def phi_factory(name, N, highest_derivative=2):
"""
Generate N+1 basis functions (phi) on [0,1] which vanish
on the boundary. Differentiate the functions up to
(and including) highest_derivative.
"""
x = sym.Symbol('x')
from sympy import sin, cos, pi
if name == 'sines':
phi = {0: [sin(pi*(i+1)*x) for i in range(N+1)]}
elif name == 'poly':
phi = {0: [x**(i+1)*(1-x) for i in range(N+1)]}
elif name == 'poly2':
phi = {0: [x**(i+2)*(1-x) for i in range(N+1)]}
elif name == 'poly3':
phi = {0: [(1-x)**(i+1) for i in range(N+1)]}
elif name == 'Lagrange':
from Lagrange import Lagrange_polynomials
# Return all Lagrange polynomials and strip off
# boundary polynomials in application code (if necessary)
phi = {0: Lagrange_polynomials(x, N, [0,1], 'uniform')[0]}
elif name == 'Lagrange_Cheb':
from Lagrange import Lagrange_polynomials
phi = {0: Lagrange_polynomials(x, N, [0,1], 'Chebyshev')}
# Compute derivatives of the basis functions
for d in range(1, highest_derivative+1):
phi[d] = [sym.diff(phi[0][i], x, d) for i in range(len(phi[0]))]
return phi
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src-approx'))
from approx1D import interpolation, comparison_plot
from Lagrange import Lagrange_polynomials
import sympy as sym
x = sym.Symbol('x')
Omega = [0, 1]
N_values = 3, 7, 11, 15
for s in 5, 20:
f = -sym.tanh(s * (x - 0.5)) # sympy expression
for distribution in 'uniform', 'Chebyshev':
for N in N_values:
phi, points = Lagrange_polynomials(x,
N,
Omega,
point_distribution=distribution)
u, c = interpolation(f, phi, points)
filename = 'tmp_tanh_%d_%d_%s' % (N, s, distribution)
comparison_plot(f,
u,
Omega,
filename,
plot_title='s=%g, N=%d, %s points' %
(s, N, distribution))
# Combine plot files (2x2)
for ext in 'png', 'pdf':
cmd = 'doconce combine_images ' + ext + ' '
cmd += ' '.join(
['tmp_tanh_%d_%d_%s' % (N, s, distribution) for N in N_values])
sys.path.insert(0, os.path.join(os.pardir, 'src'))
from approx1D import regression, comparison_plot
from Lagrange import Lagrange_polynomials
import sympy as sym
import numpy as np
x = sym.Symbol('x')
Omega = [0, 1]
N_values = 3, 7, 11, 15
for s in 5, 20:
f = -sym.tanh(s * (x - 0.5)) # sympy expression
for distribution in 'uniform', 'Chebyshev':
for N in N_values:
# Compute the points from a 2*N Lagrange polynomial
dummy, points = Lagrange_polynomials(
x, 2 * N, Omega, point_distribution=distribution)
# Compute phi from N points Lagrange polynomial
phi, dummy = Lagrange_polynomials(x,
N,
Omega,
point_distribution=distribution)
points = np.array(points, dtype=float)
point_values = -np.tanh(s * (points - 0.5))
u, c = regression(f, phi, points)
filename = 'tmp_tanh_%d_%d_%s' % (N, s, distribution)
comparison_plot(f,
u,
Omega,
filename,
plot_title='s=%g, N=%d, %s points' %