Difference between revisions of "Rational functions"

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Polynomial <math>f</math>:
 +
<html><form><input type="button" value="Add point" onClick="addPoint1()"></form>
 +
</html>
 +
<jsxgraph width="600" height="200" box="box1">
 +
var board = JXG.JSXGraph.initBoard('box1', {boundingbox: [-10, 4, 10, -4], axis:true});
 +
var p = [];
 +
var q = [];
 +
p[0] = board.create('point', [-1,2], {size:4});
 +
p[1] = board.create('point', [3,-1], {size:4});
 +
var f = JXG.Math.Numerics.lagrangePolynomial(p);
 +
var graph = board.create('functiongraph', [f,-10, 10]);
 +
 +
function addPoint1() {
 +
  p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
 +
  board.update();
 +
}
 +
 +
</jsxgraph>
 +
Polynomial <math>g</math>:
 
<html>
 
<html>
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
 
<div style="float:left"><form><input type="button" value="Add point" onClick="addPoint1()"></form>
 
<div id="box1" class="jxgbox" style="width:300px; height:300px;"></div></div>
 
 
<form><input type="button" value="Add point" onClick="addPoint2()"></form>
 
<form><input type="button" value="Add point" onClick="addPoint2()"></form>
<div><div id="box2" class="jxgbox" style="width:300px; height:300px;"></div></div>
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</html>
<div id="box3" class="jxgbox" style="width:600px; height:400px;"></div>
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<jsxgraph width="600" height="200" box="box2">
<script language="JavaScript">
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var board2 = JXG.JSXGraph.initBoard('box2', {boundingbox: [-10, 4, 10, -4], axis:true});
        board = JXG.JSXGraph.initBoard('box1', {originX: 150, originY: 200, unitX: 30, unitY: 25});
+
q[0] = board2.create('point', [-1,2], {size:4});
        // Axes
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q[1] = board2.create('point', [3,-1], {size:4});
        b1axisx = board.createElement('axis', [[0,0], [1,0]], {});
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var f2 = JXG.Math.Numerics.lagrangePolynomial(q);
        b1axisy = board.createElement('axis', [[0,0], [0,1]], {});
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var graph2 = board2.create('functiongraph', [f2, -10, 10]);
  
        var p = [];
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function addPoint2() {
        var q = [];
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  q.push(board2.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
        p[0] = board.createElement('point', [-1,2], {style:6});
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  board2.update();
        p[1] = board.createElement('point', [3,-1], {style:6});
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}
        var polynomial = function(x) {
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</jsxgraph>
                var i;
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Rational function <math>\frac{f}{g}</math>:  
                var y = 0.0;
 
                var xc = [];
 
                for (i=0;i<p.length;i++) {
 
                  xc[i] = p[i].X();
 
                }
 
                for (i=0;i<p.length;i++) {
 
                  var t = p[i].Y();
 
                  for (var k=0;k<p.length;k++) {
 
                    if (k!=i) {
 
                      t *= (x-xc[k])/(xc[i]-xc[k]);
 
                    }
 
                  }
 
                  y += t;
 
                }
 
                return y;
 
            };
 
        graph = board.createElement('curve', ['x', polynomial, 'x', -10, 10], {curveType:'graph'});
 
  
        function addPoint1() {
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<jsxgraph width="600" height="400" box="box3">
          p.push(board.createElement('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{style:6}));
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var board3 = JXG.JSXGraph.initBoard('box3', {boundingbox: [-10, 8, 10, -8], axis:true});
          board.update();
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var graph3 = board3.create('functiongraph', [function(x){return f(x)/f2(x);}, -10, 10], {strokeColor:'#ff0000'});
        }
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board.addChild(board3);
 +
board2.addChild(board3);
 +
</jsxgraph>
 +
         
