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拉格朗日插值法

 
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拉格朗日插值法<wbr style="line-height:25px"><br style="line-height:25px"><span style="color:#ff00ff; line-height:25px">拉格朗日插值法</span>可以帮助我们解决以下的问题<br style="line-height:25px"><span style="color:#000080; line-height:25px"><wbr style="line-height:25px">已知x取值0,1,-1,2时,f{x}取值2,2,0,6<br style="line-height:25px"> 求x=3时f{x}的值。</wbr></span><wbr style="line-height:25px"><br style="line-height:25px"><span style="line-height:25px">示例1</span>:<br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intxs[]={0,1,-1,2};<br style="line-height:25px"> intys[]={2,2,0,6};<br style="line-height:25px"></span><span style="color:#808080; line-height:25px">//f(3)?</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intval=lagrangePolynomial(3,xs,ys);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">staticint</span><span style="color:#3366ff; line-height:25px">lagrangePolynomial(intx,intxs[],intys[])</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intc1,c2;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">inti,j,n;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">n=xs.length;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">c1=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(i=0;i&lt;n;i++){</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">c2=ys[i]</span><span style="color:#3366ff; line-height:25px">;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(j=0;j&lt;n;j++)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">if(j!=i)</span><br style="line-height:25px"><span style="line-height:25px; font-style:normal"><span style="color:#3366ff; line-height:25px">c2=c2*(x-xs[j])/(xs<wbr style="line-height:25px">-xs[j]);<br style="line-height:25px"> }<br style="line-height:25px"> c1+=c2;<br style="line-height:25px"> }<br style="line-height:25px"></wbr></span><span style="color:#993300; line-height:25px">return</span><span style="color:#3366ff; line-height:25px">c1;<br style="line-height:25px"> }</span></span><br style="line-height:25px"><span style="line-height:25px"><wbr style="line-height:25px">Lagrange插值方法的核心就是构造一组基函数</wbr></span><wbr style="line-height:25px">。<br style="line-height:25px"><span style="color:#003366; line-height:25px">如果插值点是</span><span style="color:#0000ff; line-height:25px">{(x_i,y_i)}i=1..n</span><span style="color:#003366; line-height:25px">,那么希望构造出一组多项式</span><span style="color:#ff6600; line-height:25px">F_i(x)</span><span style="color:#003366; line-height:25px">使得</span><br style="line-height:25px"><span style="color:#0000ff; line-height:25px">F_i(x_i)=1,F_i(x_j)=0(j!=i)</span><br style="line-height:25px"><span style="color:#003366; line-height:25px">其实就是构造一个函数,这个函数在其中一点的值为1,其它点的值为0。</span><br style="line-height:25px"><span style="color:#003366; line-height:25px">这样的话把n个这样的函数加权加起来得到的函数就是在每个点上的值都是需要的了。</span><br style="line-height:25px"><span style="line-height:25px">注意1</span>:<wbr style="line-height:25px">这里的多项式是一组。函数即是指多项式。其实针对每个插入点构造一个多项式(函数)<wbr style="line-height:25px">。<br style="line-height:25px"><span style="line-height:25px"><wbr style="line-height:25px">注意2:</wbr></span>关于如何得到多项式的请参阅<a target="_blank" href="http://hwiechern.blog.163.com/blog/static/106796622007913104859712/" style="color:rgb(207,121,28); line-height:25px; text-decoration:none">http://hwiechern.blog.163.com/blog/static/106796622007913104859712/</a><br style="line-height:25px"> 也就是说要构造“只受其中一个点影响”(这种讲法比较粗糙,因为和其他点的位置还是有关系)的函数。<br style="line-height:25px"> 如果这一点能办到,那么只要取<span style="color:#ff6600; line-height:25px">f(x)=sum(y_i*F_i(x))</span>就是所要的插值多项式。<br style="line-height:25px"> Lagrange的插值方法其实就是直接构造出上述基函数:<br style="line-height:25px"><span style="color:#0000ff; line-height:25px">F_i(x)=prod(x-x_j)/prod(x_i-x_j)</span>,其中<span style="color:#800000; line-height:25px">prod</span>是关于所有不等于i的j求乘积,<br style="line-height:25px"><span style="color:#000080; line-height:25px">直接就可以验证F_i(x)满足前面提到的条件,<br style="line-height:25px"> 因为分子相当于确定了F_i(x)的所有根,分母则是归一化系数。</span><br style="line-height:25px"><br style="line-height:25px"><wbr style="line-height:25px"><span style="color:#003366; line-height:25px">对于一些很复杂的函数运算。也可以用拉格朗日插值法来计算以提高效率<wbr style="line-height:25px">。</wbr></span><br style="line-height:25px"> 我们可以先用函数运算均匀的生成一些x点和其对应得数值对。<br style="line-height:25px"> 然后就可以用拉格朗日插值法来计算以提高效率。