Orthogonal polynomials

Copyright (C) 2020 Andreas Kloeckner

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In [2]:
import numpy as np

import numpy.linalg as la

import matplotlib.pyplot as pt

Mini-Introduction to sympy

In [3]:
import sympy as sym



# Enable "pretty-printing" in IPython

sym.init_printing()

Make a new Symbol and work with it:

In [4]:
#clear

x = sym.Symbol("x")



myexpr = (x**2-3)**2

myexpr
Out[4]:
$$\left(x^{2} - 3\right)^{2}$$
In [5]:
myexpr = (x**2-3)**2

myexpr

myexpr.expand()
Out[5]:
$$x^{4} - 6 x^{2} + 9$$
In [6]:
sym.integrate(myexpr, x)
Out[6]:
$$\frac{x^{5}}{5} - 2 x^{3} + 9 x$$
In [7]:
sym.integrate(myexpr, (x, -1, 1))
Out[7]:
$$\frac{72}{5}$$

Orthogonal polynomials

Now write a function inner_product(f, g):

In [8]:
#clear

def inner_product(f, g):

    return sym.integrate(f*g, (x, -1, 1))

Show that it works:

In [9]:
#clear

inner_product(1, 1)
Out[9]:
$$2$$
In [10]:
#clear

inner_product(1, x)
Out[10]:
$$0$$

Next, define a basis consisting of a few monomials:

In [44]:
#clear

basis = [1, x, x**2, x**3]

#basis = [1, x, x**2, x**3, x**4, x**5]

And run Gram-Schmidt on it:

In [45]:
orth_basis = []



for q in basis:

    for prev_q in orth_basis:

        q = q - inner_product(prev_q, q)*prev_q / inner_product(prev_q,prev_q)

    orth_basis.append(q)



legendre_basis = [orth_basis[0],]



#to compute Legendre polynomials need to normalize so that q(1)=1 rather than ||q||=1

for q in orth_basis[1:]:

    q = q / q.subs(x,1)

    legendre_basis.append(q)
In [46]:
legendre_basis
Out[46]:
$$\left [ 1, \quad x, \quad \frac{3 x^{2}}{2} - \frac{1}{2}, \quad \frac{5 x^{3}}{2} - \frac{3 x}{2}, \quad \frac{35 x^{4}}{8} - \frac{15 x^{2}}{4} + \frac{3}{8}, \quad \frac{63 x^{5}}{8} - \frac{35 x^{3}}{4} + \frac{15 x}{8}\right ]$$

These are called the Legendre polynomials.

What do they look like?

In [47]:
mesh = np.linspace(-1, 1, 100)



pt.figure(figsize=(8,8))

for f in legendre_basis:

    f = sym.lambdify(x, f)

    pt.plot(mesh, [f(xi) for xi in mesh])

These functions are important enough to be included in scipy.special as eval_legendre:

In [48]:
import scipy.special as sps



for i in range(10):

    pt.plot(mesh, sps.eval_legendre(i, mesh))

What can we find out about the conditioning of the generalized Vandermonde matrix for Legendre polynomials?

In [49]:
#keep

n = 20

xs = np.linspace(-1, 1, n)

V = np.array([

    sps.eval_legendre(i, xs)

    for i in range(n)

]).T



la.cond(V)
Out[49]:
$$7274.598185486346$$

The Chebyshev basis can similarly be defined by Gram-Schmidt, but now with respect to a different inner-product weight function,

$$w(x) = 1/\sqrt{1-x^2}.$$
In [50]:
#clear

w = 1 / sym.sqrt(1-x**2)

def cheb_inner_product(f, g):

    return sym.integrate(w*f*g, (x, -1, 1))



orth_basis = []



for q in basis:

    for prev_q in orth_basis:

        q = q - cheb_inner_product(prev_q, q)*prev_q / cheb_inner_product(prev_q,prev_q)

    orth_basis.append(q)



cheb_basis = [1,]



#to compute Legendre polynomials need to normalize so that q(1)=1 rather than ||q||=1

for q in orth_basis[1:]:

    q = q / q.subs(x,1)

    cheb_basis.append(q)

cheb_basis
Out[50]:
$$\left [ 1, \quad x, \quad 2 x^{2} - 1, \quad 4 x^{3} - 3 x, \quad 8 x^{4} - 8 x^{2} + 1, \quad 16 x^{5} - 20 x^{3} + 5 x\right ]$$
In [51]:
for i in range(10):

    pt.plot(mesh, np.cos(i*np.arccos(mesh)))

Chebyshev polynomials achieve similar good, but imperfect conditioning on a uniform grid (but perfect conditioning on a grid of Chebyshev nodes).

In [52]:
#keep

n = 20

xs = np.linspace(-1, 1, n)

V = np.array([

    np.cos(i*np.arccos(xs))

    for i in range(n)

]).T



la.cond(V)
Out[52]:
$$4846.7105682362735$$
In [ ]: