3      0.2746    -0.2500     0.7887
4     -0.0490     0.6737     0.6617     0.5347

```