def smoothed_Heaviside(x, e=1E-2):
cond = operator.and_(-e <= x, x <= e)
r = np.zeros(len(x))
r[x < -e] = 0.0
r[cond] = 0.5 + x[cond] / (2 * e) + 1 / (2 * pi) * sin(pi * x[cond] / e)
r[x > e] = 1.0
return r
def _test1():
f = lambda x: x**5 - sin(x)
x0 = 1
x1 = 2
N = 100
epsilon = 1e-10
x, info = Secant(f, x0, x1,epsilon,N, True)
info = transpose(info)
x_values = info[0]
f_values = info[1]
figure()
plot(x_values,
xlabel='Iteration Number',
ylabel='x_i',
title='Secant method for f = x**5 - sin(x), '\
'Iteration number vs. x_i')
figure()
semilogy(f_values,
xlabel='Iteration Number',
ylabel='f(x_i)',
axes=[0, len(f_values), -1e-1,1e-1],
title='Secant method for f = x**5 - sin(x), '\
'Iteration number vs. f(x_i)')
raw_input('Press Enter to quit: ')
def _test1():
f = lambda x: x**5 - sin(x)
x0 = 1
x1 = 2
N = 100
epsilon = 1e-10
x, info = Secant(f, x0, x1, epsilon, N, True)
info = transpose(info)
x_values = info[0]
f_values = info[1]
figure()
plot(x_values,
xlabel='Iteration Number',
ylabel='x_i',
title='Secant method for f = x**5 - sin(x), '\
'Iteration number vs. x_i')
figure()
semilogy(f_values,
xlabel='Iteration Number',
ylabel='f(x_i)',
axes=[0, len(f_values), -1e-1,1e-1],
title='Secant method for f = x**5 - sin(x), '\
'Iteration number vs. f(x_i)')
raw_input('Press Enter to quit: ')
def smoothed_Heaviside(x, e=1E-2):
cond = operator.and_(-e <= x, x <= e)
r = np.zeros(len(x))
r[x < -e] = 0.0
r[cond] = 0.5 + x[cond] / (2 * e) + 1 / (2 * pi) * sin(pi * x[cond] / e)
r[x > e] = 1.0
return r
def _test2(): from scitools.std import sin f = lambda x: sin(x) x0 = 1 x1 = 2 x, info = Secant(f, x0, x1,1e-10,100, True) x_values = transpose(info)[0] f_values = transpose(info)[1] for i in range(len(x_values)): print '%10f %10f' % (x_values[i], f_values[i])
def _test2():
from scitools.std import sin
f = lambda x: sin(x)
x0 = 1
x1 = 2
x, info = Secant(f, x0, x1, 1e-10, 100, True)
x_values = transpose(info)[0]
f_values = transpose(info)[1]
for i in range(len(x_values)):
print '%10f %10f' % (x_values[i], f_values[i])
def T(z, t):
'''
Formula for temperature oscillations in the
ground.
A1 is the amplitude of annual variations, in deg C
A2 is the amplitude of the day/night variations in deg C
w1 = 2*pi*P1
w2 = 2*pi*P2
a1 = sqrt(w1/(2.*k))
a2 = sqrt(w2/(2.*k))
k is the heat conduction coefficient
T0 is the initial temperature
z is the depth in the ground (m)
t is the time in seconds
A1, A2, w1, w2, a1, a2, k, T0 are global variables
'''
from scitools.std import exp, sin
return T0 + A1*exp(-a1*z)*sin(w1*t-a1*z) \
+ A2*exp(-a2*z)*sin(w2*t-a2*z)
def T(z,t): ''' Formula for temperature oscillations in the ground. A1 is the amplitude of annual variations, in deg C A2 is the amplitude of the day/night variations in deg C w1 = 2*pi*P1 w2 = 2*pi*P2 a1 = sqrt(w1/(2.*k)) a2 = sqrt(w2/(2.*k)) k is the heat conduction coefficient T0 is the initial temperature z is the depth in the ground (m) t is the time in seconds A1, A2, w1, w2, a1, a2, k, T0 are global variables ''' from scitools.std import exp, sin return T0 + A1*exp(-a1*z)*sin(w1*t-a1*z) \ + A2*exp(-a2*z)*sin(w2*t-a2*z)
def pi_approx(N):
'''
Calculates the approximation to pi by computing
the path length of a polygon with N points inscribed
in the circle of radius 1
'''
from scitools.std import sin, pi
x = [1. / 2 * cos(2. * pi * i / (N)) for i in range(N + 1)]
y = [1. / 2 * sin(2. * pi * i / (N)) for i in range(N + 1)]
approx = pathlength(x, y)
return approx
def H(x,e): ''' A smooth, continuous version of the Heaviside function evaluated at x. e is epsilon, of which we transition from 0 to 1 from -e <= x <= e ''' from scitools.std import pi, sin # we must use there where() fn since x may be an array import operator condition = operator.and_(-e <= x, x <= e) i = where(x > e, 1, x) i = where(condition, 1./2 + x/(2.*e) + 1./(2*pi)*sin(pi*x/e), i) i = where(x < -e, 0, i) return i
def H(x, e):
'''
A smooth, continuous version of the Heaviside function
evaluated at x.
