all

69-3.py

@@ -0,0 +1,71 @@
+import math
+
+list_x = [0.0,0.2,0.4,0.6,0.8]
+list_y = [1.0,1.2214,1.4918,1.8221,2.2255]
+
+# 定义原函数和其导函数计算结果
+def FxDiff_n(x,n):
+    return math.exp(x)
+
+# 求差分表
+def GetDyList(list_y):
+    result = []
+    result.append(list_y)
+    for i in range(len(list_y)-1,0,-1):
+        tmp = []
+        for j in range(i):
+            tmp.append(result[len(list_y)-1-i][j+1] - result[len(list_y)-1-i][j])
+        result.append(tmp)
+    return result
+
+# 牛顿前插余项计算
+def NewtonForwardRegression(x,n,list_x):
+    h = list_x[1] - list_x[0]
+    t = (x - list_x[0]) / h
+    ks = max([abs(FxDiff_n(list_x[i],n+1)) for i in range(n+1)])
+    result = 1
+    for i in range(n+1):
+        result *= (t-i)*h/(i+1)
+    return  result * ks
+
+# 牛顿前插
+def NewtonForwardInterpolation(x,n,list_x,list_y):
+    list_dyk = GetDyList(list_y)
+    h = list_x[1] - list_x[0]
+    t = (x - list_x[0]) / h
+    result = list_y[0]
+    mul = 1
+    result = list_y[0]
+    for i in range(1, n):
+        mul *= ((t-i+1)/i)
+        result += list_dyk[i][0] * mul
+    r = abs(NewtonForwardRegression(x,n,list_x))
+    return (result,r)
+
+# 求差商表
+def GetDQyList(list_x,list_y):
+    result = []
+    result.append(list_y)
+    for i in range(len(list_y)-1,0,-1):
+        tmp = []
+        for j in range(i):
+            tmp.append((result[len(list_y)-1-i][j+1] - result[len(list_y)-1-i][j])/(list_x[j+len(list_y)-i] - list_x[j]))
+        result.append(tmp)
+    return result
+
+
+# 牛顿基本插值
+def NewtonBaseInterpolation(x,n,list_x,list_y):
+    list_dqy = GetDQyList(list_x,list_y)
+    result = list_dqy[0][0]
+    mul = 1
+    for i in range(1,n):
+        mul *= (x - list_x[i-1])
+        result += list_dqy[i][0] * mul
+    return result
+
+if __name__ == '__main__':
+    print("三点牛顿前插 e^0.12 结果为%f, 截断误差%f" % NewtonForwardInterpolation(0.12, 2, list_x, list_y))
+    print("四点牛顿前插 e^0.12 结果为%f, 截断误差%f" % NewtonForwardInterpolation(0.12, 3, list_x, list_y))
+    print("三点牛顿基本插值 e^0.12 结果为%f" % NewtonBaseInterpolation(0.12, 2, list_x, list_y))
+    print("四点牛顿基本插值 e^0.12 结果为%f" % NewtonBaseInterpolation(0.12, 3, list_x, list_y))