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extrema / usr / share / extrema / Help / Functions / gaussj.htm
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<TITLE>Solve system of equations</TITLE>
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<P>
<font size="+3" color="green"><B>Solve system of equations</B></font></P>
<P>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
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<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%"><CODE>
y = GAUSSJ(m,x)</CODE>
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<p>
 The <CODE>GAUSSJ</CODE> function solves the system of
 equations <code>m&lt;&gt;x = b</code>, where <code>m</code> is a square matrix and
 <code>b</code> is a vector, and returns <code>x</code>, the vector of solutions.
 This function uses the Gauss-Jordan method of elimination with full pivoting. The
 matrix must be square. The length of the input vector must be the same as the row
 dimension of the matrix.  The function returns a vector with the same length.</p>
<P>
 <font size="+1" color="green">Example 1</font></P>
<p>
 To solve the following three equations for the three unknowns
 <IMG ALIGN="bottom" SRC="gaussjI01.png">&nbsp;, <IMG ALIGN="bottom" SRC="gaussjI02.png">&nbsp;,
 and <IMG ALIGN="bottom" SRC="gaussjI03.png">&nbsp;:</p>
<p>
 <IMG ALIGN="top" SRC="gaussjI04.png"></p>
<p>
 You could use the script:</p>
<p>
 <font color="blue"><pre>
 m=[[3;10;5];[4;2;-2];[15;3;3]]
 b=[26;14;22]
 soln=gaussj(m,b)
 </pre></font></p>
<p>
 The answer: <code>soln=[1.41772;-3.77215;2.4557]</code></p>
 can be checked by using the <a href="../Operators/outerproduct.htm">outer product</a>
 operator: <code>m&lt;&gt;soln</code>
 and comparing this result with the original <code>b</code> vector.</p>
<p>
 <font size="+1" color="green">Example 2</font></P>
<p>
 The following script, <code>INVERSE.PCM</code>, will find the inverse of a square
 matrix. If you have a square matrix <code>M</code>, you could find it's inverse,
 <code>INV_M</code>, with the command: <code>@INVERSE M</code></p>
<p>
 <font color="blue"><pre>
 if ( len(?1[1,*]) ~= len(?1[*,1]) ) then
   display 'input matrix must be square'
   return
 endif
 n = len(?1[1,*])
 ! identity(n) is the identity matrix of order n
 do k = [1:n]
  inv_m[1:n,k] = gaussj(?1,identity(n)[1:n,k])
 enddo
 </pre></font></p>
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