МНК

from math import sin, cos
import numpy as np


def f1(x):
    return 1


def f2(x):
    return (x - 1) ** 2


def f3(x):
    return cos(2 * x)


def f4(x):
    return sin(3 * x)


m = 4
n = 4

points = [(0.67, 1.1), (1.4, 0.5), (2.9, 2.1), (3.3, 2.4)]
a_ij = list()

for i in range(m):
    for j in range(n):
        if j == 0:
            a_ij.append(f1(points[i][0]))
        elif j == 1:
            a_ij.append(f2(points[i][0]))
        elif j == 2:
            a_ij.append(f3(points[i][0]))
        elif j == 3:
            a_ij.append(f4(points[i][0]))

a_ij = np.array(a_ij).reshape((m, n))

a_jk_tilda = list()
b_j = list()

for j in range(n):
    for k in range(n):
        tempsum = 0
        for i in range(m):
            tempsum += a_ij[i][j] * a_ij[i][k]
        a_jk_tilda.append(tempsum * 2)

a_jk_tilda = np.array(a_jk_tilda).reshape((n, n))

for j in range(n):
    tempsum = 0
    for i in range(m):
        tempsum += points[i][1] * a_ij[i][j]
    b_j.append(tempsum * (-2))

matrix = list()

for j in range(n):
    for k in range(n):
        matrix.append(a_jk_tilda[j][k])
    matrix.append(-b_j[j])

matrix = np.array(matrix).reshape((n, n + 1))

solutions = [0.913487, 0.215415, 0.402941, 80.0784007] # https://www.wolframalpha.com/input?i=8a%2B18.3378b%2B++2.2445+c%2B+0.4778+d%3D12.2%3B18.3378a%2B+82.1073+b%2B16.19522295c+-0.13586744d%3D+40.95358%3B+2.2445+a%2B16.1952b%2B+5.2543c%2B2.3611d%3D7.8413%3B0.4778a+-0.1358+b%2B+2.3611+c%2B+4.4554d%3D+++1.7079

sum1 = 0
sum2 = 0
sum3 = 0
c = 0

for j in range(n):
    sum1 += a_jk_tilda[j][j] * (solutions[j]) ** 2

for j in range(n):
    for k in range(n):
        if j < k:
            sum2 += a_jk_tilda[j][k] * solutions[j] * solutions[k]

for j in range(n):
    sum3 += b_j[j] * solutions[j]

for i in range(m):
    c += points[i][1] ** 2

d = 0.5 * (0.5 * sum1 + sum2) + sum3 + c

print("Матрица коэффициентов a_{ij}:")
print(a_ij)

print()

print("Матрица коэффициентов ~a_{jk}:")
print(a_jk_tilda)

print()

print("Коэффициенты b_j:")
print(b_j)

print()

print("СЛАУ, решением которой являются искомые коэффициенты линейной комбинации:")
print(matrix)

print()

print("Минимальное значение D функции ошибки:")
print(d)