# Assemble matrices: O(n^2) complexity because of feval.
start = time.time()
D1 = diffmat(n, 1, [a, b])
D2 = diffmat(n, 2, [a, b])
S0 = spconvert(n, 0)
S1 = spconvert(n, 1)
M0 = multmat(n, a0, [a, b], 0)
M1 = multmat(n, a1, [a, b], 1)
M2 = multmat(n, a2, [a, b], 2)
L = M2 @ D2 + S1 @ M1 @ D1 + S1 @ S0 @ M0
L = lil_matrix(L)
for k in range(n):
T = np.zeros(n)
T[k] = 1
L[-2, k] = feval(T, 2 / (b - a) * x0 - (a + b) / (b - a))
L[-1, k] = feval(T, 2 / (b - a) * x1 - (a + b) / (b - a))
L = csr_matrix(L)
plt.figure()
plt.spy(L)
# Assemble RHS:
F = vals2coeffs(f(x))
F = S1 @ S0 @ F
F[-2] = c
F[-1] = d
F = csr_matrix(np.round(F, 13)).T
end = time.time()
print(f'Time (setup): {end-start:.5f}s')
# Sparse solve:
M0 = multmat(n, lambda r: r, [ra, rb], 0)
M2 = multmat(n, lambda r: r, [ra, rb], 2)
A1 = S1 @ S0
C1 = M2 @ D2r - S1 @ D1r
A2 = D2z
C2 = S1 @ S0 @ M0
# Assemble boundary conditions:
Bx = np.zeros([2, n])
By = np.zeros([2, n])
G = np.zeros([2, n])
H = np.zeros([2, n])
for k in range(n):
T = np.zeros(n)
T[k] = 1
Bx[0, k] = feval(T, 2 / (zb - za) * z0 - (za + zb) / (zb - za))
By[0, k] = feval(T, 2 / (rb - ra) * r0 - (ra + rb) / (rb - ra))
Bx[1, k] = feval(T, 2 / (zb - za) * z1 - (za + zb) / (zb - za))
By[1, k] = feval(T, 2 / (rb - ra) * r1 - (ra + rb) / (rb - ra))
G[0, :] = vals2coeffs(g1(z))
G[1, :] = vals2coeffs(g2(z))
H[0, :] = vals2coeffs(h1(r))
H[1, :] = vals2coeffs(h2(r))
Bx_hat = Bx[0:2, 0:2]
Bx = np.linalg.inv(Bx_hat) @ Bx
G = np.linalg.inv(Bx_hat) @ G
By_hat = By[0:2, 0:2]
By = np.linalg.inv(By_hat) @ By
H = np.linalg.inv(By_hat) @ H
# Assemble right-hand side:
S0 = spconvert(n, 0)
S1 = spconvert(n, 1)
A1 = S1 @ S0
C1 = diffmat(n, 2) + K**2 * S1 @ S0
A2 = diffmat(n, 2)
C2 = S1 @ S0
# Assemble boundary conditions:
Bx = np.zeros([2, n])
By = np.zeros([2, n])
G = np.zeros([2, n])
H = np.zeros([2, n])
for k in range(n):
T = np.zeros(n)
T[k] = 1
Bx[0, k] = feval(T, -1)
By[0, k] = feval(T, -1)
Bx[1, k] = feval(T, 1)
By[1, k] = feval(T, 1)
G[0, :] = vals2coeffs(g1(y))
G[1, :] = vals2coeffs(g2(y))
H[0, :] = vals2coeffs(h1(x))
H[1, :] = vals2coeffs(h2(x))
Bx_hat = Bx[0:2, 0:2]
Bx = np.linalg.inv(Bx_hat) @ Bx
G = np.linalg.inv(Bx_hat) @ G
By_hat = By[0:2, 0:2]
By = np.linalg.inv(By_hat) @ By
H = np.linalg.inv(By_hat) @ H
# Assemble right-hand side:
# Exact solution:
uex = ai
# Assemble matrices: O(n^2) complexity because of feval.
start = time.time()
D2 = diffmat(n, 2, [a, b])
S0 = spconvert(n, 0)
S1 = spconvert(n, 1)
M = multmat(n, lambda x: -x, [a, b])
L = eps * D2 + S1 @ S0 @ M
L = lil_matrix(L)
for k in range(n):
T = np.zeros(n)
T[k] = 1
L[-2, k] = feval(T, 2 / (b - a) * a - (a + b) / (b - a))
L[-1, k] = feval(T, 2 / (b - a) * b - (a + b) / (b - a))
L = csr_matrix(L)
plt.figure()
plt.spy(L)
# Assemble RHS:
F = vals2coeffs(f(x))
F = S1 @ S0 @ F
F[-2] = c
F[-1] = d
end = time.time()
print(f'Time (setup): {end-start:.5f}s')
# Sparse solve:
start = time.time()
# -*- coding: utf-8 -*-
"""
Created on Fri Dec 4 16:56:16 2020
Copyright 2020 by Hadrien Montanelli.
"""
# %% Imports.
# Standard library imports:
import numpy as np
# Chebpy imports:
from chebpy.cheb import chebpts, feval, vals2coeffs
# %% Evaluate f(x) = cos(x)*exp(-x^2).
# Function:
f = lambda x: np.cos(x) * np.exp(-x**2)
# Chebyshev grid:
n = 30
x = chebpts(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}')