/*
* 進行插值的類Interpolation
* 周長發編制
*/
using System;
namespace MSAlgorithm
{
public class Interpolation
{
/// <summary>
/// 一元全區間不等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)，要求x(0)..x(1)..x(n-1)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i),y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t"> 存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueLagrange(int n, double[] x, double[] y, double t)
{
int i, j, k, m;
double z, s;
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
if (n == 2)
{
z = (y[0] * (t - x[1]) - y[1] * (t - x[0])) / (x[0] - x[1]);
return (z);
}
//開始插值
i = 0;
while ((x[i] < t) && (i < n))
i = i + 1;
k = i - 4;
if (k < 0)
k = 0;
m = i + 3;
if (m > n - 1)
m = n - 1;
for (i = k; i <= m; i++)
{
s = 1.0;
for (j = k; j <= m; j++)
{
if (j != i)
s = s * (t - x[j]) / (x[i] - x[j]);//拉格朗日插值公式
}
z = z + s * y[i];
}
return (z);
}
/// <summary>
/// 一元全區間等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueLagrange(int n, double x0, double xStep, double[] y, double t)
{
int i, j, k, m;
double z, s, xi, xj;
double p, q;
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
if (n == 2)
{
z = (y[1] * (t - x0) - y[0] * (t - x0 - xStep)) / xStep;
return (z);
}
//開始插值
if (t > x0)
{
p = (t - x0) / xStep;
i = (int)p;
q = (float)i;
if (p > q)
i = i + 1;
}
else
i = 0;
k = i - 4;
if (k < 0)
k = 0;
m = i + 3;
if (m > n - 1)
m = n - 1;
for (i = k; i <= m; i++)
{
s = 1.0;
xi = x0 + i * xStep;
for (j = k; j <= m; j++)
{
if (j != i)
{
xj = x0 + j * xStep;
s = s * (t - xj) / (xi - xj);//拉格朗日插值公式
}
}
z = z + s * y[i];
}
return (z);
}
/// <summary>
/// 一元三點不等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y"> 一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueLagrange3(int n, double[] x, double[] y, double t)
{
int i, j, k, m;
double z, s;
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
if (n == 2)
{
z = (y[0] * (t - x[1]) - y[1] * (t - x[0])) / (x[0] - x[1]);
return (z);
}
//開始插值
if (t <= x[1])
{
k = 0;
m = 2;
}
else if (t >= x[n - 2])
{
k = n - 3;
m = n - 1;
}
else
{
k = 1;
m = n;
while (m - k != 1)
{
i = (k + m) / 2;
if (t < x[i - 1])
m = i;
else
k = i;
}
k = k - 1;
m = m - 1;
if (Math.Abs(t - x[k]) < Math.Abs(t - x[m]))
k = k - 1;
else
m = m + 1;
}
z = 0.0;
for (i = k; i <= m; i++)
{
s = 1.0;
for (j = k; j <= m; j++)
{
if (j != i)
s = s * (t - x[j]) / (x[i] - x[j]);//拋物線插值公式
}
z = z + s * y[i];
}
return (z);
}
/// <summary>
/// 一元三點等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueLagrange3(int n, double x0, double xStep, double[] y, double t)
{
int i, j, k, m;
double z, s, xi, xj;
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
if (n == 2)
{
z = (y[1] * (t - x0) - y[0] * (t - x0 - xStep)) / xStep;
return (z);
}
//開始插值
if (t <= x0 + xStep)
{
k = 0;
m = 2;
}
else if (t >= x0 + (n - 3) * xStep)
{
k = n - 3;
m = n - 1;
}
else
{
i = (int)((t - x0) / xStep) + 1;
if (Math.Abs(t - x0 - i * xStep) >= Math.Abs(t - x0 - (i - 1) * xStep))
{
k = i - 2;
m = i;
}
else
{
k = i - 1;
m = i + 1;
}
}
z = 0.0;
for (i = k; i <= m; i++)
{
s = 1.0;
xi = x0 + i * xStep;
for (j = k; j <= m; j++)
{
if (j != i)
{
xj = x0 + j * xStep;
//拋物線插值公式
s = s * (t - xj) / (xi - xj);
}
}
z = z + s * y[i];
}
return (z);
}
/// <summary>
