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Numerical methods challenge: Day 12

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.

Hermite interpolation: Inverting Vandermonde matrix

Today we are comparing Lagrange interpolation with Hermite interpolation. For this example we are using the confluent Vandermonde matrix \(V\). The code is based on an old code of mine that is present in this repo.

As in the case of Lagrange interpolation we 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 is not stable and should not be used for the computation of higher order interpolants, even for optimally chosed sampling. It will start failing around 20 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 conf_vander_mat(x):
    n = len(x)
    conf_van = np.zeros((2*n, 2*n))
    power = np.array(range(2*n))
    for row in range(n):
        conf_van[row, :] = x[row]**power
        conf_van[row + n, :] = power*x[row]**(power - 1)
    return conf_van


def inter_coef(x, inter_type="lagrange"):
    if inter_type == "lagrange":
        vand_mat = vander_mat(x)
    elif inter_type == "hermite":
        vand_mat = conf_vander_mat(x)
    coef = np.linalg.solve(vand_mat, np.eye(vand_mat.shape[0]))
    return coef


def compute_interp(x, f, x_eval, df=None):
    n = len(x)
    if df is None:
        coef = inter_coef(x, inter_type="lagrange")
    else:
        coef = inter_coef(x, inter_type="hermite")
    f_eval = np.zeros_like(x_eval)
    nmat = coef.shape[0]
    for row in range(nmat):
        for col in range(nmat):
            if col < n or nmat == n:
                f_eval += coef[row, col]*x_eval**row*f[col]
            else:
                f_eval += coef[row, col]*x_eval**row*df[col - n]
    return f_eval




n = 7
x = -np.cos(np.linspace(0, np.pi, n))
f = lambda x: 1/(1 + 25*x**2)
df = lambda x: -50*x/(1 + 25*x**2)**2
x_eval = np.linspace(-1, 1, 500)
interp_f = compute_interp(x, f(x), x_eval, df=df(x))
plt.plot(x_eval, f(x_eval))
plt.plot(x_eval, interp_f)
plt.plot(x, f(x), ".")
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 conf_vander_mat(x)
    n = length(x)
    conf_van = zeros(2*n, 2*n)
    power = 0:2*n-1
    for row = 1:n
        conf_van[row, :] = x[row].^power
        conf_van[row + n, :] = power.*x[row].^(power - 1)
    end
    return conf_van
end


function inter_coef(x; inter_type="lagrange")
    if inter_type == "lagrange"
        vand_mat = vander_mat(x)
    elseif inter_type == "hermite"
        vand_mat = conf_vander_mat(x)
    end
    coef = vand_mat \ eye(size(vand_mat)[1])
    return coef
end


function compute_interp(x, f, x_eval; df=nothing)
    n = length(x)
    if df == nothing
        coef = inter_coef(x, inter_type="lagrange")
    else
        coef = inter_coef(x, inter_type="hermite")
    end
    f_eval = zeros(x_eval)
    nmat = size(coef)[1]
    for row = 1:nmat
        for col = 1:nmat
            if col <= n || nmat == n
                f_eval += coef[row, col]*x_eval.^(row - 1)*f[col]
            else
                f_eval += coef[row, col]*x_eval.^(row - 1)*df[col - n]
            end
        end
    end
    return f_eval
end


n = 7
x = -cos.(linspace(0, pi, n))
f = 1./(1 + 25*x.^2)
df = -50*x./(1 + 25*x.^2).^2
x_eval = linspace(-1, 1, 500)
interp_f = compute_interp(x, f, x_eval, df=df)
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.

Hermite interpolation using Vandermonde matrix.

And, if we try with a high \(n\), say \(n=43\), we can see the problems.

Hermite interpolation using Vandermonde matrix.

Comparison Python/Julia

Regarding number of lines we have: 61 in Python and 70 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 = 7
x = -np.cos(np.linspace(0, np.pi, n))
f = lambda x: 1/(1 + 25*x**2)
df = lambda x: -50*x/(1 + 25*x**2)**2
x_eval = np.linspace(-1, 1, 500)
interp_f = compute_interp(x, f(x), x_eval, df=df(x))

with result

100 loops, best of 3: 18.1 ms per loop

For Julia:

function bench()
   n = 7
   x = -cos.(linspace(0, pi, n))
   f(x) = 1./(1 + 25*x.^2)
   df(x) = -50*x./(1 + 25*x.^2).^2
   x_eval = linspace(-1, 1, 500)
   interp_f = compute_interp(x, f(x), x_eval, df=df(x))
end
@benchmark bench()

with result

BenchmarkTools.Trial:
  memory estimate:  3.13 MiB
  allocs estimate:  836
  --------------
  minimum time:     10.318 ms (0.00% GC)
  median time:      10.449 ms (0.00% GC)
  mean time:        11.362 ms (1.74% GC)
  maximum time:     26.646 ms (0.00% GC)
  --------------
  samples:          100
  evals/sample:     1

In this case, we can say that the Python code is roughly as fast as the Julia one.

Comparison Hermite/Lagrange interpolation

We want to compare Hermite interpolation with Lagrange interpolation for the same number of degrees of freedom. We use the same function for test purposes

\begin{equation*} f(x) = \frac{1}{1 + 25x^2} \end{equation*}

This is the Python code that does that part

n_dof = np.array(range(1, 20))
error_herm = np.zeros(19)
error_lag = np.zeros(19)
for cont, n in enumerate(n_dof):
    f = lambda x: 1/(1 + 25*x**2)
    df = lambda x: -50*x/(1 + 25*x**2)**2
    x = -np.cos(np.linspace(0, np.pi, n))
    x2 = -np.cos(np.linspace(0, np.pi, 2*n))
    x_eval = np.linspace(-1, 1, 500)
    herm = compute_interp(x, f(x), x_eval, df=df(x))
    lag = compute_interp(x2, f(x2), x_eval)
    fun = f(x_eval)
    error_herm[cont] = np.linalg.norm(fun - herm)/np.linalg.norm(fun)
    error_lag[cont] = np.linalg.norm(fun - lag)/np.linalg.norm(fun)

plt.plot(2*n_dof, error_lag)
plt.plot(2*n_dof, error_herm)
plt.xlabel("Number of degrees of freedom")
plt.ylabel("Relative error")
plt.legend(["Lagrange", "Hermite"])
plt.show()

And this is the comparison of the relative errors

Comparison of errors between Lagrange and Hermite interpolation.

In general, the Lagrange approximation is closer for this function.

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