Difference between revisions of "Newton's root finding method"

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(12 intermediate revisions by the same user not shown)
Line 16: Line 16:
 
<form>
 
<form>
 
<input style="border:none; background-color:#efefef;padding:5px;margin-left:2px;" type="text" id="graphterm" value="(x-2)*(x+1)*x*0.2" size="30"/>
 
<input style="border:none; background-color:#efefef;padding:5px;margin-left:2px;" type="text" id="graphterm" value="(x-2)*(x+1)*x*0.2" size="30"/>
<input type="button" value="set term" onClick="newGraph(document.getElementById('graphterm').value);">
+
<input type="button" value="set function term" onClick="newGraph(document.getElementById('graphterm').value);">
 
</form>
 
</form>
 
</td>
 
</td>
 
<tr><td>&nbsp;</td></tr>
 
<tr><td>&nbsp;</td></tr>
 +
 
<script type="text/javascript">
 
<script type="text/javascript">
// Initial function term
+
// Get initial function term
//var term = function(x) { return (x-2)*(x+1)*x*0.2; };
 
//var graph = function(x) { return term(x); };
 
 
var term = document.getElementById('graphterm').value;
 
var term = document.getElementById('graphterm').value;
  
 
// Recursion depth
 
// Recursion depth
 
var steps = 11;
 
var steps = 11;
// Start value
+
 
var s = 3;
+
// Start value for x
 +
var x_0 = 3;
  
 
for (i = 0; i < steps; i++) {
 
for (i = 0; i < steps; i++) {
Line 39: Line 39:
 
<jsxgraph width="600" height="500">
 
<jsxgraph width="600" height="500">
 
var i;
 
var i;
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-5,5,5,-5], axis:false});
+
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-5, 5, 5, -5], axis:true});
var ax = brd.create('axis', [[0,0], [1,0]], {strokeColor: 'black'});
+
var ax = brd.defaultAxes.x;
var ay = brd.create('axis', [[0,0], [0,1]], {strokeColor: 'black'});
 
  
var g = brd.create('functiongraph', [term],{strokeWidth: 2, dash:0});
+
var g = brd.create('functiongraph', [term], {strokeWidth: 2});
var x = brd.create('glider', [s, 0, ax], {name: 'x_{0}', strokeColor: 'magenta', fillColor: 'yellow'});
+
var x = brd.create('glider', [x_0, 0, ax], {name: 'x_{0}', color: 'magenta', size: 4});
  
 
newGraph(document.getElementById('graphterm').value);
 
newGraph(document.getElementById('graphterm').value);
Line 50: Line 49:
  
 
function xval() {
 
function xval() {
     for (i = 0; i < steps; i++)
+
     for (i = 0; i < steps; i++) {
 
         document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);
 
         document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);
 +
    }
 
}
 
}
  
brd.addHook(xval);
+
brd.on('update', xval);
 +
// Initial call of xval()
 +
xval();
  
 
function newton(p, i, board) {
 
function newton(p, i, board) {
 
     board.suspendUpdate();
 
     board.suspendUpdate();
     if(i>0) {
+
     if (i > 0) {
         var f = board.create('glider',[function(){return p.X();}, function(){return graph(p.X())}, g], {name: '', style: 3, strokeColor: 'green', fillColor: 'yellow'});
+
         var f = board.create('glider', [function(){ return p.X(); }, function(){ return g.Y(p.X()) }, g], {
         var l = board.create('line', [p,f],{strokeWidth: 0.5, dash: 1, straightFirst: false, straightLast: false, strokeColor: 'black'});
+
            name: '', style: 3, color: 'green'});
         var t = board.create('tangent',[f],{strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});
+
         var l = board.create('segment', [p, f], {strokeWidth: 0.5, dash: 1, strokeColor: 'black'});
         var x = board.create('intersection',[ax,t,0],{name: 'x_{'+(steps-i+1) + '}', style: 4, strokeColor: 'magenta', fillColor: 'yellow'});
+
         var t = board.create('tangent', [f], {strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});
         newton(x,--i, board);
+
         var x = board.create('intersection', [ax, t, 0],{name: 'x_{' + (steps - i + 1) + '}', style: 4, color: 'red'});
 +
         newton(x, --i, board);
 
