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))