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Solving a Linear System Given in Triangular Form


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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 $ x_1$ from the first line.

(Authors: Wipper/Abele)

see also:


  automatically generated 5/12/2006