/// 連分式不等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValuePqs(int n, double[] x, double[] y, double t)
{
int i, j, k, m, l;
double z, h;
double[] b = new double[8];
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
//連分式插值
if (n <= 8)
{
k = 0;
m = n;
}
else if (t < x[4])
{
k = 0;
m = 8;
}
else if (t > x[n - 5])
{
k = n - 8;
m = 8;
}
else
{
k = 1;
j = n;
while (j - k != 1)
{
i = (k + j) / 2;
if (t < x[i - 1])
j = i;
else
k = i;
}
k = k - 4;
m = 8;
}
b[0] = y[k];
for (i = 2; i <= m; i++)
{
h = y[i + k - 1];
l = 0;
j = 1;
while ((l == 0) && (j <= i - 1))
{
if (Math.Abs(h - b[j - 1]) + 1.0 == 1.0)
l = 1;
else
h = (x[i + k - 1] - x[j + k - 1]) / (h - b[j - 1]);
j = j + 1;
}
b[i - 1] = h;
if (l != 0)
b[i - 1] = 1.0e+35;
}
z = b[m - 1];
for (i = m - 1; i >= 1; i--)
z = b[i - 1] + (t - x[i + k - 1]) / z;
return (z);
}
/// <summary>
/// 連分式等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValuePqs(int n, double x0, double xStep, double[] y, double t)
{
int i, j, k, m, l;
double z, hh, xi, xj;
double[] b = new double[8];
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
//連分式插值
if (n <= 8)
{
k = 0;
m = n;
}
else if (t < (x0 + 4.0 * xStep))
{
k = 0;
m = 8;
}
else if (t > (x0 + (n - 5) * xStep))
{
k = n - 8;
m = 8;
}
else
{
k = (int)((t - x0) / xStep) - 3;
m = 8;
}
b[0] = y[k];
for (i = 2; i <= m; i++)
{
hh = y[i + k - 1];
l = 0;
j = 1;
while ((l == 0) && (j <= i - 1))
{
if (Math.Abs(hh - b[j - 1]) + 1.0 == 1.0)
l = 1;
else
{
xi = x0 + (i + k - 1) * xStep;
xj = x0 + (j + k - 1) * xStep;
hh = (xi - xj) / (hh - b[j - 1]);
}
j = j + 1;
}
b[i - 1] = hh;
if (l != 0)
b[i - 1] = 1.0e+35;
}
z = b[m - 1];
for (i = m - 1; i >= 1; i--)
z = b[i - 1] + (t - (x0 + (i + k - 1) * xStep)) / z;
return (z);
}
/// <summary>
/// 埃爾米特不等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一維數組，長度爲n，存放給定的n個結點的函數導數值y'(i)，y'(i) = f'(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueHermite(int n, double[] x, double[] y, double[] dy, double t)
{
int i, j;
double z, p, q, s;
//初值
z = 0.0;
//循環插值
for (i = 1; i <= n; i++)
{
s = 1.0;
for (j = 1; j <= n; j++)
{
if (j != i)
s = s * (t - x[j - 1]) / (x[i - 1] - x[j - 1]);
}
s = s * s;
p = 0.0;
for (j = 1; j <= n; j++)
{
if (j != i)
p = p + 1.0 / (x[i - 1] - x[j - 1]);
}
q = y[i - 1] + (t - x[i - 1]) * (dy[i - 1] - 2.0 * y[i - 1] * p);
z = z + p * s;
}
return (z);
}
/// <summary>
/// 埃爾米特等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一維數組，長度爲n，存放給定的n個結點的函數導數值y'(i)，y'(i) = f'(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueHermite(int n, double x0, double xStep, double[] y, double[] dy, double t)
{
int i, j;
double z, s, p, q;
//初值
z = 0.0;
//循環插值
for (i = 1; i <= n; i++)
{
s = 1.0;
q = x0 + (i - 1) * xStep;
for (j = 1; j <= n; j++)
{
p = x0 + (j - 1) * xStep;
if (j != i)
s = s * (t - p) / (q - p);
}
s = s * s;
p = 0.0;
for (j = 1; j <= n; j++)
{
if (j != i)
p = p + 1.0 / (q - (x0 + (j - 1) * xStep));
}
q = y[i - 1] + (t - q) * (dy[i - 1] - 2.0 * y[i - 1] * p);
z = z + p * s;
}
return (z);
}
/// <summary>
/// 埃特金不等距逐步插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)， y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <param name="eps">控制精度參數</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueAitken(int n, double[] x, double[] y, double t, double eps)
{
int i, j, k, m, l = 0;
double z;
