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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.

Syntax

[ar, info] = f07wj(transr, uplo, n, ar)
[ar, info] = nag_lapack_dpftri(transr, uplo, n, ar)

Description

nag_lapack_dpftri (f07wj) is used to compute the inverse of a real symmetric positive definite matrix $A$, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The function must be preceded by a call to nag_lapack_dpftrf (f07wd), which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

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:     $\mathrm{transr}$ – string (length ≥ 1)
Specifies whether the RFP representation of $A$ is normal or transposed.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'T'}$
The matrix $A$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – double array
The Cholesky factorization of $A$ stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).

None.

Output Parameters

1:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – double array
The factorization stores the $n$ by $n$ matrix ${A}^{-1}$ stored in RFP format.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see 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.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  ${\mathbf{info}}>0$
The leading minor of order $_$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no function specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling nag_lapack_dsytri (f07mj).

Accuracy

The computed inverse $X$ satisfies
 $XA-I2≤cnεκ2A and AX-I2≤cnεκ2A ,$
where $c\left(n\right)$ is a modest function of $n$, $\epsilon$ is the machine precision and ${\kappa }_{2}\left(A\right)$ is the condition number of $A$ defined by
 $κ2A=A2A-12 .$

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

Example

This example computes the inverse of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite, stored in RFP format, and must first be factorized by nag_lapack_dpftrf (f07wd).
```function f07wj_example

fprintf('f07wj example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 0.76   0.34;
4.16   1.18;
-3.12   5.03;
0.56  -0.83;
-0.10   1.18];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

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

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

```
```f07wj example results

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
```