<br style="line-height:25px"> 具体使用见<span style="line-height:25px"><wbr style="line-height:25px">实例1</wbr></span><wbr style="line-height:25px">:<br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">staticvoid</span><span style="color:#3366ff; line-height:25px"></span><span style="color:#ff6600; line-height:25px">lagrangeDemo</span><span style="color:#3366ff; line-height:25px">()</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intk=10000;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">inta=2;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intx=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">inty=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">intm=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdata[]=newfloat[k];</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">Randomrandom=newRandom(System.currentTimeMillis());</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(inti=0;i&lt;k;i++)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">do{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">m=random.nextInt((int)(kMaxX-kMinX));</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">=m+1/random.nextInt();</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><span style="color:#993300; line-height:25px">while</span><span style="color:#3366ff; line-height:25px">(data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">&lt;kMinX||data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">&gt;kMaxX);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatres0,res1;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdiff=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdiffMax=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdiffRate=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdiffRateMax=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">floatdiffAll=0;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(inti=0;i&lt;k;i++)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">res0=count(a,data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">res1=function(a,data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diff=(res1-res0);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">if(Math.abs(diff)&gt;diffMax)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diffMax=diff;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diffAll+=diff;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">if</span><span style="color:#3366ff; line-height:25px">(res1!=0)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diffRate=diff*100/res1;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">if</span><span style="color:#3366ff; line-height:25px">(Math.abs(diffRate)&gt;diffRateMax)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diffRateMax=diffRate;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">System.out.println("diffAll:"+diffAll+"diffMax:"+(diffMax)+"diffRateMax:"+diffRateMax+"%");</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">longtime=System.currentTimeMillis();</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(inti=0;i&lt;k;i++)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">res0=count(a,data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">System.out.println("lagrangetime:"+(System.currentTimeMillis()-time));</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">time=System.currentTimeMillis();</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(inti=0;i&lt;k;i++)</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">{</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">res1=function(a,data</span><wbr style="line-height:25px"><span style="color:#3366ff; line-height:25px">);</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">System.out.println("Normaltime:"+(System.currentTimeMillis()-time));</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px">}</span><span style="color:#808080; line-height:25px"><br style="line-height:25px"></span><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">static</span><span style="color:#3366ff; line-height:25px">HashMap&lt;String,float[][]&gt;hashMap=newHashMap&lt;String,float[][]&gt;();</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">finalstaticint</span><span style="color:#3366ff; line-height:25px">kSection=100;</span><br style="line-height:25px"><span style="color:#3366ff; line-height:25px"></span><span style="color:#993300; line-height:25px">staticfloat</span><span style="color:#3366ff; line-height:25px">count(inta,floatx)<br style="line-height:25px"> {<br style="line-height:25px"> Stringkey=""+a;<br style="line-height:25px"> floatf[][]=hashMap.get(key);<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">if</span><span style="color:#3366ff; line-height:25px">(f==null)</span> <div style="line-height:25px"> <span style="color:#3366ff; line-height:25px"> {<br style="line-height:25px"> f=getPairs(a);<br style="line-height:25px"> hashMap.put(key,f);<br style="line-height:25px"> }<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">return</span><span style="color:#3366ff; line-height:25px">lagrangePolynomial(x,f[0],f[1]);<br style="line-height:25px"> }<br style="line-height:25px"> finalstaticfloatkMinX=0;<br style="line-height:25px"> finalstaticfloatkMaxX=100;<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">staticfloat</span><span