      1 ﻿
      2 /**
      3  * @name	CeL polynomial function
      4  * @fileoverview
      5  * 本檔案包含了數學多項式的 functions。
      6  * @since
      7  */
      8
      9
     10 if (typeof CeL === 'function')
     11 CeL.setup_module('data.math.polynomial',
     12 function(library_namespace, load_arguments) {
     13
     14 //	no required
     15
     16
     17 var
     18 /**
     19  * null module constructor
     20  * @class 數學多項式相關之 function。
     21  * @constructor
     22  */
     23 CeL.data.math.polynomial
     24 = function () {
     25 	//	null module constructor
     26 };
     27
     28 /**
     29  * for JSDT: 有 prototype 才會將之當作 Class
     30  */
     31 CeL.data.math.polynomial
     32 .prototype = {};
     33
     34
     35
     36
     37
     38 //	polynomial	-----------------------------------
     39
     40 /*
     41 	return [r1,r2,..[,餘式]]
     42 	** 若有無法解的餘式，會附加在最後!
     43
     44 高次代數方程數值求根解法:	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
     45 	修正牛頓法 1819年霍納法 伯努利法 勞思表格法	http://en.wikipedia.org/wiki/Ruffini%27s_rule
     46 	Newton's method牛頓法	x2=x1-f(x1)/f'(x1)	http://zh.wikipedia.org/wiki/%E7%89%9B%E9%A1%BF%E6%B3%95
     47 四次方程Finding roots	http://zh.wikipedia.org/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E7%A8%8B
     48 一元三次方程的公式解	http://en.wikipedia.org/wiki/Cubic_equation	http://math.xmu.edu.cn/jszg/ynLin/JX/jiaoxueKJ/5.ppt
     49
     50 var rootFindingFragment=1e-15;	//	因為浮點乘除法而會產生的誤差
     51 */
     52 rootFinding[generateCode.dLK]='rootFindingFragment';
     53 function rootFinding(polynomial){
     54  var r=[],a,q;
     55
     56  //alert(NewtonMethod(polynomial));
     57
     58  while(a=polynomial.length,a>1){
     59   if(a<4){
     60    if(a==2)r.push(-polynomial[1]/polynomial[0]);
     61    else{
     62     a=polynomial[1]*polynomial[1]-4*polynomial[0]*polynomial[2];	//	b^2-4ac
     63     q=2*polynomial[0];
     64     if(a<0)a=(Math.sqrt(-a)/Math.abs(q))+'i',q=-polynomial[1]/q,r.push(q+'+'+a,q+'-'+a);
     65     else a=Math.sqrt(a)/q,q=-polynomial[1]/q,r.push(q+a,q-a);
     66    }
     67    polynomial=[];break;
     68   }else if(a=NewtonMethod(polynomial),Math.abs(a[1])>rootFindingFragment){
     69    //alert('rootFinding: NewtonMethod 無法得出根!\n誤差:'+a[1]);
     70    break;
     71   }
     72   a=qNum(a[0],1e6);//alert(a[0]+'/'+a[1]);
     73   q=pLongDivision(polynomial,[a[1],-a[0]]);
     74   if(Math.abs(q[1][0])>pLongDivisionFragment){alert('rootFinding error!\n誤差:'+q[1][0]);break;}
     75   r.push(a[0]/a[1]),polynomial=q[0];
     76   //alert('get root: '+a[0]+'\n'+polynomial);
     77  }
     78
     79  if(polynomial.length==5){	//	兩對共軛虛根四次方程
     80   q=[],a=polynomial.length,i=0;
     81   while(--a)q.push(polynomial[i++]*a);	//	微分
     82   if(q=rootFinding(q),q.length>1){
     83    //a=0;for(var i=0;i<polynomial.length;i++)a=a*q[0]+polynomial[i];
     84    //	將函數上下移動至原極值有根處，則會有二重根。原函數之根應為(-b +- (b^2-4ac)^.5)/2a，則此二重根即為-b/2a（？）
     85    //	故可將原函數分解為(x^2-2*q[n]*x+&)(?x^2+?x+?)
     86    //	以長除法解之可得&有三解:a*&^2+(-2*q[n]*(b+2*a*q[n])-c)*&+e=0 or ..
