python如何正确完成拉格朗日插值

Question

I was asked to use Lagrange Interpolation to draw a line that pass through several dots on a graph (scipy.interpolate.lagrange banned).

But it doesn't seem that my code can give the correct result. To be frank, I don't even know which part of it is wrong.

And if I was asked to find the value of a specific point on the line, how should I adjust the code to do so?

The first image is the expected output. The second one is the output from my code.

Can someone please tell me how to correct it?

Source code

import numpy as np
import matplotlib.pyplot as plt

m = 4
X = np.array([0, 1, 1.5, 2, 3])
Y = np.array([2, 8, 12, 10, 20])

def p(x): 
    px = 0;
    for j in range(m+1):
        Lmj = 1
        for k in range(m+1):
            if k != j:
                Lmj *= (x - X[k])/(X[j] - X[k])
            px += Y[j] * Lmj;
        return px

plt.scatter(X, Y)
Xinterp = np.linspace(min(X), max(X), 100)
plt.plot(Xinterp, p(Xinterp))
plt.show()

Also, is that plt.show() line necessary? From my lecturer's notes, the code don't have this line and still can display a graph. But in my copied version of the source code, I need to add this line to show the graph.

Thank you very much for noticing this problem.

Answer 1

1. Lagrange Interpolation produces a polynomial from a given set of points.
2. You can then use this polynomial to find the value of a specific point on the line.
3. plt.show() is not necessary if you are running the code inside a Jupyter notebook (or similar), but is necessary if running via the terminal.

Here is some python code that generates a numpy.polynomial from your set of points:

import numpy as np
from numpy.polynomial import Polynomial
import matplotlib.pyplot as plt

X = np.array([0, 1, 1.5, 2, 3])
Y = np.array([2, 8, 12, 10, 20])

n = len(X)
poly = Polynomial(np.zeros(n))

for j in range(n):
    k = [k for k in range(n) if k != j]
    roots = -1 * X[k]

    sub_poly = Polynomial.fromroots(X[k])
    scale = Y[j] / np.prod(X[j] - X[k])
    sub_poly.coef *= scale

    poly.coef += sub_poly.coef

You can then plot it via:

plt.scatter(X, Y)
Xinterp = np.linspace(min(X), max(X), 100)
plt.plot(Xinterp, poly(Xinterp))
plt.show()

Or to find a specific value:

print(poly(0.5))

Please see here for more information: https://learn.64bitdragon.com/articles/computer-science/numerical-analysis/lagrange-interpolation

Full disclosure: I'm the author of this article.