用Python實現Newton插值法
1. n階差商實現
def diff(xi,yi,n): """ param xi:插值節點xi param yi:插值節點yi param n: 求幾階差商 return: n階差商 """ if len(xi) != len(yi): #xi和yi必須保證長度一致 return else: diff_quot = [[] for i in range(n)] for j in range(1,n+1): if j == 1: for i in range(n+1-j): diff_quot[j-1].append((yi[i]-yi[i+1]) / (xi[i] - xi[i + 1])) else: for i in range(n+1-j): diff_quot[j-1].append((diff_quot[j-2][i]-diff_quot[j-2][i+1]) / (xi[i] - xi[i + j])) return diff_quot
測試一下:
xi = [1.615,1.634,1.702,1.828] yi = [2.41450,2.46259,2.65271,3.03035] n = 3 print(diff(xi,yi,n))
返回的差商結果為:
[[2.53105263157897, 2.7958823529411716, 2.997142857142854], [3.0440197857724347, 1.0374252793901158], [-9.420631485362996]]
2. 牛頓插值實現
def Newton(x): f = yi[0] v = [] r = 1 for i in range(n): r *= (x - xi[i]) v.append(r) f += diff_quot[i][0] * v[i] return f
測試一下:
x = 1.682 print(Newton(x))
結果為:
2.5944760289639732
3.完整Python代碼
def Newton(xi,yi,n,x): """ param xi:插值節點xi param yi:插值節點yi param n: 求幾階差商 param x: 代求近似值 return: n階差商 """ if len(xi) != len(yi): #xi和yi必須保證長度一致 return else: diff_quot = [[] for i in range(n)] for j in range(1,n+1): if j == 1: for i in range(n+1-j): diff_quot[j-1].append((yi[i]-yi[i+1]) / (xi[i] - xi[i + 1])) else: for i in range(n+1-j): diff_quot[j-1].append((diff_quot[j-2][i]-diff_quot[j-2][i+1]) / (xi[i] - xi[i + j])) print(diff_quot) f = yi[0] v = [] r = 1 for i in range(n): r *= (x - xi[i]) v.append(r) f += diff_quot[i][0] * v[i] return f
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