﻿ f07te Method
f07te solves a real triangular system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$.

# Syntax

C#
```public static void f07te(
string uplo,
string trans,
string diag,
int n,
int nrhs,
double[,] a,
double[,] b,
out int info
)```
Visual Basic
```Public Shared Sub f07te ( _
uplo As String, _
trans As String, _
diag As String, _
n As Integer, _
nrhs As Integer, _
a As Double(,), _
b As Double(,), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f07te(
String^ uplo,
String^ trans,
String^ diag,
int n,
int nrhs,
array<double,2>^ a,
array<double,2>^ b,
[OutAttribute] int% info
)```
F#
```static member f07te :
uplo : string *
trans : string *
diag : string *
n : int *
nrhs : int *
a : float[,] *
b : 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 $\text{"L"}$.
trans
Type: System..::..String
On entry: indicates the form of the equations.
${\mathbf{trans}}=\text{"N"}$
The equations are of the form $AX=B$.
${\mathbf{trans}}=\text{"T"}$ or $\text{"C"}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{trans}}=\text{"N"}$, $\text{"T"}$ or $\text{"C"}$.
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: ${\mathbf{n}}\ge 0$.
nrhs
Type: System..::..Int32
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
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.
b
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 b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# Description

f07te solves a real triangular system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$.

# References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

# Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDB) 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 and the solution has not been 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 solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤cnεA,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞x∞≤cncondA,xε, provided cncondA,xε<1,$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }$.
Note that $\mathrm{cond}\left(A,x\right)\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$; $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$ and it is also possible for $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ to be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f07th, and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling (F07TGF not in this release) with ${\mathbf{norm}}=\text{"I"}$.

None.

# Further Comments

The total number of floating-point operations is approximately ${n}^{2}r$.
The complex analogue of this method is (F07TSF not in this release).

# Example

This example solves the system of equations $AX=B$, where
 $A=4.300.000.000.00-3.96-4.870.000.000.400.31-8.020.00-0.270.07-5.950.12 and B=-12.90-21.5016.7514.93-17.556.33-11.048.09.$

Example program (C#): f07tee.cs

Example program data: f07tee.d

Example program results: f07tee.r