e is epsilon, of which we transition from 0 to 1 from
-e <= x <= e
'''
from scitools.std import pi, sin
# we must use there where() fn since x may be an array
import operator
condition = operator.and_(-e <= x, x <= e)
i = where(x > e, 1, x)
i = where(condition,
1. / 2 + x / (2. * e) + 1. / (2 * pi) * sin(pi * x / e), i)
i = where(x < -e, 0, i)
return i
return s
def test_p_L(xp, yp):
n = len(xp) - 1
eps = 10e-12
count = 0 #count how many answers are not close enough
for k in range(n + 1):
if abs(p_L(xp[k], xp, yp) - yp[k]) > eps:
count += 1
if count == 0:
print 'Success! The Lagrange interpolation polynomial indeed passes',\
'through all the points (xp,yp).'
else:
print 'Oh no! There were %d points where the interpolation polynomial'\
%(count), 'fails to pass through (xp,yp).'
def graph(f, n, xmin, xmax, resolution=1001):
Xp = linspace(xmin, xmax, n)
Yp = f(Xp)
Xr = linspace(xmin, xmax, resolution) #array of values to plot
Yr = array([p_L(x, Xp, Yp) for x in Xr]) #get interpolation points
plot(Xr, Yr, '-r', Xp, Yp, 'bo', title='Lagrange Interpolation')
if __name__ == '__main__':
XP = linspace(0, pi, 5)
YP = sin(XP)
test_p_L(XP, YP)
graph(sin, 5, 0, pi)
def points(N):
x = [0.5 * cos(2 * pi * i / N) for i in range(N + 1)]
y = [0.5 * sin(2 * pi * i / N) for i in range(N + 1)]
return x, y
def _dg(x):
return -2*0.1*x*exp(-0.1*x**2)*sin(pi/2*x) + \
pi/2*exp(-0.1*x**2)*cos(pi/2*x)
"""
Exercise 5.31: Animate a wave packet
Author: Weiyun Lu
"""
from scitools.std import exp, sin, pi, linspace, plot, movie
import time
import glob
import os
f = lambda x, t: exp(-(x - 3 * t)**2) * sin(3 * pi * (x - t))
fps = float(1 / 6)
xp = linspace(-6, 6, 1001)
tp = linspace(-1, 1, 61)
counter = 0
for name in glob.glob('pix/plot_wave*.png'):
os.remove(name)
for t in tp:
yp = f(xp, t)
plot(xp, yp, '-b', axis=[xp[0], xp[-1], -1.5, 1.5], legend='t=%4.2f' % t,\
title='Evolution of wave over time', savefig='pix/plot_wave%04d.png' \
% counter)
counter += 1
time.sleep(fps)
#movie('pix/plot_wave*.png')
def value(self, x): from scitools.std import exp, sin a = self.a w = self.w return exp(-a*x)*sin(w*x)
def _g(x):
return exp(-0.1 * x**2) * sin(pi / 2 * x)
def _g(x):
return exp(-0.1*x**2)*sin(pi/2*x)
def _dg(x):
return -2*0.1*x*exp(-0.1*x**2)*sin(pi/2*x) + \
pi/2*exp(-0.1*x**2)*cos(pi/2*x)
# Make a figure of a curve and "dart throws" for illustrating # Monte Carlo integration in 2D for area computations. import numpy as np xr = np.random.uniform(0, 2, 500) yr = np.random.uniform(0, 2.4, 500) x = np.linspace(0, 2, 51) from scitools.std import exp, sin, pi, plot y = 2 + x**2 * exp(-0.5 * x) * sin(pi * x) plot(x, y, 'r', xr, yr, 'o', hardcopy='tmp.eps')
def points(N):
x = [0.5 * cos(2 * pi * i / N) for i in range(N + 1)]
y = [0.5 * sin(2 * pi * i / N) for i in range(N + 1)]
return x, y
def test_manufactured():
A = 1
B = 0
mx = 1
my = 1
b = 1
c = 1.1
#define some variables
Lx = 2.
