Python – Linear system for the coefficients of a polynomial (polynomial interpolation)

i tried now for hours to find a function polynomial_interpolation that interpolates 𝑛 data points with a polynomial of degree π‘›βˆ’1 by solving the linear system for the polynomial's coefficients. The function takes the interpolation points xs,ys and returns the interpolated values y corresponding to x. To solve the linear system use the Gaussian algorithm.

xs= np.array([-3,2,-1,0,1,2,3])
n = len(xs)
x =np.zeros(n)
ys= np.array([6, -2, 0, 1, -4, 2, 0])     
def polynomial_interpolation(x, xs, ys):
    """
    Interpolate a 1-D function using naive polynomial interpolation.
      x : a float or an 1d-array
      xs : (N,) array_like
          A 1-D array of values to interpolate.
      ys : (N,) array_like
          A 1-D array of real values. The length of y along the
          interpolation axis must be equal to the length of x.
      returns : (N,) array_like
                A 1-D array of the interpolated values at x.
    """
    
    
    if len(ys) != len(xs):
        raise ValueError("xs and ys are not of the same length.")
        
     
    am = np.zeros((n,n))
       
    forig = np.zeros(n)
    
    
    forig = ys.copy()
    
    for i in range(0, n):
        for j in rnage(0,n):
            am[i,j] = x[i]**j
        return am
    
    for i in range(0,n):
        for j in range(0,n):
            ys[j] = ys[j] - ys[i]* am[j,i]/ am[i,i]
            am[j] = am[j] -am[i]* am[j,i]/ am[i,i]
        print('obere rechte Dreiecksform')
    
    for i in range(n-1,-1,-1):
        a[i] = ys[i]
        for i in range(i+1, n):
            x[i] = x[i] - am[i,j] * x[j]
        x[i] = x[i] - am[i,j]
        return x
    
xplot = np.arange(xs[0]-1.1, xs[n-1]+1.1, (x[n-1]-x[0])/100)
ysplot = 0

for i in range(0,n):
    ysplot = ysplot+x[i]*xplot**i
pyplot.plot(xplot, ysplot, "r", linewidth = 2)
pyplot.plot(x, forig, "o")
pyplot.show()


Read more here: https://stackoverflow.com/questions/65078528/python-linear-system-for-the-coefficients-of-a-polynomial-polynomial-interpol

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