def p1_7(part=None):
"""
Complete solution to problem 1.7.
Parameters
----------
part: str, optional(default=None)
The part number you would like evaluated. If this is left blank
the default value of None is used and the entire problem will be
solved.
Returns
-------
i-dont-know-yet
"""
x, h = make_grid(0, 5, 1000, 'cell_edge') # cell-edge data
f = np.cos(x)
y = np.cos(x) + .001 * np.random.rand(x.size)
yp = np.zeros(y.shape) # Pre-allocate memory for yp and ypp
ypp = np.zeros(y.shape)
# Get centered estimate for yp(p)
yp = centered_difference(y, 1, h, 'linear')
ypp = centered_difference(y, 2, h, 'linear')
# Get real derivatives
fp = -np.sin(x)
fpp = -np.cos(x)
# Plot f, y, fp, yp
plt.figure()
plt.subplot(211)
plt.plot(x, f, 'b', x, y, 'g')
plt.legend(('Real', 'Noisy'), loc=0)
plt.title('f(x)')
plt.subplot(212)
plt.plot(x, fp, 'k', x, yp, 'r')
plt.legend(('Real', 'Estimated'), loc=0)
plt.title('fp(x)')
plt.draw()
pass # return nothing
def p1_4(part=None):
"""
Complete solution to problem 1.4
Parameters
----------
part: str, optional(default=None)
The part number you would like evaluated. If this is left blank
the default value of None is used and the entire problem will be
solved.
Returns
-------
i-dont-know-yet
TODO: Call centered difference formula for derivatives.
"""
x, h = make_grid(0, 5, 100)
f = lambda x: jn(0, x) # define function
y = f(x) # Evaluate function on grid
# Get numerical derivatives using routine from tools.py
yplin = centered_difference(y, 1, h, 'linear')
ypplin = centered_difference(y, 2, h, 'linear')
ypquad = centered_difference(y, 1, h, 'quadratic')
yppquad = centered_difference(y, 2, h, 'quadratic')
# Calculate real derivatives
fp = - jn(1, x)
fpp = 1 / 2 * (- jn(0, x) + jn(2, x))
abs_err_linp = np.mean(np.abs(yplin - fp)) # linear 1st der. error
abs_err_linpp = np.mean(np.abs(ypplin - fpp)) # linear 2nd der. error
abs_err_quadp = np.mean(np.abs(ypquad - fp)) # quad 1st der. error
abs_err_quadpp = np.mean(np.abs(yppquad - fpp)) # quad 2nd der. error
print 'Linear first derivative error: %.8e' % (abs_err_linp)
print 'Linear second derivative error: %.8e' % (abs_err_linpp)
print 'Quadratic first derivative error: %.8e' % (abs_err_quadp)
print 'Quadratic second derivative error: %.8e' % (abs_err_quadpp)
plt.figure()
plt.subplot(211)
plt.plot(x, fp, x, yplin, x, ypquad)
plt.title('First derivatives')
plt.legend(('Real Derivative',
'Linear Extrapolation',
'quadratic Extrapolation'),
loc=0)
plt.subplot(212)
plt.plot(x, fpp, x, ypplin, x, yppquad)
plt.title('Second derivatives')
plt.legend(('Real Derivative',
'Linear Extrapolation',
'quadratic Extrapolation'),
loc=0)
plt.draw()
pass # return nothing
