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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpftri (f07wj)

## Purpose

nag_lapack_dpftri (f07wj) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by nag_lapack_dpftrf (f07wd) 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] = f07wj(transr, uplo, n, a)
[a, info] = nag_lapack_dpftri(transr, uplo, n, a)

## Description

nag_lapack_dpftri (f07wj) is used to compute the inverse of a real symmetric positive definite matrix A$A$, the function must be preceded by a call to nag_lapack_dpftrf (f07wd), which computes the Cholesky factorization of A$A$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = UTU$A={U}^{\mathrm{T}}U$ and A1${A}^{-1}$ is computed by first inverting U$U$ and then forming (U1)UT$\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = LLT$A=L{L}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by first inverting L$L$ and then forming LT(L1)${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

## References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19
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 how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UTU$A={U}^{\mathrm{T}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LLT$A=L{L}^{\mathrm{T}}$, 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 Cholesky factorization of A$A$ stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).

None.

None.

### Output Parameters

1:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double array
The factorization stores the n$n$ by n$n$ matrix A1${A}^{-1}$ stored in RFP format.
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

The computed inverse X$X$ satisfies
 ‖XA − I‖2 ≤ c(n)εκ2(A)   and   ‖AX − I‖2 ≤ c(n)εκ2(A) , $‖XA-I‖2≤c(n)εκ2(A) and ‖AX-I‖2≤c(n)εκ2(A) ,$
where c(n)$c\left(n\right)$ is a modest function of n$n$, ε$\epsilon$ is the machine precision and κ2(A)${\kappa }_{2}\left(A\right)$ is the condition number of A$A$ defined by
 κ2(A) = ‖A‖2‖A − 1‖2 . $κ2(A)=‖A‖2‖A-1‖2 .$

The total number of floating point operations is approximately (2/3)n3$\frac{2}{3}{n}^{3}$.
The complex analogue of this function is nag_lapack_zpftri (f07ww).

## Example

```function nag_lapack_dpftri_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
[a, info] = nag_lapack_dpftrf(transr, uplo, n, a);

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

Inverse
1          2          3          4
1      0.6995
2      0.7769     1.4239
3      0.7508     1.8255     4.0688
4     -0.9340    -1.8841    -2.9342     3.4978

```
```function f07wj_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
[a, info] = f07wd(transr, uplo, n, a);

if info == 0
% Compute inverse of a
[a, info] = f07wj(transr, uplo, n, a);
% Convert inverse to full array form, and print it
[f, info] = f01vg(transr, uplo, n, a);
fprintf('\n');
[ifail] = x04ca(uplo, 'n', f, 'Inverse');
else
fprintf('\na is not positive definite.\n');
end
```
```

Inverse
1          2          3          4
1      0.6995
2      0.7769     1.4239
3      0.7508     1.8255     4.0688
4     -0.9340    -1.8841    -2.9342     3.4978

```