double[] xx = new double[10];
double[] yy = new double[10];
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
//開始插值
m = 10;
if (m > n)
m = n;
if (t <= x[0])
k = 1;
else if (t >= x[n - 1])
k = n;
else
{
k = 1;
j = n;
while ((k - j != 1) && (k - j != -1))
{
l = (k + j) / 2;
if (t < x[l - 1])
j = l;
else
k = l;
}
if (Math.Abs(t - x[l - 1]) > Math.Abs(t - x[j - 1]))
k = j;
}
j = 1;
l = 0;
for (i = 1; i <= m; i++)
{
k = k + j * l;
if ((k < 1) || (k > n))
{
l = l + 1;
j = -j;
k = k + j * l;
}
xx[i - 1] = x[k - 1];
yy[i - 1] = y[k - 1];
l = l + 1;
j = -j;
}
i = 0;
do
{
i = i + 1;
z = yy[i];
for (j = 0; j <= i - 1; j++)
z = yy[j] + (t - xx[j]) * (yy[j] - z) / (xx[j] - xx[i]);
yy[i] = z;
} while ((i != m - 1) && (Math.Abs(yy[i] - yy[i - 1]) > eps));
return (z);
}
/// <summary>
/// 埃特金等距逐步插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <param name="eps">控制精度參數</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueAitken(int n, double x0, double xStep, double[] y, double t, double eps)
{
int i, j, k, m, l = 0;
double z;
double[] xx = new double[10];
double[] yy = new double[10];
//初值
z = 0.0;
//特例處理
if (n < 1)
return (z);
if (n == 1)
{
z = y[0];
return (z);
}
//開始插值
m = 10;
if (m > n)
m = n;
if (t <= x0)
k = 1;
else if (t >= x0 + (n - 1) * xStep)
k = n;
else
{
k = 1;
j = n;
while ((k - j != 1) && (k - j != -1))
{
l = (k + j) / 2;
if (t < x0 + (l - 1) * xStep)
j = l;
else
k = l;
}
if (Math.Abs(t - (x0 + (l - 1) * xStep)) > Math.Abs(t - (x0 + (j - 1) * xStep)))
k = j;
}
j = 1;
l = 0;
for (i = 1; i <= m; i++)
{
k = k + j * l;
if ((k < 1) || (k > n))
{
l = l + 1;
j = -j;
k = k + j * l;
}
xx[i - 1] = x0 + (k - 1) * xStep;
yy[i - 1] = y[k - 1];
l = l + 1;
j = -j;
}
i = 0;
do
{
i = i + 1;
z = yy[i];
for (j = 0; j <= i - 1; j++)
z = yy[j] + (t - xx[j]) * (yy[j] - z) / (xx[j] - xx[i]);
yy[i] = z;
} while ((i != m - 1) && (Math.Abs(yy[i] - yy[i - 1]) > eps));
return (z);
}
/// <summary>
/// 光滑不等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <param name="s">一維數組，長度爲5，其中s(0)，s(1)，s(2)，s(3)返回三次多項式的係數，s(4)返回指定插值點t處的函數近似值f(t)（k小於0時）或任意值（k大於等於0時）</param>
/// <param name="k">控制參數，若k>=0，則只計算第k個子區間[x(k), x(k+1)]上的三次多項式的係數</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueAkima(int n, double[] x, double[] y, double t, double[] s, int k)
{
int kk, m, l;
double p, q;
double[] u = new double[5];
// 初值
s[4] = 0.0;
s[0] = 0.0;
s[1] = 0.0;
s[2] = 0.0;
s[3] = 0.0;
// 特例處理
if (n < 1)
return s[4];
if (n == 1)
{
s[0] = y[0];
s[4] = y[0];
return s[4];
}
if (n == 2)
{
s[0] = y[0];
s[1] = (y[1] - y[0]) / (x[1] - x[0]);
if (k < 0)
s[4] = (y[0] * (t - x[1]) - y[1] * (t - x[0])) / (x[0] - x[1]);
return s[4];
}
// 插值
if (k < 0)
{
if (t <= x[1])
kk = 0;
else if (t >= x[n - 1])
kk = n - 2;
else
{
kk = 1;
m = n;
while (((kk - m) != 1) && ((kk - m) != -1))
{
l = (kk + m) / 2;
if (t < x[l - 1])
m = l;
else
kk = l;
}
kk = kk - 1;
}
}
else
kk = k;
if (kk >= n - 1)
kk = n - 2;
u[2] = (y[kk + 1] - y[kk]) / (x[kk + 1] - x[kk]);
if (n == 3)
{
if (kk == 0)
{
u[3] = (y[2] - y[1]) / (x[2] - x[1]);
u[4] = 2.0 * u[3] - u[2];
u[1] = 2.0 * u[2] - u[3];
u[0] = 2.0 * u[1] - u[2];
}
else
{
u[1] = (y[1] - y[0]) / (x[1] - x[0]);
u[0] = 2.0 * u[1] - u[2];
u[3] = 2.0 * u[2] - u[1];
u[4] = 2.0 * u[3] - u[2];
}
}
else
{
if (kk <= 1)
{
