F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07UWF (ZTPTRI)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07UWF (ZTPTRI) computes the inverse of a complex triangular matrix, using packed storage.

## 2  Specification

 SUBROUTINE F07UWF ( UPLO, DIAG, N, AP, INFO)
 INTEGER N, INFO COMPLEX (KIND=nag_wp) AP(*) CHARACTER(1) UPLO, DIAG
The routine may be called by its LAPACK name ztptri.

## 3  Description

F07UWF (ZTPTRI) forms the inverse of a complex triangular matrix $A$, using packed storage. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

## 4  References

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

## 5  Parameters

1:     $\mathrm{UPLO}$ – CHARACTER(1)Input
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{UPLO}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{DIAG}$ – CHARACTER(1)Input
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{DIAG}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{DIAG}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{DIAG}}=\text{'N'}$ or $\text{'U'}$.
3:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     $\mathrm{AP}\left(*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/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 $n$ by $n$ triangular matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ 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 $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
If ${\mathbf{DIAG}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether ${\mathbf{DIAG}}=\text{'N'}$ or ‘U’.
On exit: $A$ is overwritten by ${A}^{-1}$, using the same storage format as described above.
5:     $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.
${\mathbf{INFO}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. $A$ is singular its inverse cannot be computed.

## 7  Accuracy

The computed inverse $X$ satisfies
 $XA-I≤cnεXA ,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
Note that a similar bound for $\left|AX-I\right|$ cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
 $X-A-1≤cnεA-1AX .$
See Du Croz and Higham (1992).

## 8  Parallelism and Performance

F07UWF (ZTPTRI) is not threaded by NAG in any implementation.
F07UWF (ZTPTRI) 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 $\frac{4}{3}{n}^{3}$.
The real analogue of this routine is F07UJF (DTPTRI).

## 10  Example

This example computes the inverse of the matrix $A$, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i ,$
using packed storage.

### 10.1  Program Text

Program Text (f07uwfe.f90)

### 10.2  Program Data

Program Data (f07uwfe.d)

### 10.3  Program Results

Program Results (f07uwfe.r)