cejs/data/math/polynomial.js

A JavaScript module framework that is simple to use.

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/**

 * @name	CeL polynomial function

 * @fileoverview

 * ,g�jHhS+T�Nxex[Y�_�v functions0

 * @since	

 */





'use strict';

if (typeof CeL === 'function')

CeL.run({name:'data.math.polynomial',

code:function(library_namespace ) {



//	no required





var 

/**

 * null module constructor

 * @class xex[Y�_�vܕKN function0

 * @constructor

 */

_// JSDT:_module_

= function () {

	//	null module constructor

};



/**

 * for JSDT: 	g prototype Mbg\KNvu\O Class

 */

_// JSDT:_module_

.prototype = {};











//	polynomial	-----------------------------------



/*

	return [r1,r2,..[,�_]]

	** �	g!q�l㉄v�_�gD��R(Wg�_!



ؚ!k�Nxe�ezxe<PBl9h��l:	http://www.journals.zju.edu.cn/sci/2003/200303/030305.pdf	http://tsg.gxtvu.com.cn/eduwest/web_courseware/maths/0092/2/2-3.htm

	�Ock[r��l 1819t^
�
}�l /O�R)R�l �R`h�<h�l	http://en.wikipedia.org/wiki/Ruffini%27s_rule

	Newton's method[r��l	x2=x1-f(x1)/f'(x1)	http://zh.wikipedia.org/wiki/%E7%89%9B%E9%A1%BF%E6%B3%95

�V!k�ezFinding roots	http://zh.wikipedia.org/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E7%A8%8B

NCQ	N!k�ez�vlQ_�	http://en.wikipedia.org/wiki/Cubic_equation	http://math.xmu.edu.cn/jszg/ynLin/JX/jiaoxueKJ/5.ppt



var rootFindingFragment=1e-15;	//	�V�pnmޞXNd��l�g"uu�v���]

*/

rootFinding[generateCode.dLK]='rootFindingFragment';

function rootFinding(polynomial){

 var r=[],a,q;



 //alert(NewtonMethod(polynomial));



 while(a=polynomial.length,a>1){

  if(a<4){

   if(a==2)r.push(-polynomial[1]/polynomial[0]);

   else{

    a=polynomial[1]*polynomial[1]-4*polynomial[0]*polynomial[2];	//	b^2-4ac

    q=2*polynomial[0];

    if(a<0)a=(Math.sqrt(-a)/Math.abs(q))+'i',q=-polynomial[1]/q,r.push(q+'+'+a,q+'-'+a);

    else a=Math.sqrt(a)/q,q=-polynomial[1]/q,r.push(q+a,q-a);

   }

   polynomial=[];break;

  }else if(a=NewtonMethod(polynomial),Math.abs(a[1])>rootFindingFragment){

   //alert('rootFinding: NewtonMethod !q�l�_�Q9h!\n���]:'+a[1]);

   break;

  }

  a=qNum(a[0],1e6);//alert(a[0]+'/'+a[1]);

  q=pLongDivision(polynomial,[a[1],-a[0]]);

  if(Math.abs(q[1][0])>pLongDivisionFragment){alert('rootFinding error!\n���]:'+q[1][0]);break;}

  r.push(a[0]/a[1]),polynomial=q[0];

  //alert('get root: '+a[0]+'\n'+polynomial);

 }



 if(polynomial.length==5){	//	iQ
\qQێ[�9h�V!k�ez

  q=[],a=polynomial.length,i=0;

  while(--a)q.push(polynomial[i++]*a);	//	�_R

  if(q=rootFinding(q),q.length>1){

   //a=0;for(var i=0;i<polynomial.length;i++)a=a*q[0]+polynomial[i];

   //	\�Qxe
NN�y�R�Sui<P	g9hU��GRg	g�N͑9h0�S�QxeKN9h�a�p(-b +- (b^2-4ac)^.5)/2a�GRdk�N͑9hsS�p-b/2a��	�

   //	Ee�S\�S�QxeR㉺p(x^2-2*q[n]*x+&)(?x^2+?x+?)

