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))