u[3] = (y[kk + 2] - y[kk + 1]) / (x[kk + 2] - x[kk + 1]);
if (kk == 1)
{
u[1] = (y[1] - y[0]) / (x[1] - x[0]);
u[0] = 2.0 * u[1] - u[2];
if (n == 4)
u[4] = 2.0 * u[3] - u[2];
else
u[4] = (y[4] - y[3]) / (x[4] - x[3]);
}
else
{
u[1] = 2.0 * u[2] - u[3];
u[0] = 2.0 * u[1] - u[2];
u[4] = (y[3] - y[2]) / (x[3] - x[2]);
}
}
else if (kk >= (n - 3))
{
u[1] = (y[kk] - y[kk - 1]) / (x[kk] - x[kk - 1]);
if (kk == (n - 3))
{
u[3] = (y[n - 1] - y[n - 2]) / (x[n - 1] - x[n - 2]);
u[4] = 2.0 * u[3] - u[2];
if (n == 4)
u[0] = 2.0 * u[1] - u[2];
else
u[0] = (y[kk - 1] - y[kk - 2]) / (x[kk - 1] - x[kk - 2]);
}
else
{
u[3] = 2.0 * u[2] - u[1];
u[4] = 2.0 * u[3] - u[2];
u[0] = (y[kk - 1] - y[kk - 2]) / (x[kk - 1] - x[kk - 2]);
}
}
else
{
u[1] = (y[kk] - y[kk - 1]) / (x[kk] - x[kk - 1]);
u[0] = (y[kk - 1] - y[kk - 2]) / (x[kk - 1] - x[kk - 2]);
u[3] = (y[kk + 2] - y[kk + 1]) / (x[kk + 2] - x[kk + 1]);
u[4] = (y[kk + 3] - y[kk + 2]) / (x[kk + 3] - x[kk + 2]);
}
}
s[0] = Math.Abs(u[3] - u[2]);
s[1] = Math.Abs(u[0] - u[1]);
if ((s[0] + 1.0 == 1.0) && (s[1] + 1.0 == 1.0))
p = (u[1] + u[2]) / 2.0;
else
p = (s[0] * u[1] + s[1] * u[2]) / (s[0] + s[1]);
s[0] = Math.Abs(u[3] - u[4]);
s[1] = Math.Abs(u[2] - u[1]);
if ((s[0] + 1.0 == 1.0) && (s[1] + 1.0 == 1.0))
q = (u[2] + u[3]) / 2.0;
else
q = (s[0] * u[2] + s[1] * u[3]) / (s[0] + s[1]);
s[0] = y[kk];
s[1] = p;
s[3] = x[kk + 1] - x[kk];
s[2] = (3.0 * u[2] - 2.0 * p - q) / s[3];
s[3] = (q + p - 2.0 * u[2]) / (s[3] * s[3]);
if (k < 0)
{
p = t - x[kk];
s[4] = s[0] + s[1] * p + s[2] * p * p + s[3] * p * p * p;
}
return s[4];
}
/// <summary>
/// 光滑等距插值
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x0">存放等距n個結點中第一個結點的值</param>
/// <param name="xStep">等距結點的步長</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值點的值</param>
/// <param name="s">一維數組，長度爲5，其中s(0)，s(1)，s(2)，s(3)返回三次多項式的係數，s(4)返回指定插值點t處的函數近似值f(t)（k小於0時）或任意值（k大於等於0時）</param>
/// <param name="k">控制參數，若k>=0，則只計算第k個子區間[x(k), x(k+1)]上的三次多項式的係數</param>
/// <returns>double 型，指定的查指點t的函數近似值f(t)</returns>
public static double GetValueAkima(int n, double x0, double xStep, double[] y, double t, double[] s, int k)
{
int kk, m, l;
double[] u = new double[5];
double p, q;
// 初值
s[4] = 0.0;
s[0] = 0.0;
s[1] = 0.0;
s[2] = 0.0;
s[3] = 0.0;
// 特例處理
if (n < 1)
return s[4];
if (n == 1)
{
s[0] = y[0];
s[4] = y[0];
return s[4];
}
if (n == 2)
{
s[0] = y[0];
s[1] = (y[1] - y[0]) / xStep;
if (k < 0)
s[4] = (y[1] * (t - x0) - y[0] * (t - x0 - xStep)) / xStep;
return s[4];
}
// 插值
if (k < 0)
{
if (t <= x0 + xStep)
kk = 0;
else if (t >= x0 + (n - 1) * xStep)
kk = n - 2;
else
{
kk = 1;
m = n;
while (((kk - m) != 1) && ((kk - m) != -1))
{
l = (kk + m) / 2;
if (t < x0 + (l - 1) * xStep)
m = l;
else
kk = l;
}
kk = kk - 1;
}
}
else
kk = k;
if (kk >= n - 1)
kk = n - 2;
u[2] = (y[kk + 1] - y[kk]) / xStep;
if (n == 3)
{
if (kk == 0)
{
u[3] = (y[2] - y[1]) / xStep;
u[4] = 2.0 * u[3] - u[2];
u[1] = 2.0 * u[2] - u[3];
u[0] = 2.0 * u[1] - u[2];
}
else
{
u[1] = (y[1] - y[0]) / xStep;
u[0] = 2.0 * u[1] - u[2];
u[3] = 2.0 * u[2] - u[1];
u[4] = 2.0 * u[3] - u[2];
}
}
else
{
if (kk <= 1)
{
u[3] = (y[kk + 2] - y[kk + 1]) / xStep;
if (kk == 1)
{
u[1] = (y[1] - y[0]) / xStep;
u[0] = 2.0 * u[1] - u[2];
if (n == 4)
u[4] = 2.0 * u[3] - u[2];
else
u[4] = (y[4] - y[3]) / xStep;
