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1、分段线性插值
2、三次样条插值
3、拉格朗日插值
#xff08;1#xff09;一元全区间不等距插值
#xff08;2#xff09;一元全区间等距插值
4、埃尔米特插值
#xff08;1#xff09;埃尔米特不等距插值
#xff08;2#xff09;埃尔米特等距插值 1、…目录
1、分段线性插值
2、三次样条插值
3、拉格朗日插值
1一元全区间不等距插值
2一元全区间等距插值
4、埃尔米特插值
1埃尔米特不等距插值
2埃尔米特等距插值 1、分段线性插值 /// summary /// 分段线性插值将一组数插值为所需点数 /// /summary /// param namedataIn待插值的数据数组/param /// param namen插值点数/param /// returns插值后的数据数组/returns public static float[] Interpolation(float[] dataIn, int n){float[] dataOut new float[n];int lenIn dataIn.Length;float[] a new float[lenIn];float[] divIn new float[lenIn];float[] divOut new float[n];divIn[0] 0;for (int i 1; i lenIn; i){divIn[i] divIn[i - 1] 1;}divOut[0] 0;for (int i 1; i n; i){divOut[i] divOut[i - 1] lenIn / (float)n;}int k 0;for (int i k; i n; i){for (int j 0; j lenIn - 1; j){if (divOut[i] divIn[j] divOut[i] divIn[j 1]){dataOut[i] (dataIn[j 1] - dataIn[j]) * (divOut[i] - divIn[j]) / (divIn[j 1] - divIn[j]) dataIn[j];k i;}}}return dataOut;}
2、三次样条插值
三次样条插值 C#代码实现_c# 三次样条插值_Big_潘大师的博客-CSDN博客 /// summary/// 三次样条插值/// /summary/// param namepoints排序好的x、y点集合/param/// param namexs输入x轴数据插值计算出对应的y轴点/param/// param namechf写1/param/// returns返回计算好的Y轴数值/returnspublic static double[] SplineInsertPoint(PointClass[] points, double[] xs, int chf){int plength points.Length;double[] h new double[plength];double[] f new double[plength];double[] l new double[plength];double[] v new double[plength];double[] g new double[plength];for (int i 0; i plength - 1; i){h[i] points[i 1].x - points[i].x;f[i] (points[i 1].y - points[i].y) / h[i];}for (int i 1; i plength - 1; i){l[i] h[i] / (h[i - 1] h[i]);v[i] h[i - 1] / (h[i - 1] h[i]);g[i] 3 * (l[i] * f[i - 1] v[i] * f[i]);}double[] b new double[plength];double[] tem new double[plength];double[] m new double[plength];double f0 (points[0].y - points[1].y) / (points[0].x - points[1].x);double fn (points[plength - 1].y - points[plength - 2].y) / (points[plength - 1].x - points[plength - 2].x);b[1] v[1] / 2;for (int i 2; i plength - 2; i){// Console.Write( i);b[i] v[i] / (2 - b[i - 1] * l[i]);}tem[1] g[1] / 2;for (int i 2; i plength - 1; i){//Console.Write( i);tem[i] (g[i] - l[i] * tem[i - 1]) / (2 - l[i] * b[i - 1]);}m[plength - 2] tem[plength - 2];for (int i plength - 3; i 0; i--){//Console.Write( i);m[i] tem[i] - b[i] * m[i 1];}m[0] 3 * f[0] / 2.0;m[plength - 1] fn;int xlength xs.Length;double[] insertRes new double[xlength];for (int i 0; i xlength; i){int j 0;for (j 0; j plength; j){if (xs[i] points[j].x)break;}j j - 1;Console.WriteLine(j);if (j -1 || j points.Length - 1){if (j -1)throw new Exception(插值下边界超出);if (j points.Length - 1 xs[i] points[j].x)insertRes[i] points[j].y;elsethrow new Exception(插值下边界超出);}else{double p1;p1 (xs[i] - points[j 1].x) / (points[j].x - points[j 1].x);p1 p1 * p1;double p2; p2 (xs[i] - points[j].x) / (points[j 1].x - points[j].x);p2 p2 * p2;double p3; p3 p1 * (1 2 * (xs[i] - points[j].x) / (points[j 1].x - points[j].x)) * points[j].y p2 * (1 2 * (xs[i] - points[j 1].x) / (points[j].x - points[j 1].x)) * points[j 1].y;double p4; p4 p1 * (xs[i] - points[j].x) * m[j] p2 * (xs[i] - points[j 1].x) * m[j 1];// Console.WriteLine(m[j] m[j 1] j);p4 p4 p3;insertRes[i] p4;//Console.WriteLine(f( xs[i] ) p4);}}//Console.ReadLine();return insertRes;}
