F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07TWF (ZTRTRI)

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

F07TWF (ZTRTRI) computes the inverse of a complex triangular matrix.

## 2  Specification

 SUBROUTINE F07TWF ( UPLO, DIAG, N, A, LDA, INFO)
 INTEGER N, LDA, INFO COMPLEX (KIND=nag_wp) A(LDA,*) CHARACTER(1) UPLO, DIAG
The routine may be called by its LAPACK name ztrtri.

## 3  Description

F07TWF (ZTRTRI) forms the inverse of a complex triangular matrix $A$. 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:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ triangular matrix $A$.
• If ${\mathbf{UPLO}}=\text{'U'}$, $A$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{UPLO}}=\text{'L'}$, $A$ is lower triangular and the elements of the array above the diagonal are not referenced.
• If ${\mathbf{DIAG}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
On exit: $A$ is overwritten by ${A}^{-1}$, using the same storage format as described above.
5:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07TWF (ZTRTRI) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
6:     $\mathrm{INFO}$ – 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

F07TWF (ZTRTRI) is not threaded by NAG in any implementation.
F07TWF (ZTRTRI) 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 F07TJF (DTRTRI).

## 10  Example

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

### 10.1  Program Text

Program Text (f07twfe.f90)

### 10.2  Program Data

Program Data (f07twfe.d)

### 10.3  Program Results

Program Results (f07twfe.r)