# NAG CL Interfacef07fwc (zpotri)

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## 1Purpose

f07fwc computes the inverse of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by f07frc.

## 2Specification

 #include
 void f07fwc (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, NagError *fail)
The function may be called by the names: f07fwc, nag_lapacklin_zpotri or nag_zpotri.

## 3Description

f07fwc is used to compute the inverse of a complex Hermitian positive definite matrix $A$, the function must be preceded by a call to f07frc, which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $A={U}^{\mathrm{H}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{H}}$.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $A=L{L}^{\mathrm{H}}$ 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: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{uplo}$Nag_UploType Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: the upper triangular matrix $U$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or the lower triangular matrix $L$ if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, as returned by f07frc.
On exit: $U$ is overwritten by the upper triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$; $L$ is overwritten by the lower triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$.
5: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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

f07fwc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this function is f07fjc.

## 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 and must first be factorized by f07frc.

### 10.1Program Text

Program Text (f07fwce.c)

### 10.2Program Data

Program Data (f07fwce.d)

### 10.3Program Results

Program Results (f07fwce.r)