}
else
{
u[1] = 2.0 * u[2] - u[3];
u[0] = 2.0 * u[1] - u[2];
u[4] = (y[3] - y[2]) / xStep;
}
}
else if (kk >= (n - 3))
{
u[1] = (y[kk] - y[kk - 1]) / xStep;
if (kk == (n - 3))
{
u[3] = (y[n - 1] - y[n - 2]) / xStep;
u[4] = 2.0 * u[3] - u[2];
if (n == 4)
u[0] = 2.0 * u[1] - u[2];
else
u[0] = (y[kk - 1] - y[kk - 2]) / xStep;
}
else
{
u[3] = 2.0 * u[2] - u[1];
u[4] = 2.0 * u[3] - u[2];
u[0] = (y[kk - 1] - y[kk - 2]) / xStep;
}
}
else
{
u[1] = (y[kk] - y[kk - 1]) / xStep;
u[0] = (y[kk - 1] - y[kk - 2]) / xStep;
u[3] = (y[kk + 2] - y[kk + 1]) / xStep;
u[4] = (y[kk + 3] - y[kk + 2]) / xStep;
}
}
s[0] = Math.Abs(u[3] - u[2]);
s[1] = Math.Abs(u[0] - u[1]);
if ((s[0] + 1.0 == 1.0) && (s[1] + 1.0 == 1.0))
p = (u[1] + u[2]) / 2.0;
else
p = (s[0] * u[1] + s[1] * u[2]) / (s[0] + s[1]);
s[0] = Math.Abs(u[3] - u[4]);
s[1] = Math.Abs(u[2] - u[1]);
if ((s[0] + 1.0 == 1.0) && (s[1] + 1.0 == 1.0))
q = (u[2] + u[3]) / 2.0;
else
q = (s[0] * u[2] + s[1] * u[3]) / (s[0] + s[1]);
s[0] = y[kk];
s[1] = p;
s[3] = xStep;
s[2] = (3.0 * u[2] - 2.0 * p - q) / s[3];
s[3] = (q + p - 2.0 * u[2]) / (s[3] * s[3]);
if (k < 0)
{
p = t - (x0 + kk * xStep);
s[4] = s[0] + s[1] * p + s[2] * p * p + s[3] * p * p * p;
}
return s[4];
}
/// <summary>
/// 第一種邊界條件的三次樣條函數插值、微商與積分
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一維數組，長度爲n，調用時，dy(0)存放給定區間的左端點處的一階導數值，dy(n-1)存放給定區間的右端點處的一階導數值。
/// 返回時，存放n個給定點處的一階導數值y'(i)，i=0,1,...,n-1</param>
/// <param name="ddy">一維數組，長度爲n，返回時，存放n個給定點處的二階導數值y''(i)，i=0,1,...,n-1</param>
/// <param name="m">指定插值點的個數</param>
/// <param name="t">一維數組，長度爲m，存放m個指定的插值點的值。</param>
/// <param name="z">一維數組，長度爲m，存放m個指定的插值點處的函數值</param>
/// <param name="dz">一維數組，長度爲m，存放m個指定的插值點處的一階導數值</param>
/// <param name="ddz">一維數組，長度爲m，存放m個指定的插值點處的二階導數值</param>
/// <returns>double 型，指定函數的x(0)到x(n-1)的定積分值</returns>
public static double GetValueSpline1(int n, double[] x, double[] y, double[] dy, double[] ddy,
int m, double[] t, double[] z, double[] dz, double[] ddz)
{
int i, j;
double h0, h1, alpha, beta, g;
// 初值
double[] s = new double[n];
s[0] = dy[0];
dy[0] = 0.0;
h0 = x[1] - x[0];
for (j = 1; j <= n - 2; j++)
{
h1 = x[j + 1] - x[j];
alpha = h0 / (h0 + h1);
beta = (1.0 - alpha) * (y[j] - y[j - 1]) / h0;
beta = 3.0 * (beta + alpha * (y[j + 1] - y[j]) / h1);
dy[j] = -alpha / (2.0 + (1.0 - alpha) * dy[j - 1]);
s[j] = (beta - (1.0 - alpha) * s[j - 1]);
s[j] = s[j] / (2.0 + (1.0 - alpha) * dy[j - 1]);
h0 = h1;
}
for (j = n - 2; j >= 0; j--)
dy[j] = dy[j] * dy[j + 1] + s[j];
for (j = 0; j <= n - 2; j++)
s[j] = x[j + 1] - x[j];
for (j = 0; j <= n - 2; j++)
{
h1 = s[j] * s[j];
ddy[j] = 6.0 * (y[j + 1] - y[j]) / h1 - 2.0 * (2.0 * dy[j] + dy[j + 1]) / s[j];
}
h1 = s[n - 2] * s[n - 2];
ddy[n - 1] = 6.0 * (y[n - 2] - y[n - 1]) / h1 + 2.0 * (2.0 * dy[n - 1] + dy[n - 2]) / s[n - 2];
g = 0.0;
for (i = 0; i <= n - 2; i++)
{
h1 = 0.5 * s[i] * (y[i] + y[i + 1]);
h1 = h1 - s[i] * s[i] * s[i] * (ddy[i] + ddy[i + 1]) / 24.0;
g = g + h1;
}
for (j = 0; j <= m - 1; j++)
{
if (t[j] >= x[n - 1])
i = n - 2;
else
{
i = 0;
while (t[j] > x[i + 1])
i = i + 1;
}
h1 = (x[i + 1] - t[j]) / s[i];
h0 = h1 * h1;
z[j] = (3.0 * h0 - 2.0 * h0 * h1) * y[i];
z[j] = z[j] + s[i] * (h0 - h0 * h1) * dy[i];
dz[j] = 6.0 * (h0 - h1) * y[i] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i];