排序计算 public class PointClass{public double x 0;public double y 0;public PointClass(){x 0; y 0;}//-------写一个排序函数使得输入的点按顺序排列是因为插值算法的要求是x轴递增有序的---------public static PointClass[] DeSortX(PointClass[] points){int length points.Length;double temx, temy;for (int i 0; i length - 1; i){for (int j 0; j length - i - 1; j)if (points[j].x points[j 1].x){temx points[j 1].x;points[j 1].x points[j].x;points[j].x temx;temy points[j 1].y;points[j 1].y points[j].y;points[j].y temy;}}return points;}}
3、拉格朗日插值
1一元全区间不等距插值 /// summary/// 一元全区间不等距插值/// 拉格朗日插值算法/// /summary/// param namex一维数组长度为n存放给定的n个结点的值x(i)要求x(0)x(1)...x(n-1)/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Lagrange(double[] x, double[] y, double t){// x,y点数int n x.Length;double z 0.0;// 特例处理if (n 1){return (z);}else if (n 1){z y[0];return (z);}else if (n 2){z (y[0] * (t - x[1]) - y[1] * (t - x[0])) / (x[0] - x[1]);return (z);}// 开始插值int ik 0;while ((x[ik] t) (ik n)){ik ik 1;}int k ik - 4;if (k 0){k 0;}int m ik 3;if (m n - 1){m n - 1;}for (int i k; i m; i){double s 1.0;for (int j k; j m; j){if (j ! i){// 拉格朗日插值公式s s * (t - x[j]) / (x[i] - x[j]);}}z z s * y[i];}return (z);} /// summary/// 一元全区间不等距插值/// /summary/// param namepoints点集含XY坐标/param/// param namet/param/// returns/returnspublic static double Lagrange(PointF[] points, double t){double[] x new double[points.Length];double[] y new double[points.Length];for (int i 0; i points.Length; i){x[i] points[i].X;y[i] points[i].Y;}return Lagrange(x, y, t);} /// summary/// 一元全区间不等距插值/// /summary/// param namepoints二元组类型的点集含XY坐标/param/// param namet/param/// returns/returnspublic static double Lagrange(ListTupledouble, double points, double t){double[] x new double[points.Count];double[] y new double[points.Count];for (int i 0; i points.Count; i){x[i] points[i].Item1;y[i] points[i].Item2;}return Lagrange(x, y, t);} /// summary/// 一元全区间不等距插值获得插值后的曲线折线拟合数据/// /summary/// param namepoints点集含XY坐标/param/// param namesegment_count每数据段的分割数/param/// returns/returnspublic static PointF[] Lagrange_Curve(PointF[] points, int segment_count 10){int n points.Length;PointF[] segments new PointF[n * segment_count 1];for (int i 0; i points.Length - 1; i){double dt (points[i 1].X - points[i].X) / segment_count;double t points[i].X;for (int j 0; j segment_count; j, t dt){PointF p new PointF(0.0F, 0.0F);p.X (float)t;if (j 0) p.Y points[i].Y;else if (j segment_count) p.Y points[i 1].Y;else p.Y (float)(Lagrange(points, t));segments[i] p;}}return segments;} /// summary/// 一元全区间等距插值/// (使用非等距插值的方法)/// /summary/// param namex0存放等距n个结点中第一个结点的值/param/// param namestep等距结点的步长/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Lagrange(double x0, double step, double[] y, double t){double[] x new double[y.Length];for (int i 0; i y.Length; i, x0 step){x[i] x0;}return Lagrange(x, y, t);}
