DEMAPP06¶
Chebychev and cubic spline derivative approximation errors¶
In [1]:
from demos.setup import np, plt, demo
from compecon import BasisChebyshev, BasisSpline, nodeunif
%matplotlib inline
Function to be approximated¶
In [2]:
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
In [3]:
a = -1 # left endpoint
b = 1 # right endpoint
n = 10 # order of interpolatioin
Construct refined uniform grid for error ploting
In [4]:
x = nodeunif(1001, a, b)
Compute actual and fitted values on grid
In [5]:
y, d, s = f(x) # actual
Construct and evaluate Chebychev interpolant¶
In [6]:
C = BasisChebyshev(n, a, b, f=f) # chose basis functions
yc, dc, sc = C(x, [[0, 1, 2]]) # values, first derivative, second derivative
Construct and evaluate cubic spline interpolant¶
In [7]:
S = BasisSpline(n, a, b, f=f) # chose basis functions
ys, ds, ss = S(x, [[0, 1, 2]]) # values, first derivative, second derivative
Plot function approximation error¶
In [8]:
plt.figure()
demo.subplot(2, 1, 1, 'Function Approximation Error', '', 'Chebychev')
plt.plot(x, y - yc[0])
demo.subplot(2, 1, 2, '', 'x', 'Cubic Spline')
plt.plot(x, y - ys[0])
Out[8]:
Plot first derivative approximation error¶
In [9]:
plt.figure()
demo.subplot(2, 1, 1, 'First Derivative Approximation Error', '','Chebychev'),
plt.plot(x, d - dc[0])
demo.subplot(2, 1, 2, '', 'x', 'Cubic Spline')
plt.plot(x, d - ds[0], 'm')
Out[9]:
Plot second derivative approximation error¶
In [10]:
plt.figure()
demo.subplot(2, 1, 1, 'Second Derivative Approximation Error', '', 'Chebychev')
plt.plot(x, s - sc[0])
demo.subplot(2, 1, 2, '', 'x', 'Cubic Spline')
plt.plot(x, s - ss[0], 'm')
Out[10]:
