# 3: y = 1./(1+25*x.^2).
# 4: y = sqrt(abs(x))
## Functions to be approximated
funcs = [
lambda x: 1 + x + 2 * x**2 - 3 * x**3, lambda x: np.exp(-x), lambda x: 1 /
(1 + 25 * x**2), lambda x: np.sqrt(np.abs(x))
]
# Set degree of approximation and endpoints of approximation interval
n = 7 # degree of approximation
a = -1 # left endpoint
b = 1 # right endpoint
# Construct uniform grid for error ploting
x = nodeunif(2001, a, b)
def subfig(k, x, y, xlim, ylim, title):
plt.subplot(2, 2, k)
plt.plot(x, y)
plt.xlim(xlim)
plt.ylim(ylim)
plt.title(title)
for ifunc, ff in enumerate(funcs):
# Construct interpolants
C = BasisChebyshev(n, a, b, f=ff)
S = BasisSpline(n, a, b, f=ff)
y1 = 1 + 3 * x
y2 = 1 + 3 * x + 8 * x**2
plt.figure(figsize=[6, 6])
plt.plot(x, y, 'k', linewidth=3, label='Function')
plt.plot(x, y1, 'b', linewidth=3, label='1st order approximation')
plt.plot(x, y2, 'r', linewidth=3, label='2nd order approximation')
plt.legend()
plt.xticks([-1, 0, 1])
plt.show()
## Bivariate Taylor approximation
nplot = [101, 101]
a = [0, -1]
b = [2, 1]
x1, x2 = nodeunif(nplot, a, b)
x1.shape = nplot
x2.shape = nplot
y = np.exp(x2) * x1**2
y1 = 2 * x1 - x2 - 1
y2 = x1**2 - 2 * x1 * x2 + 0.5 * x2**2 + x2
def newPlot(title, Y, k):
ax = fig.add_subplot(1, 3, k, projection='3d')
ax.plot_surface(x1,
x2,
Y,
rstride=1,
cstride=1,
ax2 = fig.add_subplot(122, title='Derivative approximation error', **axopts)
ax2.plot(x, np.zeros_like(x), '--', color='gray', linewidth=2)
ax2.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax2.plot(x, f1fit(x, 1) - d1(x))
''' Bivariate Interpolation '''
# Define function
f2 = lambda x: np.cos(x[0]) / np.exp(x[1])
# Set degree and domain interpolation
n, a, b = 7, 0.0, 1.0
f2fit = BasisChebyshev([n, n], a, b, f=f2)
# Nice plot of function approximation error
nplot = [101, 101]
x = nodeunif(nplot, a, b)
x1, x2 = x
error = f2fit(x) - f2(x)
error.shape = nplot
x1.shape = nplot
x2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.gca(projection='3d',
title='Chebyshev Approximation Error',
xlabel='$x_1$',
ylabel='$x_2$',
zlabel='Error')
ax.plot_surface(x1,
x2,
error,
'The exact and approximate second partial derivatives of f w.r.t. x1 at x=[0 0] is'
)
print('{:4.0f} {:20.15f}\n'.format(0, d11))
print(
'The exact and approximate second partial derivatives of f w.r.t. x2 at x=[0 0] is'
)
print('{:4.0f} {:20.15f}\n'.format(0, d22))
print(
'The exact and approximate second cross partial derivatives of f at x=[0 0] is'
)
print('{:4.0f} {:20.15f}\n'.format(-1, d12))
# One may evaluate the accuracy of the Chebychev polynomial approximant by computing the approximation error on a
# highly refined grid of points:
nplot = [101, 101] # chose grid discretization
X = nodeunif(nplot, [a, a], [b, b]) # generate refined grid for plotting
yapp = F(X) # approximant values at grid nodes
yact = f(X) # actual function values at grid points
error = (yapp - yact).reshape(nplot)
X1, X2 = X
X1.shape = nplot
X2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.add_subplot(1, 2, 1, projection='3d')
ax.plot_surface(X1,
X2,
error,
rstride=1,
cstride=1,
cmap=cm.coolwarm,
y1 = 1 + 3 * x
y2 = 1 + 3 * x + 8 * x ** 2
plt.figure(figsize=[6, 6])
plt.plot(x, y, 'k', linewidth=3, label='Function')
plt.plot(x, y1, 'b', linewidth=3, label='1st order approximation')
plt.plot(x, y2, 'r', linewidth=3, label='2nd order approximation')
plt.legend()
plt.xticks([-1, 0, 1])
plt.show()
## Bivariate Taylor approximation
nplot = [101, 101]
