Mathematics-Online lexicon: Solving a Linear System Given in Triangular Form

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	Mathematics-Online lexicon:
	Solving a Linear System Given in Triangular Form

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overview

A linear system of equations given in triangular form

$\begin{displaymath} \begin{array}{rcrcrcccccl} a_{11}x_1 &+& a_{12}x_2 &+& a_{13... ...ots \\ & & & & & & & & a_{nn}x_n &=& f_n \\ \par \end{array}\end{displaymath}$

can be uniquely solved, if $a_{ii}\neq 0$ für $i=1,2,3,\dots,n$ .

The solution can be determined by back substitution. So from the last line it follows that $x_n=f_n/a_{nn}$ . Inserting that into the second but last line allows to calculate $x_{n-1}$ . Successively moving upwards, one finally obtains from the first line.

(Authors: Wipper/Abele)

see also:

Keyword: System of equations, linear: direct methods

automatically generated 5/12/2006