 +
=== References ===
 +
* [http://en.wikipedia.org/wiki/Rational_function http://en.wikipedia.org/wiki/Rational_function]
  
        board2 = JXG.JSXGraph.initBoard('box2', {originX: 150, originY: 200, unitX: 30, unitY: 25});
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=== The underlying JavaScript source code ===
        // Axes
 
        b1axisx2 = board2.createElement('axis', [[0,0], [1,0]], {});
 
        b1axisy2 = board2.createElement('axis', [[0,0], [0,1]], {});
 
  
        q[0] = board2.createElement('point', [-1,2], {style:6});
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<source lang="javascript">
        q[1] = board2.createElement('point', [3,-1], {style:6});
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var board = JXG.JSXGraph.initBoard('box1', {boundingbox: [-10, 4, 10, -4], axis:true});
        var polynomial2 = function(x) {
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var p = [];
                var i;
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var q = [];
                var y = 0.0;
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p[0] = board.create('point', [-1,2], {size:4});
                var xc = [];
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p[1] = board.create('point', [3,-1], {size:4});
                for (i=0;i<q.length;i++) {
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var f = JXG.Math.Numerics.lagrangePolynomial(p);
                  xc[i] = q[i].X();
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var graph = board.create('functiongraph', [f,-10, 10]);
                }
 
                for (i=0;i<q.length;i++) {
 
                  var t = q[i].Y();
 
                  for (var k=0;k<q.length;k++) {
 
                    if (k!=i) {
 
                      t *= (x-xc[k])/(xc[i]-xc[k]);
 
                    }
 
                  }
 
                  y += t;
 
                }
 
                return y;
 
            };
 
        graph2 = board2.createElement('curve', ['x', polynomial2, 'x', -10, 10], {curveType:'graph'});
 
  
        function addPoint2() {
+
function addPoint1() {
          q.push(board2.createElement('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{style:6}));
+
  p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
          board2.update();
+
  board.update();
        }
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}
  
        board3 = JXG.JSXGraph.initBoard('box3', {originX: 300, originY: 200, unitX: 30, unitY: 25});
+
var board2 = JXG.JSXGraph.initBoard('box2', {boundingbox: [-10, 4, 10, -4], axis:true});
        // Axes
+
q[0] = board2.create('point', [-1,2], {size:4});
        b1axisx3 = board3.createElement('axis', [[0,0], [1,0]], {});
+
q[1] = board2.create('point', [3,-1], {size:4});
        b1axisy3 = board3.createElement('axis', [[0,0], [0,1]], {});
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var f2 = JXG.Math.Numerics.lagrangePolynomial(q);
        graph3 = board3.createElement('curve', ['x', function(x){return polynomial(x)/polynomial2(x);}, 'x', -10, 10], {curveType:'graph'});
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var graph2 = board2.create('functiongraph', [f2, -10, 10]);
 +
 
 +
function addPoint2() {
 +
  q.push(board2.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
 +
  board2.update();
 +
}
 +
 
 +
var board3 = JXG.JSXGraph.initBoard('box3', {boundingbox: [-10, 8, 10, -8], axis:true});
 +
var graph3 = board3.create('functiongraph', [function(x){return f(x)/f2(x);}, -10, 10], {strokeColor:'#ff0000'});
 +
 
 +
board.addChild(board3);
 +
board2.addChild(board3);
 
            
 
            
</script>
+
</source>
</html>
+
 
 +
[[Category:Examples]]
 +
[[Category:Calculus]]

Latest revision as of 17:47, 20 February 2013

Polynomial [math]f[/math]:

Polynomial [math]g[/math]:

Rational function [math]\frac{f}{g}[/math]:

References

The underlying JavaScript source code

var board = JXG.JSXGraph.initBoard('box1', {boundingbox: [-10, 4, 10, -4], axis:true});
var p = [];
var q = [];
p[0] = board.create('point', [-1,2], {size:4});
p[1] = board.create('point', [3,-1], {size:4});
var f = JXG.Math.Numerics.lagrangePolynomial(p);
var graph = board.create('functiongraph', [f,-10, 10]);

function addPoint1() {
   p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
   board.update();
}

var board2 = JXG.JSXGraph.initBoard('box2', {boundingbox: [-10, 4, 10, -4], axis:true});
q[0] = board2.create('point', [-1,2], {size:4});
q[1] = board2.create('point', [3,-1], {size:4});
var f2 = JXG.Math.Numerics.lagrangePolynomial(q);
var graph2 = board2.create('functiongraph', [f2, -10, 10]);

function addPoint2() {
   q.push(board2.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{size:4}));
   board2.update();
}

var board3 = JXG.JSXGraph.initBoard('box3', {boundingbox: [-10, 8, 10, -8], axis:true});
var graph3 = board3.create('functiongraph', [function(x){return f(x)/f2(x);}, -10, 10], {strokeColor:'#ff0000'});

board.addChild(board3);
board2.addChild(board3);