def test_cheby_shape(self):
n, a, b = 7, 0, 2
basis = BasisChebyshev(n, a, b)
assert_equal(basis.nodes.size, n)
assert_equal(basis.Phi().shape, (n, n))
assert_equal(basis._diff(0, 2).shape, (n - 2, n))
assert_equal(basis._diff(0, -3).shape, (n + 3, n))
assert_equal(basis().shape, (n, ))
nx = 24
x = np.linspace(a, b, nx)
assert_equal(basis.Phi(x).shape, (nx, n))
assert_equal(basis.Phi(x, 1).shape, (nx, n))
assert_equal(basis.Phi(x, -1).shape, (nx, n))
assert_equal(basis(x).shape, (nx, ))
m = 1001 x = np.linspace(a, b, m) # ### % Plot monomial basis functions # In[6]: Phi = np.array([x**j for j in np.arange(n)]) basisplot(x, Phi, 'Monomial Basis Functions on [0,1]') # ### % Plot Chebychev basis functions and nodes # In[7]: B = BasisChebyshev(n, a, b) basisplot(x, B.Phi(x).T, 'Chebychev Polynomial Basis Functions on [0,1]') nodeplot(B.nodes, 'Chebychev Nodes on [0,1]') # ### % Plot linear spline basis functions and nodes # In[8]: L = BasisSpline(n, a, b, k=1) basisplot(x, L.Phi(x).T.toarray(), 'Linear Spline Basis Functions on [0,1]') nodeplot(L.nodes, 'Linear Spline Standard Nodes on [0,1]') # ### % Plot cubic spline basis functions and nodes # In[9]: C = BasisSpline(n, a, b, k=3)
# Set the points of approximation interval:
a = -1 # left points
b = 1 # right points
# Choose an approximation scheme. In this case, let us use an 11 by 11 Chebychev approximation scheme:
n = 11 # order of approximation
basis = BasisChebyshev(
[n, n], a,
b) # write n twice to indicate the two dimensions. a and b are expanded
# Compute the basis coefficients c. There are various way to do this:
# One may compute the standard approximation nodes x and corresponding interpolation matrix Phi and function values y
# and use:
x = basis.nodes
Phi = basis.Phi(x) # input x may be omitted if evaluating at the basis nodes
y = f(x)
c = np.linalg.solve(Phi, y)
print('Interpolation coeff =\n ', c)
# Alternatively, one may compute the standard approximation nodes x and corresponding function values y and use these
# values to create an BasisChebyshev object with keyword argument y:
x = basis.nodes
y = f(x)
fa = BasisChebyshev([n, n], a, b, y=y)
print('Interpolation coeff =\n ', fa.c) # attribute c returns the coefficients
# ... or one may simply pass the function directly to BasisChebyshev using keyword 'f', which by default
# will evaluate it at the basis nodes
F = BasisChebyshev([n, n], a, b, f=f)
print('Interpolation coeff =\n ', F.c)
for i in range(nn):
# Uniform-node monomial-basis approximant
xnodes = np.linspace(a, b, n[i])
c = np.polyfit(xnodes, runge(xnodes), n[i])
yfit = np.polyval(c, x)
phi = xnodes.reshape(-1, 1)**np.arange(n[i])
errunif[i] = yfit - y
nrmunif[i] = np.log10(norm(yfit - y, np.inf))
conunif[i] = np.log10(cond(phi, 2))
# Chebychev-node Chebychev-basis approximant
yapprox = BasisChebyshev(n[i], a, b, f=runge)
yfit = yapprox(
x) # [0] no longer needed? # index zero is to eliminate one dimension
phi = yapprox.Phi()
errcheb[i] = yfit - y
nrmcheb[i] = np.log10(norm(yfit - y, np.inf))
concheb[i] = np.log10(cond(phi, 2))
# Plot Chebychev- and uniform node polynomial approximation errors
ax2 = fig1.add_subplot(
212,
title="Runge's Function $11^{th}$-Degree\nPolynomial Approximation Error.",
xlabel='x',
ylabel='Error')
ax2.axhline(color='gray', linestyle='--')
ax2.plot(x, errcheb[4], label='Chebychev Nodes')
ax2.plot(x, errunif[4], label='Uniform Nodes')
ax2.legend(loc='upper center', frameon=False)