ddz[j] = (6.0 - 12.0 * h1) * y[i] / (s[i] * s[i]);
ddz[j] = ddz[j] + (2.0 - 6.0 * h1) * dy[i] / s[i];
h1 = (t[j] - x[i]) / s[i];
h0 = h1 * h1;
z[j] = z[j] + (3.0 * h0 - 2.0 * h0 * h1) * y[i + 1];
z[j] = z[j] - s[i] * (h0 - h0 * h1) * dy[i + 1];
dz[j] = dz[j] - 6.0 * (h0 - h1) * y[i + 1] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i + 1];
ddz[j] = ddz[j] + (6.0 - 12.0 * h1) * y[i + 1] / (s[i] * s[i]);
ddz[j] = ddz[j] - (2.0 - 6.0 * h1) * dy[i + 1] / s[i];
}
return (g);
}
/// <summary>
/// 第二種邊界條件的三次樣條函數插值、微商與積分
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一維數組，長度爲n，調用時，dy(0)存放給定區間的左端點處的一階導數值，
/// dy(n-1)存放給定區間的右端點處的一階導數值。返回時，存放n個給定點處的一階導數值y'(i)，i=0,1,...,n-1</param>
/// <param name="ddy">一維數組，長度爲n，返回時，存放n個給定點處的二階導數值y''(i)，i=0,1,...,n-1</param>
/// <param name="m">指定插值點的個數</param>
/// <param name="t">一維數組，長度爲m，存放m個指定的插值點的值。</param>
/// <param name="z">一維數組，長度爲m，存放m個指定的插值點處的函數值</param>
/// <param name="dz">一維數組，長度爲m，存放m個指定的插值點處的一階導數值</param>
/// <param name="ddz">一維數組，長度爲m，存放m個指定的插值點處的二階導數值</param>
/// <returns>double 型，指定函數的x(0)到x(n-1)的定積分值</returns>
public static double GetValueSpline2(int n, double[] x, double[] y, double[] dy, double[] ddy,
int m, double[] t, double[] z, double[] dz, double[] ddz)
{
int i, j;
double h0, h1 = 0, alpha, beta, g;
// 初值
double[] s = new double[n];
dy[0] = -0.5;
h0 = x[1] - x[0];
s[0] = 3.0 * (y[1] - y[0]) / (2.0 * h0) - ddy[0] * h0 / 4.0;
for (j = 1; j <= n - 2; j++)
{
h1 = x[j + 1] - x[j];
alpha = h0 / (h0 + h1);
beta = (1.0 - alpha) * (y[j] - y[j - 1]) / h0;
beta = 3.0 * (beta + alpha * (y[j + 1] - y[j]) / h1);
dy[j] = -alpha / (2.0 + (1.0 - alpha) * dy[j - 1]);
s[j] = (beta - (1.0 - alpha) * s[j - 1]);
s[j] = s[j] / (2.0 + (1.0 - alpha) * dy[j - 1]);
h0 = h1;
}
dy[n - 1] = (3.0 * (y[n - 1] - y[n - 2]) / h1 + ddy[n - 1] * h1 / 2.0 - s[n - 2]) / (2.0 + dy[n - 2]);
for (j = n - 2; j >= 0; j--)
dy[j] = dy[j] * dy[j + 1] + s[j];
for (j = 0; j <= n - 2; j++)
s[j] = x[j + 1] - x[j];
for (j = 0; j <= n - 2; j++)
{
h1 = s[j] * s[j];
ddy[j] = 6.0 * (y[j + 1] - y[j]) / h1 - 2.0 * (2.0 * dy[j] + dy[j + 1]) / s[j];
}
h1 = s[n - 2] * s[n - 2];
ddy[n - 1] = 6.0 * (y[n - 2] - y[n - 1]) / h1 + 2.0 * (2.0 * dy[n - 1] + dy[n - 2]) / s[n - 2];
g = 0.0;
for (i = 0; i <= n - 2; i++)
{
h1 = 0.5 * s[i] * (y[i] + y[i + 1]);
h1 = h1 - s[i] * s[i] * s[i] * (ddy[i] + ddy[i + 1]) / 24.0;
g = g + h1;
}
for (j = 0; j <= m - 1; j++)
{
if (t[j] >= x[n - 1])
i = n - 2;
else
{
i = 0;
while (t[j] > x[i + 1])
i = i + 1;
}
h1 = (x[i + 1] - t[j]) / s[i];
h0 = h1 * h1;
z[j] = (3.0 * h0 - 2.0 * h0 * h1) * y[i];
z[j] = z[j] + s[i] * (h0 - h0 * h1) * dy[i];
dz[j] = 6.0 * (h0 - h1) * y[i] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i];
ddz[j] = (6.0 - 12.0 * h1) * y[i] / (s[i] * s[i]);
ddz[j] = ddz[j] + (2.0 - 6.0 * h1) * dy[i] / s[i];
h1 = (t[j] - x[i]) / s[i];
h0 = h1 * h1;
z[j] = z[j] + (3.0 * h0 - 2.0 * h0 * h1) * y[i + 1];
z[j] = z[j] - s[i] * (h0 - h0 * h1) * dy[i + 1];
dz[j] = dz[j] - 6.0 * (h0 - h1) * y[i + 1] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i + 1];
ddz[j] = ddz[j] + (6.0 - 12.0 * h1) * y[i + 1] / (s[i] * s[i]);
ddz[j] = ddz[j] - (2.0 - 6.0 * h1) * dy[i + 1] / s[i];
}
return (g);
}
/// <summary>
/// 第三種邊界條件的三次樣條函數插值、微商與積分