Ly = 2.
T = 1
C = 0.3
dt = 0.1
#help varabeles
kx = pi * mx / Lx
ky = pi * my / Ly
w = 1
#Exact solution
ue = lambda x, y, t: A * cos(x * kx) * cos(y * ky) * cos(t * w) * exp(-c *
t)
I = lambda x, y: A * cos(x * kx) * cos(y * ky)
V = lambda x, y: -c * A * cos(x * kx) * cos(y * ky)
q = lambda x, y: x**2 + y**2
f = lambda x, y, t: A * (
-b * (c * cos(t * w) + w * sin(t * w)) * cos(kx * x) * cos(ky * y) + c
**2 * cos(kx * x) * cos(ky * y) * cos(t * w) + 2 * c * w * sin(
t * w) * cos(kx * x) * cos(ky * y) + kx**2 *
(x**2 + y**2) * cos(kx * x) * cos(ky * y) * cos(t * w) + 2 * kx * x *
sin(kx * x) * cos(ky * y) * cos(t * w) + ky**2 *
(x**2 + y**2) * cos(kx * x) * cos(ky * y) * cos(t * w) + 2 * ky * y *
sin(ky * y) * cos(kx * x) * cos(t * w) - w**2 * cos(kx * x) * cos(
ky * y) * cos(t * w)) * exp(-c * t)
#factor dt decreeses per step
step = 0.5
#number of steps I want to do
val = 5
#array to store errors
E = zeros(val)
for i in range(val):
v = 'vector'
#solve eqation
u, x, y, t, e = solver(I,
V,
f,
q,
b,
Lx,
Ly,
dt * step**(i),
T,
C,
8,
mode=v,
ue=ue)
E[i] = e
#find convergence rate between diffrent dt values
r = zeros(val - 1)
r = log(E[1:] / E[:-1]) / log(step)
print
print "Convergence rates for manufactured solution:"
for i in range(val):
if i == 0:
print "dt: ", dt, " Error: ", E[i]
else:
print "dt: ", dt * step**(
i), " Error: ", E[i], "rate of con.: ", r[i - 1]
#requiere "close" to 2 in convergence rate for last r.
assert abs(r[-1] - 2) < 0.5
def test_manufactured():
A=1
B=0
mx=1
my=1
b=1
c=1.1
#define some variables
Lx = 2.
Ly = 2.
T = 1
C = 0.3
dt= 0.1
#help varabeles
kx = pi*mx/Lx
ky = pi*my/Ly
w=1
#Exact solution
ue = lambda x,y,t: A*cos(x*kx)*cos(y*ky)*cos(t*w)*exp(-c*t)
I = lambda x,y: A*cos(x*kx)*cos(y*ky)
V = lambda x,y: -c*A*cos(x*kx)*cos(y*ky)
q = lambda x,y: x**2+y**2
f = lambda x,y,t:A*(-b*(c*cos(t*w) + w*sin(t*w))*cos(kx*x)*cos(ky*y) + c**2*cos(kx*x)*cos(ky*y)*cos(t*w) + 2*c*w*sin(t*w)*cos(kx*x)*cos(ky*y) + kx**2*(x**2 + y**2)*cos(kx*x)*cos(ky*y)*cos(t*w) + 2*kx*x*sin(kx*x)*cos(ky*y)*cos(t*w) + ky**2*(x**2 + y**2)*cos(kx*x)*cos(ky*y)*cos(t*w) + 2*ky*y*sin(ky*y)*cos(kx*x)*cos(t*w) - w**2*cos(kx*x)*cos(ky*y)*cos(t*w))*exp(-c*t)
#factor dt decreeses per step
step=0.5
#number of steps I want to do
val=5
#array to store errors
E=zeros(val)
for i in range(val):
v='vector'
#solve eqation
u,x,y,t,e=solver(I,V,f,q,b,Lx,Ly,dt*step**(i),T,C,8,mode=v,ue=ue)
E[i]=e
#find convergence rate between diffrent dt values
r =zeros(val-1)
r = log(E[1:]/E[:-1])/log(step)
print
print "Convergence rates for manufactured solution:"
for i in range(val):
if i==0:
print "dt: ",dt," Error: ",E[i]
else:
print "dt: ",dt*step**(i)," Error: ",E[i], "rate of con.: ", r[i-1]
#requiere "close" to 2 in convergence rate for last r.
