aaa

按方法整理/插值-牛顿前插-牛顿基本插值.py

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+import math
+
+# 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,FxDiff_n):
+    h = list_x[1] - list_x[0]
+    t = (x - list_x[0]) / h
+    ks = max([abs(FxDiff_n(list_x[i],n)) for i in range(n)])
+    result = 1
+    for i in range(n):
+        result *= (t-i)*h/(i+1)
+    return  result * ks
+
+# 牛顿前插
+def NewtonForwardInterpolation(x,n,list_x,list_y,FxDiff_n):
+    list_dyk = GetDyList(list_y)
+    print("\n差分表")
+    print("--------------------------------------------------")
+    for i in range(len(list_dyk)):
+        print(list_dyk[i])
+    print("--------------------------------------------------")
+    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,FxDiff_n))
+    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)
+    print("\n差商表")
+    print("--------------------------------------------------")
+    for i in range(len(list_dqy)):
+        print(list_dqy[i])
+    print("--------------------------------------------------")
+    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
+
+
+
+# 定义原函数和其导函数计算结果，用于计算牛顿前插的截断误差，如果不需要则不用管
+def FxDiff_n1(x,n):
+    result = 0
+    if n == 0:
+        # 下面改成原函数 ############################################################
+        result = math.exp(x)
+    else:
+        # 下面改成n阶导数 ##############################################################################
+        result = math.exp(x)
+    return result
+
+
+if __name__ == '__main__':
+    ##############################################################################################################
+    list_x = [0.0,0.2,0.4,0.6,0.8]                 # 已给出的x数值，与y数值对应
+    list_y = [1.0,1.2214,1.4918,1.8221,2.2255]     # 已给出的y数值，与x数值对应
+    x_to_predict = 0.12                            # 要预测的x值
+    # 记得改下面的点数(第二个参数)，n是点数，阶数为n-1 ################################
+    print("三点牛顿前插结果为%f, 截断误差%f" % NewtonForwardInterpolation(x_to_predict, 3, list_x, list_y,FxDiff_n1))
+    print("四点牛顿前插结果为%f, 截断误差%f" % NewtonForwardInterpolation(x_to_predict, 4, list_x, list_y,FxDiff_n1))
+    print("三点牛顿基本插值结果为%f" % NewtonBaseInterpolation(x_to_predict, 3, list_x, list_y))
+    print("四点牛顿基本插值结果为%f" % NewtonBaseInterpolation(x_to_predict, 4, list_x, list_y))