import numpy as np
import matplotlib.pyplot as plt
from math import *
import pandas as pd
import titleprint as pt
from sympy import *

def xi(i):
	return a + i * h


def func_value(x):
	return x**3 - sin(x)


def print_value_table():
	values = []
	for i in range(0, 11):
		x = xi(i)
		y = round(func_value(x), 2)
		values.append({'x' : x, 'y' : y})
	print(pd.DataFrame(values))


def L1(i):
	return func_value(xi(i)) * (inter_point - xi(i + 1)) \
	/ (xi(i) - xi(i + 1)) + func_value(xi(i + 1)) \
	* (inter_point - xi(i)) / (xi(i + 1) - xi(i))


def find_der2_val(i):
	val = xi(i)
	return expr_der2.subs(x, val)


def R1(w2, i):
	return abs(find_der2_val(i) * w2 / 2)


def W2(i):
	return round((inter_point - xi(i)) * (inter_point - xi(i + 1)), 4)


def check(min_value, max_value, true_value):
	if true_value >= max_value or true_value <= min_value:
		return false
	return true


def L2(i):
	return func_value(xi(i - 1)) * ((inter_point - xi(i)) * (inter_point - xi(i + 1))) / ((xi(i - 1) - xi(i)) * (xi(i - 1) - xi(i + 1))) \
           + func_value(xi(i)) * ((inter_point - xi(i - 1)) * (inter_point - xi(i + 1))) / ((xi(i) - xi(i - 1)) * (xi(i) - xi(i + 1))) \
           + func_value(xi(i + 1)) * ((inter_point - xi(i - 1)) * (inter_point - xi(i))) / ((xi(i + 1) - xi(i - 1)) * (xi(i + 1) - xi(i)))


def find_der3_val(i):
	val = xi(i)
	return expr_der3.subs(x, val)


def R2(w3, i):
	return abs(find_der3_val(i) * w3 / 6)


def W3(i):
	return round((inter_point - xi(i - 1)) * (inter_point - xi(i)) * (inter_point - xi(i + 1)), 6)


def difference1(_x_i, _x_j):
    return (func_value(_x_j) - func_value(_x_i)) / (_x_j - _x_i)


def difference2(_x_i, _x_j, _x_k):
    return (difference1(_x_j, _x_k) - difference1(_x_i, _x_j)) / (_x_k - _x_i)


def N1(i):
    return func_value(xi(i)) + difference1(xi(i), xi(i + 1)) * (inter_point - xi(i))


def N2(i):
    return func_value(xi(i - 1)) + difference1(xi(i - 1), xi(i)) * (inter_point - xi(i - 1)) \
           + difference2(xi(i - 1), xi(i), xi(i + 1)) * (inter_point - xi(i - 1)) * (inter_point - xi(i))


a = 0.6
b = 1.1
h = round((b - a) / 10, 2)

# TASK 1
pt.printTitle(1)
print_value_table()


# TASK 2
pt.printTitle(2)
inter_point = 0.92
index = floor((inter_point - a) / h)
lagrange1_value = L1(index)
print('Значение функции по интерполяционной формуле Лагранжа первого порядка = ', lagrange1_value)


# TASK 3.1
pt.printTitle(3.1)
x = symbols('x')
expr = func_value(x)
expr_der2 = diff(expr, x, 2)
print('Функция: ', expr)
print('Вторая производная', expr_der2)
der2_val1 = find_der2_val(index)
der2_val2 = find_der2_val(index+1)
print('Минимальное значение производной: ', min(der2_val1, der2_val2))
print('Максимальное значение производной: ', max(der2_val1, der2_val2))

# TASK 3.2
pt.printTitle(3.2)
w2 = W2(index)
remainder1 = R1(w2, index)
remainder2 = R1(w2, index + 1)
min_remainder = min(remainder1, remainder2)
max_remainder = max(remainder1, remainder2)
print('Минимальное значение остаточного члена: ', min_remainder)
print('Максимальное значение остаточного члена: ', max_remainder)


# TASK 4
pt.printTitle(4)
true_value = func_value(inter_point)
R1_inter_point = abs(lagrange1_value - true_value)
print(min_remainder, '<', R1_inter_point, '<', max_remainder, 'is ', check(min_remainder, max_remainder, R1_inter_point))


# TASK 5
pt.printTitle(5)
lagrange2_value = L2(index)
print('Значение функции по интерполяционной формуле Лагранжа второго порядка = ', lagrange2_value)


# TASK 6.1
pt.printTitle(6.1)
expr_der3 = diff(expr, x, 3)
print('Третья производная: ', expr_der3)
der3_val1 = find_der3_val(index - 1)
der3_val2 = find_der3_val(index + 1)
print('Минимальное значение производной: ', min(der3_val1, der3_val2))
print('Максимальное значение производной: ', max(der3_val1, der3_val2))

# TASK 6.2
pt.printTitle(6.2)
w3 = W3(index)
remainder1 = R2(w3, index - 1)
remainder2 = R2(w3, index + 1)
min_remainder = min(remainder1, remainder2)
max_remainder = max(remainder1, remainder2)
print('Минимальное значение остаточного члена:', min_remainder)
print('Максимальное значение остаточного члена:', max_remainder)


# TASK 7
pt.printTitle(7)
R2_inter_point = abs(lagrange2_value - true_value)
print(min_remainder, '<', R2_inter_point, '<', max_remainder, 'is ', check(min_remainder, max_remainder, R2_inter_point))


# TASK 8.1
pt.printTitle(8.1)
differ = []
x_i, x_j, x_k = symbols('x_i, x_j, x_k')
differ.append({
    'f(x_i)': func_value(xi(index)),
    'f(x_i, x_j)': difference1(xi(index), xi(index + 1)),
    'f(x_i, x_j, x_k)': difference2(xi(index - 1), xi(index), xi(index + 1))
})
print(pd.DataFrame(differ))

# TASK 8.2
pt.printTitle(8.2)
newton1 = N1(index)
print('Значение функции по формуле Ньютона1:', newton1)
newton2 = N2(index)
print('Значение функции по формуле Ньютона2:', newton2)

# TASK 8.3
print('Разность интерполяционных формул Лагранжа1 и Ньютона1:', abs(lagrange1_value - newton1))
print('Разность интерполяционных формул Лагранжа2 и Ньютона2:', abs(lagrange2_value - newton2))