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Numerical methods challenge: Day 11

Nicolás Guarín-Zapata

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During October (2017) I will write a program per day for some well-known numerical methods in both Python and Julia. It is intended to be an exercise then don't expect the code to be good enough for real use. Also, I should mention that I have almost no experience with Julia, so it probably won't be idiomatic Julia but more Python-like Julia.

Lagrange interpolation: Inverting Vandermonde matrix

Today we have Lagrange interpolation, yet one more time. I will have a different approach to compute the interpolation; I will form the Vandermonde matrix \(V\) and solve the system

\begin{equation*} V\mathbf{c} = I \end{equation*}

where \(\mathbf{c}\) is the vector of coefficients and \(I\) is the identity matrix. This method, and the previous one are not stable and should not be used for the computation of higher order interpolants, even for optimally chosed sampling. It will start failing around 40 points. A better approach is to use the barycentric form of the interpolation.

In the example below we use Chebyshev nodes. The nodes are given by

\begin{equation*} x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k = 1, \ldots, n \end{equation*}

where \(n\) is the degree of the polynomial.

Following are the codes.

Python

from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as plt


def vander_mat(x):
    n = len(x)
    van = np.zeros((n, n))
    power = np.array(range(n))
    for row in range(n):
        van[row, :] = x[row]**power
    return van


def inter_coef(x):
    vand_mat = vander_mat(x)
    coef = np.linalg.solve(vand_mat, np.eye(len(x)))
    return coef


def compute_interp(x, f, x_eval):
    n = len(x)
    coef = inter_coef(x)
    f_eval = np.zeros_like(x_eval)
    for row in range(n):
        for col in range(n):
            f_eval += coef[row, col]*x_eval**row*f[col]
    return f_eval


n = 11
x = -np.cos(np.linspace(0, np.pi, n))
f = 1/(1 + 25*x**2)
x_eval = np.linspace(-1, 1, 500)
interp_f = compute_interp(x, f, x_eval)
plt.figure()
plt.plot(x_eval, 1/(1 + 25*x_eval**2))
plt.plot(x_eval, interp_f)
plt.plot(x, f, ".")
plt.ylim(0, 1.2)
plt.show()

Julia

using PyPlot


function vander_mat(x)
    n = length(x)
    van = zeros(n, n)
    power = 0:n-1
    for row = 1:n
        van[row, :] = x[row].^power
    end
    return van
end


function inter_coef(x)
    vand_mat = vander_mat(x)
    coef = vand_mat \ eye(length(x))
    return coef
end


function compute_interp(x, f, x_eval)
    n = length(x)
    coef = inter_coef(x)
    f_eval = zeros(x_eval)
    for row = 1:n
        for col = 1:n
            f_eval += coef[row, col]*x_eval.^(row - 1)*f[col]
        end
    end
    return f_eval
end


n = 11
x = - cos.(linspace(0, pi, n))
f = 1./(1 + 25*x.^2)
x_eval = linspace(-1, 1, 500)
interp_f = compute_interp(x, f, x_eval)
plot(x_eval, 1./(1 + 25*x_eval.^2))
plot(x_eval, interp_f)
plot(x, f, ".")
ylim(0, 1.2)
show()

In both cases the result is the plot below.

Lagrange interpolation using Vandermonde matrix.

And, if we try with a high \(n\), say \(n=45\), we can see the problems.

Lagrange interpolation using Vandermonde matrix.

Comparison Python/Julia

Regarding number of lines we have: 41 in Python and 44 in Julia. The comparison in execution time is done with %%timeit magic command in IPython and @benchmark in Julia.

For Python:

%%timeit -n 100
n = 11
x = -np.cos(np.linspace(0, np.pi, n))
f = 1/(1 + 25*x**2)
x_eval = np.linspace(-1, 1, 500)
interp_f = compute_interp(x, f, x_eval)

with result

100 loops, best of 3: 7.86 ms per loop

For Julia:

function bench()
   x = - cos.(linspace(0, pi, n))
   f = 1./(1 + 25*x.^2)
   x_eval = linspace(-1, 1, 500)
   interp_f = compute_interp(x, f, x_eval)
   return nothing
end
@benchmark bench()

with result

BenchmarkTools.Trial:
  memory estimate:  32.23 MiB
  allocs estimate:  8277
  --------------
  minimum time:     114.282 ms (1.50% GC)
  median time:      122.061 ms (1.46% GC)
  mean time:        129.733 ms (1.90% GC)
  maximum time:     163.716 ms (1.98% GC)
  --------------
  samples:          39
  evals/sample:     1

In this case, we can say that the Python code is roughly 16 times faster than the Julia one.

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