assert abs(r[-1]-2)<0.5
# Make a figure of a curve and "dart throws" for illustrating # Monte Carlo integration in 2D for area computations. import numpy as np xr = np.random.uniform(0, 2, 500) yr = np.random.uniform(0, 2.4, 500) x = np.linspace(0, 2, 51) from scitools.std import exp, sin, pi, plot y = 2 + x**2*exp(-0.5*x)*sin(pi*x) plot(x, y, 'r', xr, yr, 'o', hardcopy='tmp.eps')
s = 0 #initialize sum
n = len(xp) - 1
for k in range(n + 1):
s += yp[k] * L_k(x, k, xp, yp)
return s
def test_p_L(xp, yp):
n = len(xp) - 1
eps = 10e-12
count = 0 #count how many answers are not close enough
for k in range(n + 1):
if abs(p_L(xp[k], xp, yp) - yp[k]) > eps:
count += 1
if count == 0:
print 'Success! The Lagrange interpolation polynomial indeed passes',\
'through all the points (xp,yp).'
else:
print 'Oh no! There were %d points where the interpolation polynomial'\
%(count), 'fails to pass through (xp,yp).'
if __name__ == '__main__':
Xp = linspace(0, pi, 5)
Yp = sin(Xp)
test_p_L(Xp, Yp)
x = Xp[2] - Xp[1] / 2.0
y = sin(x)
y_inter = p_L(x, Xp, Yp)
print 'Take the point between Xp[1] and Xp[2]. Evaluating the sin function',\
'directly yields %.6f, whilst the interpolated value is %.6f.' %(y, y_inter)
"""
Exercise 5.6: Simulate by hand a vectorized expression
Author: Weiyun Lu
"""
from scitools.std import array, sin, cos, exp, zeros
x = array([0, 2])
t = array([1, 1.5])
y = zeros((2, 2))
f = lambda x, t: cos(sin(x)) + exp(1.0 / t)
for i in range(2):
for j in range(2):
y[i][j] = f(x[i], t[j])
print y
for name in glob.glob('pix/planet*.png'):
os.remove(name)
n = 100 #we want 100 frames
a = 1 #set parameters for the orbit
b = 1
w = 1
delta_t = (2 * pi) / (w * n)
counter = 0
X = []
Y = []
for k in range(n + 1):
tk = k * delta_t #each time through we need to add a new k value
x = a * cos(w * tk)
y = b * sin(w * tk)
X.append(x)
Y.append(y)
XP = array(X)
YP = array(Y)
XF = array([x])
YF = array([y])
velo = w * sqrt(a**2 * sin(w * tk)**2 + b**2 * cos(w * tk)**2)
plot(XP,
YP,
'-r',
XF,
YF,
'bo',
axis=[-1.2, 1.2, -1.2, 1.2],
title='Planetary orbit',
Calculates the approximation to pi by computing
the path length of a polygon with N points inscribed
in the circle of radius 1
'''
from scitools.std import sin, pi
x = [1. / 2 * cos(2. * pi * i / (N)) for i in range(N + 1)]
y = [1. / 2 * sin(2. * pi * i / (N)) for i in range(N + 1)]
approx = pathlength(x, y)
return approx
N = array(range(3, 31))
count = 0
C = 1000 # points with which we draw the circle
xcircle = [1. / 2 * cos(2. * pi * i / C) for i in range(C + 1)]
ycircle = [1. / 2 * sin(2. * pi * i / C) for i in range(C + 1)]
xmin = -.55
xmax = .55
ymin = xmin
ymax = .65
for n in N:
x = [1. / 2 * cos(2. * pi * i / n) for i in range(n + 1)]
y = [1. / 2 * sin(2. * pi * i / n) for i in range(n + 1)]
plot(x,
y,
'-o-',
xcircle,
ycircle,
axis=[xmin, xmax, ymin, ymax],
xlabel='x',
ylabel='y',