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