# Standard library imports:
from math import pi
import numpy as np
# Chebpy imports:
from chebpy.trig import trigpts, quadwts
# %% Integrate f(x) = 1 + sin(4*pi*x) on [-1,1].
# Function:
f = lambda x: 1 + np.sin(4 * pi * x)
# Grid and weights:
n = 12
x = trigpts(n)
w = quadwts(n)
# Compute integral with quadrature:
I = w @ f(x)
# Compute error:
Iex = 2
error = np.abs(I - Iex)
print(f'Error: {error:.2e}')
# %% Integrate f(x) = 1 + sin(4*x) on [-pi,pi].
# Function:
f = lambda x: 1 + np.sin(4 * x)
# Standard library imports: from math import pi from matplotlib import cm import matplotlib.pyplot as plt import numpy as np # Chebpy imports: from chebpy.trig import trigpts from chebpy.sph import vals2coeffs, feval, mean, spharm # %% Test. # Grid points: n = 32 dom = [-pi, pi] lam = trigpts(n, dom) tt = trigpts(n, dom) LAM, TT = np.meshgrid(lam, tt) # Spherical harmonic: l = 4 # has to be >= 0 m = -3 # -l <= m <= +l Y = spharm(l, m) # Plot: plt.figure() plt.contourf(LAM, TT, feval(Y, LAM, TT), 40, cmap=cm.coolwarm) plt.colorbar() # Check orthonormality: F = vals2coeffs(feval(Y, LAM, TT) * feval(Y, LAM, TT))
# Standard library imports: from math import pi import matplotlib.pyplot as plt import numpy as np from scipy.sparse.linalg import spsolve import time # Chebpy imports: from chebpy.trig import coeffs2vals, trigpts, vals2coeffs from chebpy.trig import diffmat, multmat # %% Solve u''(x) + sin(x)*u'(x) + 1000*cos(2x)*u(x) = f on [0,2*pi]. # Grid: n = 10000 x = trigpts(n, [0, 2 * pi]) # Right-hand side f: f = lambda x: 100 + 0 * x # Assemble matrices: start = time.time() D1 = diffmat(n, 1, [0, 2 * pi]) D2 = diffmat(n, 2, [0, 2 * pi]) M0 = multmat(n, lambda x: 1000 * np.cos(2 * x), [0, 2 * pi]) M1 = multmat(n, lambda x: np.sin(x), [0, 2 * pi]) L = D2 + M1 @ D1 + M0 plt.figure() plt.spy(L) # Assemble RHS:
Copyright 2020 by Hadrien Montanelli.
"""
# %% Imports.
# Standard library import:
from math import pi
import numpy as np
# Chebpy imports:
from chebpy.trig import coeffs2vals, trigpts, vals2coeffs
# %% Transforms (1D) on [-1,1].
f = lambda x: np.cos(10000 * pi * x) + np.cos(5 * pi * x)
n = 100
x = trigpts(n)
error = coeffs2vals(vals2coeffs(f(x))) - f(x)
print(f'Error (1D): {np.max(np.abs(error)):.2e}')
# %% Transforms (1D) on [0,2*pi].
f = lambda x: np.exp(np.cos(10 * pi * x)**2)
n = 120
x = trigpts(n, [0, 2 * pi])
error = coeffs2vals(vals2coeffs(f(x))) - f(x)
print(f'Error (1D): {np.max(np.abs(error)):.2e}')
# %% Transforms (2D) on [-1,1]x[-1,1].
f = lambda x, y: np.exp(np.cos(10 * pi * x)**2) * np.sin(pi * y)**2
n = 100
"""
Created on Fri Dec 4 16:56:16 2020
Copyright 2020 by Hadrien Montanelli.
"""
# %% Imports.
# Standard library imports:
from math import pi
import numpy as np
# Chebpy imports:
from chebpy.trig import feval, trigpts, vals2coeffs
# %% Evaluate f(x) = cos(pi*x)*exp(-sin(pi*x)^2).
# Function:
f = lambda x: np.cos(pi*x)*np.exp(-np.sin(pi*x)**2)
# Equispaced grid:
n = 100
x = trigpts(n)
F = vals2coeffs(f(x))
# Evaluation grid:
xx = np.linspace(-1, 1, 100)
vals = feval(F, xx)
# Error:
error = np.max(np.abs(vals - f(xx)))/np.max(np.abs(f(xx)))
print(f'Error: {error:.2e}')
import time
# Chebpy imports:
from chebpy.trig import coeffs2vals, diffmat, trigpts, vals2coeffs
# %% First-order differentiation on [-pi, pi].
# Function:
w = 100
f = lambda x: np.cos(w * x)
dfex = lambda x: -w * np.sin(w * x)
# Grid:
n = 4 * w
dom = [-pi, pi]
x = trigpts(n, dom)
# Compute coeffs of f:
F = vals2coeffs(f(x))
# Differentiation matrix in coefficient space:
start = time.time()
D = diffmat(n, 1, dom)
end = time.time()
print(f'Time (setup): {end-start:.5f}s')
plt.figure()
plt.spy(D)
# Differentiate:
start = time.time()
DF = D @ F
