﻿ f07tj Method
f07tj computes the inverse of a real triangular matrix.

# Syntax

C#
```public static void f07tj(
string uplo,
string diag,
int n,
double[,] a,
out int info
)```
Visual Basic
```Public Shared Sub f07tj ( _
uplo As String, _
diag As String, _
n As Integer, _
a As Double(,), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f07tj(
String^ uplo,
String^ diag,
int n,
array<double,2>^ a,
[OutAttribute] int% info
)```
F#
```static member f07tj :
uplo : string *
diag : string *
n : int *
a : float[,] *
info : int byref -> unit
```

#### Parameters

uplo
Type: System..::..String
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 $\text{"L"}$.
diag
Type: System..::..String
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 $\text{"U"}$.
n
Type: System..::..Int32
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
a
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# Description

f07tj forms the inverse of a real triangular matrix $A$. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

# References

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

# Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
${\mathbf{info}}<0$
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.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# 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).

# Parallelism and Performance

None.

The total number of floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
The complex analogue of this method is (F07TWF not in this release).

# Example

This example computes the inverse of the matrix $A$, where
 $A=4.300.000.000.00-3.96-4.870.000.000.400.31-8.020.00-0.270.07-5.950.12.$

Example program (C#): f07tje.cs

Example program data: f07tje.d

Example program results: f07tje.r