# NAG FL Interfacef07gwf (zpptri)

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## 1Purpose

f07gwf computes the inverse of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by f07grf, using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07gwf ( uplo, n, ap, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: ap(*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07gwf_ (const char *uplo, const Integer *n, Complex ap[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07gwf, nagf_lapacklin_zpptri or its LAPACK name zpptri.

## 3Description

f07gwf is used to compute the inverse of a complex Hermitian positive definite matrix $A$, the routine must be preceded by a call to f07grf, which computes the Cholesky factorization of $A$, using packed storage.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{H}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{H}}\left({L}^{-1}\right)$.

## 4References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## 5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{ap}\left(*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the Cholesky factor of $A$ stored in packed form, as returned by f07grf.
On exit: the factorization is overwritten by the $n×n$ matrix ${A}^{-1}$.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
4: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and 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.
${\mathbf{info}}>0$
Diagonal element $⟨\mathit{\text{value}}⟩$ of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of $A$ cannot be computed.

## 7Accuracy

The computed inverse $X$ satisfies
 $‖XA-I‖2≤c(n)εκ2(A) and ‖AX-I‖2≤c(n)εκ2(A) ,$
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
 $κ2(A)=‖A‖2‖A-1‖2 .$

## 8Parallelism and Performance

f07gwf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this routine is f07gjf.

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) .$
Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by f07grf.

### 10.1Program Text

Program Text (f07gwfe.f90)

### 10.2Program Data

Program Data (f07gwfe.d)

### 10.3Program Results

Program Results (f07gwfe.r)