/// </summary>
/// <param name="n">結點的個數</param>
/// <param name="x">一維數組，長度爲n，存放給定的n個結點的值x(i)</param>
/// <param name="y">一維數組，長度爲n，存放給定的n個結點的函數值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一維數組，長度爲n，調用時，dy(0)存放給定區間的左端點處的一階導數值，
/// dy(n-1)存放給定區間的右端點處的一階導數值。返回時，存放n個給定點處的一階導數值y'(i)，i=0,1,...,n-1</param>
/// <param name="ddy">一維數組，長度爲n，返回時，存放n個給定點處的二階導數值y''(i)，i=0,1,...,n-1</param>
/// <param name="m">指定插值點的個數</param>
/// <param name="t">一維數組，長度爲m，存放m個指定的插值點的值。</param>
/// <param name="z">一維數組，長度爲m，存放m個指定的插值點處的函數值</param>
/// <param name="dz">一維數組，長度爲m，存放m個指定的插值點處的一階導數值</param>
/// <param name="ddz">一維數組，長度爲m，存放m個指定的插值點處的二階導數值</param>
/// <returns>double 型，指定函數的x(0)到x(n-1)的定積分值</returns>
public static double GetValueSpline3(int n, double[] x, double[] y, double[] dy, double[] ddy,
int m, double[] t, double[] z, double[] dz, double[] ddz)
{
int i, j;
double h0, y0, h1, y1, alpha = 0, beta = 0, u, g;
// 初值
double[] s = new double[n];
h0 = x[n - 1] - x[n - 2];
y0 = y[n - 1] - y[n - 2];
dy[0] = 0.0; ddy[0] = 0.0; ddy[n - 1] = 0.0;
s[0] = 1.0; s[n - 1] = 1.0;
for (j = 1; j <= n - 1; j++)
{
h1 = h0; y1 = y0;
h0 = x[j] - x[j - 1];
y0 = y[j] - y[j - 1];
alpha = h1 / (h1 + h0);
beta = 3.0 * ((1.0 - alpha) * y1 / h1 + alpha * y0 / h0);
if (j < n - 1)
{
u = 2.0 + (1.0 - alpha) * dy[j - 1];
dy[j] = -alpha / u;
s[j] = (alpha - 1.0) * s[j - 1] / u;
ddy[j] = (beta - (1.0 - alpha) * ddy[j - 1]) / u;
}
}
for (j = n - 2; j >= 1; j--)
{
s[j] = dy[j] * s[j + 1] + s[j];
ddy[j] = dy[j] * ddy[j + 1] + ddy[j];
}
dy[n - 2] = (beta - alpha * ddy[1] - (1.0 - alpha) * ddy[n - 2]) /
(alpha * s[1] + (1.0 - alpha) * s[n - 2] + 2.0);
for (j = 2; j <= n - 1; j++)
dy[j - 2] = s[j - 1] * dy[n - 2] + ddy[j - 1];
dy[n - 1] = dy[0];
for (j = 0; j <= n - 2; j++)
s[j] = x[j + 1] - x[j];
for (j = 0; j <= n - 2; j++)
{
h1 = s[j] * s[j];
ddy[j] = 6.0 * (y[j + 1] - y[j]) / h1 - 2.0 * (2.0 * dy[j] + dy[j + 1]) / s[j];
}
h1 = s[n - 2] * s[n - 2];
ddy[n - 1] = 6.0 * (y[n - 2] - y[n - 1]) / h1 + 2.0 * (2.0 * dy[n - 1] + dy[n - 2]) / s[n - 2];
g = 0.0;
for (i = 0; i <= n - 2; i++)
{
h1 = 0.5 * s[i] * (y[i] + y[i + 1]);
h1 = h1 - s[i] * s[i] * s[i] * (ddy[i] + ddy[i + 1]) / 24.0;
g = g + h1;
}
for (j = 0; j <= m - 1; j++)
{
h0 = t[j];
while (h0 >= x[n - 1])
h0 = h0 - (x[n - 1] - x[0]);
while (h0 < x[0])
h0 = h0 + (x[n - 1] - x[0]);
i = 0;
while (h0 > x[i + 1])
i = i + 1;
u = h0;
h1 = (x[i + 1] - u) / s[i];
h0 = h1 * h1;
z[j] = (3.0 * h0 - 2.0 * h0 * h1) * y[i];
z[j] = z[j] + s[i] * (h0 - h0 * h1) * dy[i];
dz[j] = 6.0 * (h0 - h1) * y[i] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i];
ddz[j] = (6.0 - 12.0 * h1) * y[i] / (s[i] * s[i]);
ddz[j] = ddz[j] + (2.0 - 6.0 * h1) * dy[i] / s[i];
h1 = (u - x[i]) / s[i];
h0 = h1 * h1;
z[j] = z[j] + (3.0 * h0 - 2.0 * h0 * h1) * y[i + 1];
z[j] = z[j] - s[i] * (h0 - h0 * h1) * dy[i + 1];
dz[j] = dz[j] - 6.0 * (h0 - h1) * y[i + 1] / s[i];
dz[j] = dz[j] + (3.0 * h0 - 2.0 * h1) * dy[i + 1];
ddz[j] = ddz[j] + (6.0 - 12.0 * h1) * y[i + 1] / (s[i] * s[i]);
ddz[j] = ddz[j] - (2.0 - 6.0 * h1) * dy[i + 1] / s[i];
}
return (g);
}
/// <summary>
/// 二元三點插值
/// </summary>
/// <param name="n">x方向上給定結點的點數</param>
/// <param name="x">一維數組，長度爲n，存放給定n x m 個結點x方向上的n個值x(i)</param>
/// <param name="m">y方向上給定結點的點數</param>