     }
 
     }
 
     board.unsuspendUpdate();     
 
     board.unsuspendUpdate();     
}
+
}
 +
 
function newGraph(v) {
 
function newGraph(v) {
/*
+
    g.generateTerm('x', 'x', v);
eval("term = function(x){ return "+v+";}");
+
    //g.updateCurve();
graph = function(x) { return term(x); };
+
    brd.update();
g.Y = function(x){ return term(x); };
 
g.updateCurve();
 
*/
 
        g.generateTerm('x', 'x', v);
 
        brd.update();
 
 
}
 
}
 
</jsxgraph>
 
</jsxgraph>
  
 
===The underlying JavaScript code===
 
===The underlying JavaScript code===
 +
 
<source lang="xml">
 
<source lang="xml">
 
<table width="600" border="0" cellpadding="0" cellspacing="0">
 
<table width="600" border="0" cellpadding="0" cellspacing="0">
Line 95: Line 95:
 
<tr><td>&nbsp;</td></tr>
 
<tr><td>&nbsp;</td></tr>
 
<script type="text/javascript">
 
<script type="text/javascript">
// Initial function term
+
// Get initial function term
var term = function(x) { return x*x*x/5; };
+
var term = document.getElementById('graphterm').value;
var graph = function(x) { return term(x); };
+
 
 
 
// Recursion depth
 
// Recursion depth
 
var steps = 11;
 
var steps = 11;
// Start value
+
 
var s = 3;
+
// Start value for x
 +
var x_0 = 3;
  
 
for (i = 0; i < steps; i++) {
 
for (i = 0; i < steps; i++) {
 
     document.write('<tr><td><nobr>x<sub>' + i + '</sub> = </nobr></td><td><font id="xv' + i + '"></font></td></tr>');
 
     document.write('<tr><td><nobr>x<sub>' + i + '</sub> = </nobr></td><td><font id="xv' + i + '"></font></td></tr>');
 
}
 
}
</script>
+
<</script>
 
</table>  
 
</table>  
 
</source>
 
</source>
Line 113: Line 113:
 
<source lang="javascript">
 
<source lang="javascript">
 
var i;
 
var i;
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-5,5,5,-5], axis:false});
+
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-5, 5, 5, -5], axis:true});
var ax = brd.create('axis', [[0,0], [1,0]], {strokeColor: 'black'});
+
var ax = brd.defaultAxes.x;
var ay = brd.create('axis', [[0,0], [0,1]], {strokeColor: 'black'});
 
  
var g = brd.create('functiongraph', [function(x){return graph(x);}],{strokeWidth: 2, dash:0});
+
var g = brd.create('functiongraph', [term], {strokeWidth: 2});
var x = brd.create('glider',[s,0,ax], {name: 'x_{0}', strokeColor: 'magenta', fillColor: 'yellow'});
+
var x = brd.create('glider', [x_0, 0, ax], {name: 'x_{0}', color: 'magenta', size: 4});
  
 
newGraph(document.getElementById('graphterm').value);
 
newGraph(document.getElementById('graphterm').value);
Line 124: Line 123:
  
 
function xval() {
 
function xval() {
     for (i = 0; i < steps; i++)
+
     for (i = 0; i < steps; i++) {
 
         document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);
 
         document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);
 +
    }
 
}
 
}
  
brd.addHook(xval);
+
brd.on('update', xval);
 +
// Initial call of xval()
 +
xval();
  
 
function newton(p, i, board) {
 
function newton(p, i, board) {
 
     board.suspendUpdate();
 