a = [0, -1]
b = [2, 1]
x1, x2 = nodeunif(nplot, a, b)
x1.shape = nplot
x2.shape = nplot
y = np.exp(x2) * x1 ** 2
y1 = 2 * x1 - x2 - 1
y2 = x1 ** 2 - 2 * x1 * x2 + 0.5 * x2 ** 2 + x2
def newPlot(title, Y, k):
ax = fig.add_subplot(1, 3, k, projection='3d')
ax.plot_surface(x1, x2, Y, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$y$')
plt.title(title)
ax2.plot(x, np.zeros_like(x), '--', color='gray', linewidth=2)
ax2.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax2.plot(x, f1fit(x, 1) - d1(x))
''' Bivariate Interpolation '''
# Define function
f2 = lambda x: np.cos(x[0]) / np.exp(x[1])
# Set degree and domain interpolation
n, a, b = 7, 0.0, 1.0
f2fit = BasisChebyshev([n, n], a, b, f=f2)
# Nice plot of function approximation error
nplot = [101, 101]
x = nodeunif(nplot, a, b)
x1, x2 = x
error = f2fit(x) - f2(x)
error.shape = nplot
x1.shape = nplot
x2.shape = nplot
fig = plt.figure(figsize=[15, 6])
ax = fig.gca(projection='3d', title='Chebyshev Approximation Error',
xlabel='$x_1$', ylabel='$x_2$', zlabel='Error')
ax.plot_surface(x1, x2, error, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
plt.show()
# Compute partial Derivatives
x = np.array([[0.5], [0.5]])
# Function to be approximated
def f(x):
g = np.zeros((3, x.size))
g[0], g[1], g[2] = np.exp(-x), -np.exp(-x), np.exp(-x)
return g
# Set degree of approximation and endpoints of approximation interval
a = -1 # left endpoint
b = 1 # right endpoint
n = 10 # order of interpolatioin
# Construct refined uniform grid for error ploting
x = nodeunif(1001, a, b)
# Compute actual and fitted values on grid
y, d, s = f(x) # actual
# Construct and evaluate Chebychev interpolant
C = BasisChebyshev(n, a, b, f=f) # chose basis functions
yc = C(x) # values
dc = C(x, 1) # first derivative
sc = C(x, 2) # second derivative
# Construct and evaluate cubic spline interpolant
S = BasisSpline(n, a, b, f=f) # chose basis functions
ys = S(x) # values
ds = S(x, 1) # first derivative
ss = S(x, 2) # second derivative
# Function to be approximated
def f(x):
g = np.zeros((3, x.size))
g[0], g[1], g[2] = np.exp(-x), -np.exp(-x), np.exp(-x)
return g
# Set degree of approximation and endpoints of approximation interval
a = -1 # left endpoint
b = 1 # right endpoint
n = 10 # order of interpolatioin
# Construct refined uniform grid for error ploting
x = nodeunif(1001, a, b)
# Compute actual and fitted values on grid
y, d, s = f(x) # actual
# Construct and evaluate Chebychev interpolant
C = BasisChebyshev(n, a, b, f=f) # chose basis functions
yc = C(x) # values
dc = C(x, 1) # first derivative
sc = C(x, 2) # second derivative
# Construct and evaluate cubic spline interpolant
S = BasisSpline(n, a, b, f=f) # chose basis functions
ys = S(x) # values
ds = S(x, 1) # first derivative
# 3: y = 1./(1+25*x.^2).
# 4: y = sqrt(abs(x))
## Functions to be approximated
funcs = [lambda x: 1 + x + 2 * x ** 2 - 3 * x ** 3,
lambda x: np.exp(-x),
lambda x: 1 / ( 1 + 25 * x ** 2),
lambda x: np.sqrt(np.abs(x))]
# Set degree of approximation and endpoints of approximation interval
n = 7 # degree of approximation
a = -1 # left endpoint
b = 1 # right endpoint
# Construct uniform grid for error ploting
x = nodeunif(2001, a, b)
def subfig(k, x, y, xlim, ylim, title):
plt.subplot(2, 2, k)
plt.plot(x, y)
plt.xlim(xlim)
plt.ylim(ylim)
plt.title(title)
for ifunc, ff in enumerate(funcs):
# Construct interpolants
C = BasisChebyshev(n, a, b, f=ff)
S = BasisSpline(n, a, b, f=ff)
L = BasisSpline(n, a, b, k=1, f=ff)