/// <param name="y">一維數組，長度爲m，存放給定n x m 個結點y方向上的m個值y(i)</param>
/// <param name="z">一維數組，長度爲n x m，存放給定的n x m個結點的函數值z(i,j)，
/// z(i,j) = f(x(i), y(j)), i=0,1,...,n-1, j=0,1,...,m-1</param>
/// <param name="u">存放插值點x座標</param>
/// <param name="v">存放插值點y座標</param>
/// <returns>double 型，指定函數值f(u, v)</returns>
public static double GetValueTqip(int n, double[] x, int m, double[] y, double[] z, double u, double v)
{
int nn, mm, ip, iq, i, j, k, l;
double[] b = new double[3];
double h, w;
// 初值
nn = 3;
// 特例
if (n <= 3)
{
ip = 0;
nn = n;
}
else if (u <= x[1])
ip = 0;
else if (u >= x[n - 2])
ip = n - 3;
else
{
i = 1; j = n;
while (((i - j) != 1) && ((i - j) != -1))
{
l = (i + j) / 2;
if (u < x[l - 1])
j = l;
else
i = l;
}
if (Math.Abs(u - x[i - 1]) < Math.Abs(u - x[j - 1]))
ip = i - 2;
else
ip = i - 1;
}
mm = 3;
if (m <= 3)
{
iq = 0;
mm = m;
}
else if (v <= y[1])
iq = 0;
else if (v >= y[m - 2])
iq = m - 3;
else
{
i = 1;
j = m;
while (((i - j) != 1) && ((i - j) != -1))
{
l = (i + j) / 2;
if (v < y[l - 1])
j = l;
else
i = l;
}
if (Math.Abs(v - y[i - 1]) < Math.Abs(v - y[j - 1]))
iq = i - 2;
else
iq = i - 1;
}
for (i = 0; i <= nn - 1; i++)
{
b[i] = 0.0;
for (j = 0; j <= mm - 1; j++)
{
k = m * (ip + i) + (iq + j);
h = z[k];
for (k = 0; k <= mm - 1; k++)
{
if (k != j)
h = h * (v - y[iq + k]) / (y[iq + j] - y[iq + k]);
}
b[i] = b[i] + h;
}
}
w = 0.0;
for (i = 0; i <= nn - 1; i++)
{
h = b[i];
for (j = 0; j <= nn - 1; j++)
{
if (j != i)
h = h * (u - x[ip + j]) / (x[ip + i] - x[ip + j]);
}
w = w + h;
}
return (w);
}
/// <summary>
/// 二元全區間插值
/// </summary>
/// <param name="n">x方向上給定結點的點數</param>
/// <param name="x">一維數組，長度爲n，存放給定n x m 個結點x方向上的n個值x(i)</param>
/// <param name="m">y方向上給定結點的點數</param>
/// <param name="y">一維數組，長度爲m，存放給定n x m 個結點y方向上的m個值y(i)</param>
/// <param name="z">一維數組，長度爲n x m，存放給定的n x m個結點的函數值z(i,j)，
/// z(i,j) = f(x(i), y(j)), i=0,1,...,n-1, j=0,1,...,m-1</param>
/// <param name="u">存放插值點x座標</param>
/// <param name="v">存放插值點y座標</param>
/// <returns>double 型，指定函數值f(u, v)</returns>
public static double GetValueLagrange2(int n, double[] x, int m, double[] y, double[] z, double u, double v)
{
int ip, ipp, i, j, l, iq, iqq, k;
double h, w;
double[] b = new double[10];
// 特例
if (u <= x[0])
{
ip = 1;
ipp = 4;
}
else if (u >= x[n - 1])
{
ip = n - 3;
ipp = n;
}
else
{
i = 1;
j = n;
while (((i - j) != 1) && ((i - j) != -1))
{
l = (i + j) / 2;
if (u < x[l - 1])
j = l;
else
i = l;
}
ip = i - 3;
ipp = i + 4;
}
if (ip < 1)
ip = 1;
if (ipp > n)
ipp = n;
if (v <= y[0])
{
iq = 1;
iqq = 4;
}
else if (v >= y[m - 1])
{
iq = m - 3;
iqq = m;
}
else
{
i = 1;
j = m;
while (((i - j) != 1) && ((i - j) != -1))
{
l = (i + j) / 2;
if (v < y[l - 1])
j = l;
else
i = l;
}
iq = i - 3;
iqq = i + 4;
}
if (iq < 1)
iq = 1;
if (iqq > m)
iqq = m;
for (i = ip - 1; i <= ipp - 1; i++)
{
b[i - ip + 1] = 0.0;
for (j = iq - 1; j <= iqq - 1; j++)
{
h = z[m * i + j];
for (k = iq - 1; k <= iqq - 1; k++)
{
if (k != j)
h = h * (v - y[k]) / (y[j] - y[k]);
}
b[i - ip + 1] = b[i - ip + 1] + h;
}
}
w = 0.0;
for (i = ip - 1; i <= ipp - 1; i++)
{
h = b[i - ip + 1];
for (j = ip - 1; j <= ipp - 1; j++)
{
if (j != i)
h = h * (u - x[j]) / (x[i] - x[j]);
}
w = w + h;
}
return (w);
}
}
}
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