   //	�Nw�d��l�KN�S�_&	g	N�:a*&^2+(-2*q[n]*(b+2*a*q[n])-c)*&+e=0 or ..

   q=q[0],a=4*polynomial[0]*q+polynomial[1];

   if(a==0){a=rootFinding([polynomial[0],-2*q*(polynomial[1]+2*q*polynomial[0])-polynomial[2],polynomial[4]]);if(a.length<2)a=null;else a=a[0];}

   else a=(2*polynomial[2]*q+polynomial[3]-2*polynomial[0]*q*(2*polynomial[0]*q+polynomial[1]))/a;

   var o;

   if(!isNaN(a)&&(q=pLongDivision(polynomial,o=[1,-2*q,a]),Math.abs(q[1][0])<pLongDivisionFragment&&Math.abs(q[1][1])<pLongDivisionFragment))

    a=rootFinding(q[0]),r.push(a[0],a[1]),a=rootFinding(o),r.push(a[0],a[1]),polynomial=[];

  }

 }



 if(polynomial.length>1){

  r.push(polynomial);

  //if(polynomial.length%2==1)alert('rootFinding error!');

 }

 return r;

}

//alert(rootFinding(getPbyR([1,4/3,5,2,6])).join('\n'));

//alert(NewtonMethod(getPbyR([1,4,5,2,6])).join('\n'));

//alert(rootFinding([1,4,11,14,10]).join('\n'));

//alert(rootFinding([1,2,3,2,1]).join('\n'));



/*	w�d��l polynomial long division	http://en.wikipedia.org/wiki/Polynomial_long_division	2005/3/4 18:48

	dividend/divisor=quotient..remainder



	input	(dividend,divisor)

	return	[FU,�_]



var pLongDivisionFragment=1e-13;	//	�V�pnmޞXNd��l�g"uu�v���]

*/

pLongDivision[generateCode.dLK]='pLongDivisionFragment';

function pLongDivision(dividend,divisor){

 if(typeof dividend!='object'||typeof divisor!='object')return;

 while(!dividend[0])dividend.shift();while(!divisor[0])dividend.shift();

 if(!dividend.length||!divisor.length)return;



 var quotient=[],remainder=[],r,r0=divisor[0],c=-1,l2=divisor.length,l=dividend.length-l2+1,i;

 for(i=0;i<dividend.length;i++)remainder.push(dividend[i]);

 while(++c<l)

  for(quotient.push(r=remainder[c]/r0),i=1;i<l2;i++){

   remainder[c+i]-=r*divisor[i];

   //if(Math.abs(remainder[c+i])<Math.abs(.00001*divisor[i]*r))remainder[c+i]=0;

  }

 return [quotient,remainder.slice(l)];

}

//alert(pLongDivision([4,-5,3,1/3+2/27-1],[3,-1]).join('\n'));



/*

//	polynomial multiplicationXN�l

function polynomialMultiplication(pol1,pol2){

 //for()

}

*/



/*	Newton Iteration Function	2005/2/26 1:4

	return [root,���]]

*/

function NewtonMethod(polynomial,init,diff,count){

 var x=0,d,i,t,l,o,dp=[];

 if(!polynomial||!(d=l=polynomial.length))return;

 while(--d)dp.push(polynomial[x++]*d);	//	dp:�_Rderivative

 if(!diff)diff=rootFindingFragment;diff=Math.abs(diff);

 if(!count)count=15;

 x=init||0,o=diff+1,l--;

 //alert(polynomial+'\n'+dp+'\n'+diff+',l:'+l);

 while(o>diff&&count--){

  //alert(count+':'+x+','+d);

  for(d=t=i=0;i<l;i++)d=d*x+polynomial[i],t=t*x+dp[i];

  d=d*x+polynomial[l];

  //alert(d+'/'+t);

  if(t)d/=t;else d=1;//alert();

  t=Math.abs(d);

  if(o<=t)if(o<rootFindingFragment)break;else x++;	//	test

  o=t,x-=d;

 }

 return [x,d];

}



//	�_roots�_0RY�_	2005/2/26 0:45

function getPbyR(roots){

 var p,r,i,c=0,l;

 if(!roots||!(l=roots.length))return;

 p=[1,-roots.pop()];

 while(++c<l)

  if(r=roots.pop()){p.push(-r*p[i=c]);while(i)p[i]-=p[--i]*r;}

  else p.push(0);

 return p;

}



//alert(getPbyR([1,2,3]));

//document.write(Newton1(getPbyR([2,32,5,3])));



//	�!polynomial	-----------------------------------













return (

	_// JSDT:_module_

);

}





});