     board.suspendUpdate();
     if(i>0) {
+
     if (i > 0) {
         var f = board.create('glider',[function(){return p.X();}, function(){return graph(p.X())},g], {name: '', style: 3, strokeColor: 'green', fillColor: 'yellow'});
+
         var f = board.create('glider', [function(){ return p.X(); }, function(){ return g.Y(p.X()) }, g], {
         var l = board.create('line', [p,f],{strokeWidth: 0.5, dash: 1, straightFirst: false, straightLast: false, strokeColor: 'black'});
+
            name: '', style: 3, color: 'green'});
         var t = board.create('tangent',[f],{strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});
+
         var l = board.create('segment', [p, f], {strokeWidth: 0.5, dash: 1, strokeColor: 'black'});
         var x = board.create('intersection',[ax,t,0],{name: 'x_{'+(steps-i+1) + '}', style: 4, strokeColor: 'magenta', fillColor: 'yellow'});
+
         var t = board.create('tangent', [f], {strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});
         newton(x,--i, board);
+
         var x = board.create('intersection', [ax, t, 0],{name: 'x_{' + (steps - i + 1) + '}', style: 4, color: 'red'});
 +
         newton(x, --i, board);
 
     }
 
     }
 
     board.unsuspendUpdate();     
 
     board.unsuspendUpdate();     
}
+
}
 +
 
function newGraph(v) {
 
function newGraph(v) {
eval("term = function(x){ return "+v+";}");
+
    g.generateTerm('x', 'x', v);
graph = function(x) { return term(x); };
+
    //g.updateCurve();
g.Y = function(x){ return term(x); };
+
    brd.update();
g.updateCurve();
 
        brd.update();
 
 
}
 
}
 
</source>
 
</source>

Latest revision as of 15:54, 15 January 2021

xo is the start value. Drag it.

You may change the function term here, Try also the following function terms:
  • sin(x)
  • exp(x)
  • 2^x
  • 1-2/(x*x)

f(x) =
 

The underlying JavaScript code

<table width="600" border="0" cellpadding="0" cellspacing="0">
x<sub>o</sub> is the start value. Drag it.
<p></p>
You may change the function term here:
<br>
<td><nobr>f(x) = </nobr></td>
<td>
<form>
<input style="border:none; background-color:#efefef;padding:5px;margin-left:2px;" type="text" id="graphterm" value="x*x*x/5" size="30"/>
<input type="button" value="set term" onClick="newGraph(document.getElementById('graphterm').value);">
</form>
</td>
<tr><td>&nbsp;</td></tr>
<script type="text/javascript">
// Get initial function term
var term = document.getElementById('graphterm').value;

// Recursion depth
var steps = 11;

// Start value for x
var x_0 = 3;

for (i = 0; i < steps; i++) {
     document.write('<tr><td><nobr>x<sub>' + i + '</sub> = </nobr></td><td><font id="xv' + i + '"></font></td></tr>');
}
<</script>		
</table>
var i;
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-5, 5, 5, -5], axis:true});
var ax = brd.defaultAxes.x;

var g = brd.create('functiongraph', [term], {strokeWidth: 2});
var x = brd.create('glider', [x_0, 0, ax], {name: 'x_{0}', color: 'magenta', size: 4});

newGraph(document.getElementById('graphterm').value);
newton(x, steps, brd);	

function xval() {
    for (i = 0; i < steps; i++) {
        document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);
    }
}

brd.on('update', xval);
// Initial call of xval()
xval();

function newton(p, i, board) {	
    board.suspendUpdate();	
    if (i > 0) {
        var f = board.create('glider', [function(){ return p.X(); }, function(){ return g.Y(p.X()) }, g], {
            name: '', style: 3, color: 'green'});
        var l = board.create('segment', [p, f], {strokeWidth: 0.5, dash: 1, strokeColor: 'black'});
        var t = board.create('tangent', [f], {strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});
        var x = board.create('intersection', [ax, t, 0],{name: 'x_{' + (steps - i + 1) + '}', style: 4, color: 'red'});
        newton(x, --i, board);
    }
    board.unsuspendUpdate();    
}
	
function newGraph(v) {
    g.generateTerm('x', 'x', v);
    //g.updateCurve();
    brd.update();
}