def naive(f, s, Omega, N=10):
psi = []
for i in range(N + 1):
psi.append(sym.sin((2 * i + 1) * x))
u, c = least_squares_orth(f, psi, Omega, symbolic=False)
comparison_plot(f,
u,
Omega,
'tmp_sin%02dx' % i,
legend_loc='upper left',
show=True)
def efficient(f, s, Omega, N=10):
u = 0
for i in range(N + 1):
psi = [sym.sin((2 * i + 1) * x)]
next_term, c = least_squares_orth(f, psi, Omega, False)
u = u + next_term
comparison_plot(f,
u,
Omega,
'tmp_sin%02dx' % i,
legend_loc='upper left',
show=False,
plot_title='s=%g, i=%d' % (s, i))
def efficient(f, B, s, Omega, N=10, basis='a'):
u = B
for i in range(N+1):
if basis == 'a':
psi = [sym.sin((i+1)*x)]
elif basis == 'b':
psi = [sym.sin((2*i+1)*x)]
elif basis == 'c':
psi = [sym.sin(2*(i+1)*x)]
next_term, c = least_squares_orth(f-B, psi, Omega, False)
u = u + next_term
# Make only plot for i even
if i % 2 == 0:
comparison_plot(f, u, Omega, 'tmp_sin%02dx' % i,
legend_loc='upper left', show=False,
plot_title='s=%g, i=%d' % (s, i))
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])
cmd += " tanh_Lagrange_%s_s%s" % (distribution, s)
os.system(cmd)
for k in range(2, 6):
if symbolic:
Omega = [0, k*sym.pi] if domain_no == 1 else \
[-k*sym.pi/2, k*sym.pi/2]
else:
# cannot use sym.pi with numerical sympy computing
Omega = [0, k*pi] if domain_no == 1 else \
[-k*pi/2, k*pi/2]
u, c = least_squares(f, psi, Omega, symbolic=symbolic)
comparison_plot(f,
u,
Omega,
ymin=-2,
ymax=2,
filename='tmp_N%d_V%dOmega%dk%d' %
(N, V, k, domain_no),
plot_title='sin(x) on [0,%d*pi/2] by %s' %
(k, ','.join([str(p) for p in psi])))
# Need to kill the plot to proceed!
for ext in 'png', 'pdf':
cmd = 'doconce combine_images -2 ' + \
' '.join(['tmp_N%d_V%dOmega%dk%d.' %
(N, V, k, domain_no) + ext
for k in range(2, 6)]) + \
' sin_powers_N%d_V%d_Omega%d.' % (N, V, domain_no) + ext
print(cmd)
os.system(cmd)
# Show the standard Taylor series approximation
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])
cmd += ' tanh_Lagrange_%s_s%s' % (distribution, s)
os.system(cmd)
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src-approx'))
from approx1D import least_squares, comparison_plot
import matplotlib.pyplot as plt
import sympy as sym
from math import factorial
import numpy as np
x = sym.Symbol('x')
f = sym.exp(-x)
Omega = [0, 8]
for N in 2,4,6:
psi = [x**i for i in range(N+1)]
u, c = least_squares(f,psi,Omega)
print N, u
plt.figure()
comparison_plot(f, u, Omega, filename='tmp_exp_%d' % N,
plot_title='N=%d' % N)
for domain_no in range(1, 3):
for k in range(2, 6):
if symbolic:
Omega = [0, k*sym.pi] if domain_no == 1 else \
[-k*sym.pi/2, k*sym.pi/2]
else:
# cannot use sym.pi with numerical sympy computing
Omega = [0, k*pi] if domain_no == 1 else \
[-k*pi/2, k*pi/2]
u, c = least_squares(f, psi, Omega, symbolic=symbolic)
comparison_plot(
f, u, Omega,
ymin=-2, ymax=2,
filename='tmp_N%d_V%dOmega%dk%d' %
(N, V, k, domain_no),
plot_title='sin(x) on [0,%d*pi/2] by %s' %
(k, ','.join([str(p) for p in psi])))
# Need to kill the plot to proceed!
for ext in 'png', 'pdf':
cmd = 'doconce combine_images -2 ' + \
' '.join(['tmp_N%d_V%dOmega%dk%d.' %
(N, V, k, domain_no) + ext
for k in range(2, 6)]) + \
' sin_powers_N%d_V%d_Omega%d.' % (N, V, domain_no) + ext
print cmd
os.system(cmd)
# Show the standard Taylor series approximation
from math import factorial, pi
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' %
(s, N, distribution),
points=points,
point_values=point_values,
points_legend='%s points' % (2 * N))
# 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])
cmd += ' tanh_Lagrange_regr_%s_s%s' % (distribution, s)
os.system(cmd)
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' %
(s, N, distribution),
points=points, point_values=point_values,
points_legend='%s points' % (2*N))
# 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])
cmd += ' tanh_Lagrange_regr_%s_s%s' % (distribution, s)
os.system(cmd)