Lagrange interpolation (dup): Difference between revisions

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Construct a polynomial of degree 3 through four given points.
Constructs a polynomial of degree <math>n</math> through <math>n+1</math> given points.
Points can be added by clicking on "Add point".
<html>
<html>
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<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>
<!--<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>-->
<form><input type="button" value="Add point" onClick="addPoint()"></form>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/~alfred/jsxgraph/branches/0.70/src/loadjsxgraph.js"></script>
<div id="box" class="jxgbox" style="width:600px; height:400px;"></div>
<div id="box" class="jxgbox" style="width:600px; height:400px;"></div>
<script language="JavaScript">
<script language="JavaScript">
        board = JXG.JSXGraph.initBoard('box', {originX: 250, originY: 250, unitX: 50, unitY: 25});
var board = JXG.JSXGraph.initBoard('box', {originX: 250, originY: 250, unitX: 50, unitY: 25, axis:true});
        // Axes
var p = [];
        b1axisx = board.createElement('axis', [[0,0], [1,0]], {});
p[0] = board.create('point', [-1,2], {style:6});
        b1axisy = board.createElement('axis', [[0,0], [0,1]], {});
p[1] = board.create('point', [3,-1], {style:6});
p[2] = board.create('point', [-3,0], {style:6});
var graph = board.create('functiongraph', [board.lagrangePolynomial(p), -10, 10]);


        var p = [];
function addPoint() {
        p[0] = board.createElement('point', [-1,2], {style:6});
      p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{style:6}));
        p[1] = board.createElement('point', [0,-3], {style:6});
      board.update();
        p[2] = board.createElement('point', [1,4], {style:6});
}
        p[3] = board.createElement('point', [2,1], {style:6});
       
        var polynomial = function(x) {
                var y = p[0].Y()*(x-p[1].X())*(x-p[2].X())*(x-p[3].X())/
                        ((p[0].X()-p[1].X())*(p[0].X()-p[2].X())*(p[0].X()-p[3].X()))+
                        p[1].Y()*(x-p[0].X())*(x-p[2].X())*(x-p[3].X())/
                        ((p[1].X()-p[0].X())*(p[1].X()-p[2].X())*(p[1].X()-p[3].X()))+
                        p[2].Y()*(x-p[0].X())*(x-p[1].X())*(x-p[3].X())/
                        ((p[2].X()-p[0].X())*(p[2].X()-p[1].X())*(p[2].X()-p[3].X()))+
                        p[3].Y()*(x-p[0].X())*(x-p[1].X())*(x-p[2].X())/
                        ((p[3].X()-p[0].X())*(p[3].X()-p[1].X())*(p[3].X()-p[2].X()));
                return y;
            };
        graph = board.createElement('curve', ['x', polynomial, 'x', -10, 10], {curveType:'graph'});
</script>
</script>
</html>
</html>
=== References ===
* [http://en.wikipedia.org/wiki/Lagrange_polynomial http://en.wikipedia.org/wiki/Lagrange_polynomial]
=== The underlying JavaScript code ===
<source lang="html4strict">
<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/jsxgraphcore.js"></script>
<div id="box" class="jxgbox" style="width:600px; height:400px;"></div>
</source>


<source lang="javascript">
<source lang="javascript">
        board = JXG.JSXGraph.initBoard('box', {originX: 250, originY: 250, unitX: 50, unitY: 25});
var board = JXG.JSXGraph.initBoard('box', {originX: 250, originY: 250, unitX: 50, unitY: 25, axis:true});
        // Axes
var p = [];
        b1axisx = board.createElement('axis', [[0,0], [1,0]], {});
p[0] = board.create('point', [-1,2], {style:6});
        b1axisy = board.createElement('axis', [[0,0], [0,1]], {});
p[1] = board.create('point', [3,-1], {style:6});
p[2] = board.create('point', [-3,0], {style:6});
var graph = board.create('functiongraph', [board.lagrangePolynomial(p), -10, 10]);


        var p = [];
function addPoint() {
        p[0] = board.createElement('point', [-1,2], {style:6});
      p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{style:6}));
        p[1] = board.createElement('point', [0,-3], {style:6});
      board.update();
        p[2] = board.createElement('point', [1,4], {style:6});
}
        p[3] = board.createElement('point', [2,1], {style:6});
       
        var polynomial = function(x) {
                var y = p[0].Y()*(x-p[1].X())*(x-p[2].X())*(x-p[3].X())/
                        ((p[0].X()-p[1].X())*(p[0].X()-p[2].X())*(p[0].X()-p[3].X()))+
                        p[1].Y()*(x-p[0].X())*(x-p[2].X())*(x-p[3].X())/
                        ((p[1].X()-p[0].X())*(p[1].X()-p[2].X())*(p[1].X()-p[3].X()))+
                        p[2].Y()*(x-p[0].X())*(x-p[1].X())*(x-p[3].X())/
                        ((p[2].X()-p[0].X())*(p[2].X()-p[1].X())*(p[2].X()-p[3].X()))+
                        p[3].Y()*(x-p[0].X())*(x-p[1].X())*(x-p[2].X())/
                        ((p[3].X()-p[0].X())*(p[3].X()-p[1].X())*(p[3].X()-p[2].X()));
                return y;
            };
        graph = board.createElement('curve', ['x', polynomial, 'x', -10, 10], {curveType:'graph'});
</source>
</source>




[[Category:Examples]]
[[Category:Austragungsstueberl]]
Category:Examples
Category:Calculus
Category:Interpolation

Latest revision as of 08:05, 8 June 2011

Constructs a polynomial of degree [math]\displaystyle{ n }[/math] through [math]\displaystyle{ n+1 }[/math] given points. Points can be added by clicking on "Add point".

References

The underlying JavaScript code

<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/jsxgraphcore.js"></script>
<div id="box" class="jxgbox" style="width:600px; height:400px;"></div>

var board = JXG.JSXGraph.initBoard('box', {originX: 250, originY: 250, unitX: 50, unitY: 25, axis:true});
var p = [];
p[0] = board.create('point', [-1,2], {style:6});
p[1] = board.create('point', [3,-1], {style:6});
p[2] = board.create('point', [-3,0], {style:6});
var graph = board.create('functiongraph', [board.lagrangePolynomial(p), -10, 10]);

function addPoint() {
      p.push(board.create('point',[(Math.random()-0.5)*10,(Math.random()-0.5)*3],{style:6}));
      board.update();
}

Category:Examples Category:Calculus Category:Interpolation

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