style="color:#3366ff; line-height:25px">[][]getPairs(inta)<br style="line-height:25px"> {<br style="line-height:25px"> floatpairs[][]=newfloat[2][kSection+1];<br style="line-height:25px"> floatxs[]=pairs[0];<br style="line-height:25px"> floatys[]=pairs[1];<br style="line-height:25px"> intn=kSection;<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(inti=0;i&lt;=n;i++)<br style="line-height:25px"> {<br style="line-height:25px"> xs<wbr style="line-height:25px">=kMinX+(kMaxX-kMinX)*i/n;<br style="line-height:25px"> ys<wbr style="line-height:25px">=function(a,xs<wbr style="line-height:25px">);<br style="line-height:25px"> }<br style="line-height:25px"></wbr></wbr></wbr></span><span style="color:#993300; line-height:25px">return</span><span style="color:#3366ff; line-height:25px">pairs;<br style="line-height:25px"> }<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">staticfloat</span><span style="color:#3366ff; line-height:25px">function(inta,floatx)<br style="line-height:25px"> {<br style="line-height:25px"> floaty=x*x*x*x/(a*a+x*x*x*x+x*x);<br style="line-height:25px"> intn=1000;<br style="line-height:25px"> inti,j;<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(i=0;i&lt;n;i++)<br style="line-height:25px"> {<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(j=0;j&lt;n;j++)<br style="line-height:25px"> {<br style="line-height:25px"> y=y*n*(n+2)/((n+1)*(n+3))-1;<br style="line-height:25px"> }<br style="line-height:25px"> }<br style="line-height:25px"> returny;<br style="line-height:25px"> }<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">staticint</span><span style="color:#3366ff; line-height:25px">lagrangePolynomial(intx,intxs[],intys[])<br style="line-height:25px"> {<br style="line-height:25px"> intc1,c2;<br style="line-height:25px"> inti,j,n;<br style="line-height:25px"> n=xs.length;<br style="line-height:25px"> c1=0;<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(i=0;i&lt;n;i++){<br style="line-height:25px"> c2=ys<wbr style="line-height:25px">;<br style="line-height:25px"></wbr></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(j=0;j&lt;n;j++)<br style="line-height:25px"> {<br style="line-height:25px"> if(j!=i)<br style="line-height:25px"> c2=c2*(x-xs[j])/(xs<wbr style="line-height:25px">-xs[j]);<br style="line-height:25px"> }<br style="line-height:25px"> c1+=c2;<br style="line-height:25px"> }<br style="line-height:25px"></wbr></span><span style="color:#993300; line-height:25px">return</span><span style="color:#3366ff; line-height:25px">c1;<br style="line-height:25px"> }<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">staticfloat</span><span style="color:#3366ff; line-height:25px"></span><span style="color:#ff6600; line-height:25px">lagrangePolynomial</span><span style="color:#3366ff; line-height:25px">(floatx,floatxs[],floatys[])<br style="line-height:25px"> {<br style="line-height:25px"> floatc1,c2;<br style="line-height:25px"> inti,j,n;<br style="line-height:25px"> n=xs.length;<br style="line-height:25px"> c1=0;<br style="line-height:25px"></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(i=0;i&lt;n;i++){<br style="line-height:25px"> c2=ys<wbr style="line-height:25px">;<br style="line-height:25px"></wbr></span><span style="color:#993300; line-height:25px">for</span><span style="color:#3366ff; line-height:25px">(j=0;j&lt;n;j++)<br style="line-height:25px"> {<br style="line-height:25px"> if(j!=i)<br style="line-height:25px"> c2=c2*(x-xs[j])/(xs<wbr style="line-height:25px">-xs[j]);<br style="line-height:25px"> }<br style="line-height:25px"> c1+=c2;<br style="line-height:25px"> }<br style="line-height:25px"></wbr></span><span style="color:#993300; line-height:25px">return</span><span style="color:#3366ff; line-height:25px">c1;<br style="line-height:25px"> }</span><br style="line-height:25px"><span style="line-height:25px"><wbr style="line-height:25px">运行结果</wbr></span><wbr style="line-height:25px">:<br style="line-height:25px"><span style="color:#3366ff; line-height:25px">diffAll:NaNdiffMax:4.8828125E-4diffRateMax:-6.088556E-6%<br style="line-height:25px"> lagrangetime:687<br style="line-height:25px"> Normaltime:99359</span><br style="line-height:25px"><span style="line-height:25px">注意</span>:只有函数f(x)非常复杂,且f(x)的值随X变化不是很大时,用拉格朗日插值法来计算才有意义。<br style="line-height:25px"> 否则拉格朗日插值法的效率比直接用函数计算还要低。<br style="line-height:25px"><wbr style="line-height:25px">关于拉格朗日插值法的更多数学知识请参考<br style="line-height:25px"><a target="_blank" href="http://hwiechern.blog.163.com/blog/static/106796622007913104859712/" style="color:rgb(207,121,28); line-height:25px; text-decoration:none">http://hwiechern.blog.163.com/blog/static/106796622007913104859712/</a></wbr></wbr> </div> </wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr></wbr>
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