2一元全区间等距插值 /// summary/// 一元全区间等距插值/// /summary/// param namex0存放等距n个结点中第一个结点的值/param/// param namestep等距结点的步长/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Lagrange_Equidistant(double x0, double step, double[] y, double t){int n y.Length;double z 0.0;// 特例处理if (n 1){return (z);}else if (n 1){z y[0];return (z);}else if (n 2){z (y[1] * (t - x0) - y[0] * (t - x0 - step)) / step;return (z);}// 开始插值int ik 0;if (t x0){double p (t - x0) / step;ik (int)p;double q (float)ik;if (p q){ik ik 1;}}else{ik 0;}int k ik - 4;if (k 0){k 0;}int m ik 3;if (m n - 1){m n - 1;}for (int i k; i m; i){double s 1.0;double xi x0 i * step;for (int j k; j m; j){if (j ! i){double xj x0 j * step;// 拉格朗日插值公式s s * (t - xj) / (xi - xj);}}z z s * y[i];}return (z);}
4、埃尔米特插值
1埃尔米特不等距插值 /// summary/// 埃尔米特不等距插值/// /summary/// param namex一维数组长度为n存放给定的n个结点的值x(i)要求x(0)x(1)...x(n-1)/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namedy一维数组长度为n存放给定的n个结点的函数导数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Hermite(double[] x, double[] y, double[] dy, double t){int n x.Length;double z 0.0;// 循环插值for (int i 1; i n; i){double s 1.0;for (int j 1; j n; j){if (j ! i){s s * (t - x[j - 1]) / (x[i - 1] - x[j - 1]);}}s s * s;double p 0.0;for (int j 1; j n; j){if (j ! i){p p 1.0 / (x[i - 1] - x[j - 1]);}}double q y[i - 1] (t - x[i - 1]) * (dy[i - 1] - 2.0 * y[i - 1] * p);z z q * s;}return (z);} /// summary/// 埃尔米特等距插值/// (使用非等距插值的方法)/// /summary/// param namex0存放等距n个结点中第一个结点的值/param/// param namestep等距结点的步长/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namedy一维数组长度为n存放给定的n个结点的函数导数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Hermite(double x0, double step, double[] y, double[] dy, double t){double[] x new double[y.Length];for (int i 0; i y.Length; i, x0 step){x[i] x0;}return Hermite(x, y, dy, t);}
2埃尔米特等距插值 /// summary/// 埃尔米特等距插值/// /summary/// param namex0等距n个结点中第一个结点的值/param/// param namestep等距结点的步长/param/// param namey一维数组长度为n存放给定的n个结点的函数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namedy一维数组长度为n存放给定的n个结点的函数导数值y(i)y(i) f(x(i)), i0,1,...,n-1/param/// param namet存放指定的插值点的x值/param/// returns指定的查指点t的函数近似值yf(t)/returnspublic static double Hermite(double x0, double step, double[] y, double[] dy, double t){int n y.Length;double z 0.0;// 循环插值for (int i 1; i n; i){double s 1.0;double q x0 (i - 1) * step;double p;for (int j 1; j n; j){p x0 (j - 1) * step;if (j ! i){s s * (t - p) / (q - p);}}s s * s;p 0.0;for (int j 1; j n; j){if (j ! i){p p 1.0 / (q - (x0 (j - 1) * step));}}q y[i - 1] (t - q) * (dy[i - 1] - 2.0 * y[i - 1] * p);z z q * s;}return (z);}
