C#实现拉格朗日、牛顿、Hermite插值

拉格朗日、牛顿插值法

初始时需要至少两个参考坐标点，在参考点的基础之上构造插值函数y=f(x)，然后由插值函数确定需要求解的x坐标的函数值。相较于拉格朗日插值，牛顿插值公式更加复杂，但是在增加一个参考点时，不需要像拉格朗日插值那样从头再来。具体数学公式及背景介绍见参考1与参考2。

Hermite插值法

初始时不仅需要至少两个参考点坐标，还需要在参考点上的一阶导数值。除此之外，Hermite插值可以保证插值函数在参考点上一阶可导，且导数值与初始值一致。

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Collections;
namespace CsharpPro1.scienceCalculate
{class interpolation{private List<double> xx = new List<double>();  //存储自变量private List<double> known = new List<double>();private List<double> knownDerivatives = new List<double>();  //用来存储Hermite插值时的已知量的导函数值private List<double> tempW = new List<double>();   //用来存储牛顿插值时的中间变量private List<double> tempF = new List<double>();   //用来存储牛顿插值法需要的中间变量private List<double> tempL = new List<double>();   //用来存储Hermite插值时的中间变量private List<double> tempLL = new List<double>();   //用来存储Hermite插值法需要的中间变量private int knownCount;    //已知数数量private double pn;   //最终结果public static double angle2arc(double rangle){return rangle * Math.PI / 180;}public double fx(double rangle){rangle = angle2arc(rangle);return Math.Cos(rangle);}public interpolation(int knownCount, int sign)    //牛顿对象与拉格朗日对象构造函数{this.knownCount = knownCount;this.pn = 0;for (int i = 0; i < knownCount; i++){Console.WriteLine("请输入第{0}个已知点,以Enter结束一行的输入", i);known.Add(double.Parse(Console.ReadLine()));}}public interpolation(int knownCount){this.knownCount = knownCount;this.pn = 0;for (int i = 0; i < knownCount; i++){Console.WriteLine("请输入第{0}个已知点,以Enter结束一行的输入", i);xx.Add(double.Parse(Console.ReadLine()));Console.WriteLine("请输入第{0}个已知点的函数值,以Enter结束一行的输入", i);known.Add(double.Parse(Console.ReadLine()));Console.WriteLine("请输入第{0}个已知点的函数值，如不需要，则输入零即可,以Enter结束一行的输入", i);knownDerivatives.Add(double.Parse(Console.ReadLine()));}}public void lagrange(double x)//插值方法，拉格朗日插值{double tempNum=1;for(int i = 0; i < knownCount; i++){tempNum = fx(known[i]);for(int j = 0; j < knownCount; j++){if (j != i){tempNum *= (x - known[j]) / (known[i] - known[j]);}}pn += tempNum;}}public void setTempW(double x)     //tempW的set方法,最好由对象调用{tempW.Add(1);for(int i = 1; i < this.knownCount; i++){tempW.Add(tempW[i - 1] * (x - known[i - 1]));}}public double setTempF(int headNum,int endNum,double x){double tempresult;if (headNum == endNum){tempresult = fx(known[headNum]);}else{tempresult= (setTempF(headNum, endNum-1, x) - setTempF(headNum+1, endNum, x)) / (known[headNum] - known[endNum]);}if (headNum == 0)tempF.Insert(endNum,tempresult);return tempresult;}public void newton(double x)   //牛顿插值法{setTempW(x);setTempF(0, knownCount - 1, x);for (int i = 0; i < knownCount; i++){this.pn += tempW[i] * tempF[i];}}public void getlxAndLx(double x)  //hermite基函数与导基函数{double temp = 1, temp1 = 0;for(int i = 0; i < knownCount; i++){for(int j = 0; j < knownCount; j++){if (i != j){temp *= (x - xx[j]) / (xx[i] - xx[j]);temp1 += 1 / (xx[i] - xx[j]);}}tempL.Add(temp);tempLL.Add(temp1);temp = 1;temp1 = 0;}}public void hermite(double x)  //Hermite插值方法的实现,不适用于正余弦函数的插值{getlxAndLx(x);for (int i = 0; i < knownCount; i++){pn += known[i] * (1 - 2 * (x - xx[i]) * tempLL[i]) * Math.Pow(tempL[i], 2) +knownDerivatives[i] * (x - xx[i]) * tempL[i]*tempL[i];}}//static void Main(string[] agrs)//{//    Console.WriteLine("请输入已知参数个数");//    int knownCount = int.Parse(Console.ReadLine());//    interpolation hermiteDemo = new interpolation(knownCount);//    Console.WriteLine("请输入需要计算的自变量值");//    double rangle = double.Parse(Console.ReadLine());//    //rangle = angle2arc(rangle);//    hermiteDemo.hermite(rangle);//    Console.WriteLine("{0}的Cos值为{1}", rangle, hermiteDemo.pn);//}}
}

<参考1>http://baike.baidu.com/link?url=xEG2Gx2YJw5rC5yRiwi8Ngpt2kbabpz6ouRIRtmx4BX0KpI-3cZb0L4vFU3XgHzAEYjaAY1Kw3uBRP7RivuOLqu32-SkhFvw-t3h7sUOJuVXVhChCs1mMrZUazbk8gNMBhMRIBceS5jegTJfyAJmFKbdgFD3PshNWx-QkMykJ8_
<参考2>http://baike.baidu.com/link?url=UcEI_khjobQ5C1e_roIFE2krJ9p50KPgceHCiZA_0VG8aGtdgFPj2IC_B1dTfQPI-WgBqOVTC2eHibu_FMRs9YBzorzAkuz3x0IeFfTBFv5A0mf4w4sRewFA9MvZ_-TpLDuA10xVSHJpAWK1tieMo_