import numpy as np
import matplotlib.pyplot as plt

def leastSquares(x, y, deg=1):
    '''Compute the coefficients of a polynomial of degree deg
    that approximate the dependence y = y(x) by using the
    Least Squares method.
    Input: x is an array of values of independent variable;
           y is an array of measured values of dependent variable y=y(x);
           deg is the degree of the approximating polynomial.
    Return: the coefficients of the approximating polynomial
            in descending order.'''

    m = len(x)
    assert len(y) == m

    a = np.vander(x, deg + 1)
    pinv_a = np.linalg.pinv(a)
    w = pinv_a.dot(y)
    return w

def showLeastSquares(x, y, deg=1):
    fig = plt.figure()
    fig.set_size_inches(10/2.54, 10/2.54)
    fig.set_dpi(100)
    ax = fig.add_subplot(1, 1, 1)
    ax.set_aspect(1.)
    xmin = -5
    xmax = 5
    ax.set_xlim(xmin, xmax)
    ax.set_ylim(xmin, xmax)
    ax.grid()
    ax.axhline(0, color="black")
    ax.axvline(0, color="black")

    ax.plot(x, y, "ro")

    w = leastSquares(x, y, deg)

    n = 100; dx = (xmax - xmin)/n
    xx = [ xmin + i*dx for i in range(n+1) ]
    yy = np.polyval(w, xx)
    ax.plot(xx, yy)
    plt.show()