     87    q=q[0],a=4*polynomial[0]*q+polynomial[1];
     88    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];}
     89    else a=(2*polynomial[2]*q+polynomial[3]-2*polynomial[0]*q*(2*polynomial[0]*q+polynomial[1]))/a;
     90    var o;
     91    if(!isNaN(a)&&(q=pLongDivision(polynomial,o=[1,-2*q,a]),Math.abs(q[1][0])<pLongDivisionFragment&&Math.abs(q[1][1])<pLongDivisionFragment))
     92     a=rootFinding(q[0]),r.push(a[0],a[1]),a=rootFinding(o),r.push(a[0],a[1]),polynomial=[];
     93   }
     94  }
     95
     96  if(polynomial.length>1){
     97   r.push(polynomial);
     98   //if(polynomial.length%2==1)alert('rootFinding error!');
     99  }
    100  return r;
    101 }
    102 //alert(rootFinding(getPbyR([1,4/3,5,2,6])).join('\n'));
    103 //alert(NewtonMethod(getPbyR([1,4,5,2,6])).join('\n'));
    104 //alert(rootFinding([1,4,11,14,10]).join('\n'));
    105 //alert(rootFinding([1,2,3,2,1]).join('\n'));
    106
    107 /*	長除法 polynomial long division	http://en.wikipedia.org/wiki/Polynomial_long_division	2005/3/4 18:48
    108 	dividend/divisor=quotient..remainder
    109
    110 	input	(dividend,divisor)
    111 	return	[商,餘式]
    112
    113 var pLongDivisionFragment=1e-13;	//	因為浮點乘除法而會產生的誤差
    114 */
    115 pLongDivision[generateCode.dLK]='pLongDivisionFragment';
    116 function pLongDivision(dividend,divisor){
    117  if(typeof dividend!='object'||typeof divisor!='object')return;
    118  while(!dividend[0])dividend.shift();while(!divisor[0])dividend.shift();
    119  if(!dividend.length||!divisor.length)return;
    120
    121  var quotient=[],remainder=[],r,r0=divisor[0],c=-1,l2=divisor.length,l=dividend.length-l2+1,i;
    122  for(i=0;i<dividend.length;i++)remainder.push(dividend[i]);
    123  while(++c<l)
    124   for(quotient.push(r=remainder[c]/r0),i=1;i<l2;i++){
    125    remainder[c+i]-=r*divisor[i];
    126    //if(Math.abs(remainder[c+i])<Math.abs(.00001*divisor[i]*r))remainder[c+i]=0;
    127   }
    128  return [quotient,remainder.slice(l)];
    129 }
    130 //alert(pLongDivision([4,-5,3,1/3+2/27-1],[3,-1]).join('\n'));
    131
    132 /*
    133 //	polynomial multiplication乘法
    134 function polynomialMultiplication(pol1,pol2){
    135  //for()
    136 }
    137 */
    138
    139 /*	Newton Iteration Function	2005/2/26 1:4
    140 	return [root,誤差]
    141 */
    142 function NewtonMethod(polynomial,init,diff,count){
    143  var x=0,d,i,t,l,o,dp=[];
    144  if(!polynomial||!(d=l=polynomial.length))return;
    145  while(--d)dp.push(polynomial[x++]*d);	//	dp:微分derivative
    146  if(!diff)diff=rootFindingFragment;diff=Math.abs(diff);
    147  if(!count)count=15;
    148  x=init||0,o=diff+1,l--;
    149  //alert(polynomial+'\n'+dp+'\n'+diff+',l:'+l);
    150  while(o>diff&&count--){
    151   //alert(count+':'+x+','+d);
    152   for(d=t=i=0;i<l;i++)d=d*x+polynomial[i],t=t*x+dp[i];
    153   d=d*x+polynomial[l];
    154   //alert(d+'/'+t);
    155   if(t)d/=t;else d=1;//alert();
    156   t=Math.abs(d);
    157   if(o<=t)if(o<rootFindingFragment)break;else x++;	//	test
    158   o=t,x-=d;
    159  }
    160  return [x,d];
    161 }
    162
    163 //	從roots得到多項式	2005/2/26 0:45
    164 function getPbyR(roots){
    165  var p,r,i,c=0,l;
    166  if(!roots||!(l=roots.length))return;
    167  p=[1,-roots.pop()];
    168  while(++c<l)
    169   if(r=roots.pop()){p.push(-r*p[i=c]);while(i)p[i]-=p[--i]*r;}
    170   else p.push(0);
    171  return p;
    172 }
    173
    174 //alert(getPbyR([1,2,3]));
    175 //document.write(Newton1(getPbyR([2,32,5,3])));
    176
    177 //	↑polynomial	-----------------------------------
    178
    179
    180
    181
    182
    183
    184 return (
    185 	CeL.data.math.polynomial
    186 );
    187 }
    188
    189
    190 );
    191
    192