import math

# 定义原函数和其导函数计算结果
def FxDiff_n(x,n):
    result = 0
    if n == 0:
        # 下面改成原函数 ####################3########################################
        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

# 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):
    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):
    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):
    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)
    return (result,r)

# 抛物线插值
def ParabolaInterpolation(x,list_x,list_y):
    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)
    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


list_x = [10,11,12,13]  #改成题干的数值#############################
list_y = [2.3026,2.3979,2.4849,2.5649]  #改成题干的数值#############################
if __name__ == "__main__":
    print("线性插值 ln11.75 结果为%f, 截断误差%f" % LinearInterpolation(11.75,list_x,list_y))  #改成题干的数值#############################
    print("抛物线插值 ln11.75 结果为%f, 截断误差%f" % ParabolaInterpolation(11.75,list_x,list_y))  #改成题干的数值#############################

    # print(LagrangeInterpolation(11.75,list_x,list_y))