aaa

按方法整理/插值-线性-抛物线-拉格朗日多项式.py

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+import math
+
+# 获取与待求x最接近的两个点
+def GetClosestTwo(x,list_x):
+    for i in range(0, len(list_x)):
+        if x < list_x[i]:
+            return i-1, i
+    return len(list_x)-2, len(list_x)-1
+
+# 获取与待求x最接近的三个点
+def GetClosestThree(x,list_x):
+    if x < list_x[1]:
+        return 0, 1, 2
+    for i in range(3, len(list_x)):
+        if x < list_x[i]:
+            return i-2, i-1, i
+    return len(list_x)-3, len(list_x)-2, len(list_x)-1
+
+# 线性插值余项计算
+def LinearRegression(x,list_x,FxDiff_n):
+    i, j = GetClosestTwo(x,list_x)
+    ks = max([abs(FxDiff_n(list_x[i],2)), abs(FxDiff_n(list_x[j],2))])
+    omg = (x-list_x[i])*(x-list_x[j])
+    return abs(ks*omg/2)
+
+# 抛物线插值余项计算
+def ParabolaRegression(x,list_x,FxDiff_n):
+    i, j, k = GetClosestThree(x,list_x)
+    ks = max([abs(FxDiff_n(list_x[i],3)), abs(FxDiff_n(list_x[j],3)), abs(FxDiff_n(list_x[k],3))])
+    omg = (x-list_x[i])*(x-list_x[j])*(x-list_x[k])
+    return abs(ks*omg/6)
+
+# 线性插值
+def LinearInterpolation(x,list_x,list_y,FxDiff_n):
+    i, j = GetClosestTwo(x,list_x)
+    result = list_y[i] + (x - list_x[i]) * (list_y[j] - list_y[i]) / (list_x[j] - list_x[i])
+    r = LinearRegression(x,list_x,FxDiff_n)
+    return (result,r)
+
+# 抛物线插值
+def ParabolaInterpolation(x,list_x,list_y,FxDiff_n):
+    i, j, k = GetClosestThree(x,list_x)
+    result  = list_y[i] * (x - list_x[j]) * (x - list_x[k]) / (list_x[i] - list_x[j]) / (list_x[i] - list_x[k])
+    result += list_y[j] * (x - list_x[i]) * (x - list_x[k]) / (list_x[j] - list_x[i]) / (list_x[j] - list_x[k])
+    result += list_y[k] * (x - list_x[i]) * (x - list_x[j]) / (list_x[k] - list_x[i]) / (list_x[k] - list_x[j])
+    r = ParabolaRegression(x,list_x,FxDiff_n)
+    return (result,r)
+
+# 拉格朗日插值
+def LagrangeInterpolation(x,list_x,list_y):
+    result = 0
+    for i in range(0, len(list_x)):
+        temp = 1
+        for j in range(0, len(list_x)):
+            if i != j:
+                temp *= (x - list_x[j]) / (list_x[i] - list_x[j])
+        result += temp * list_y[i]
+    return result
+
+# 定义原函数和其导函数计算结果，用于计算插值的截断误差，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.log(x)
+    else:
+        # 下面改成n阶导数 ##############################################################################
+        result = (-1)**(n+1) * math.factorial(n-1) / (x**n)
+    return result
+
+if __name__ == "__main__":
+    ##############################################################################
+    list_x = [10,11,12,13]                    # 已给出的x数值，与y数值对应
+    list_y = [2.3026,2.3979,2.4849,2.5649]    # 已给出的y数值，与x数值对应
+    x_to_predict = 11.75                      # 要预测的x值
+    print("线性插值结果为%f, 截断误差%f" % LinearInterpolation(x_to_predict,list_x,list_y,FxDiff_n1))
+    print("抛物线插值结果为%f, 截断误差%f" % ParabolaInterpolation(x_to_predict,list_x,list_y,FxDiff_n1))
+
+    # print(LagrangeInterpolation(x_to_predict,list_x,list_y))