def area(self):
from scitools.std import array, sqrt
from numpy import dot, cross
v1 = array(self.v1)
v2 = array(self.v2)
v3 = array(self.v3)
v2 = v2 - v1
v3 = v3 - v1
v1 = v1 - v1 # now v1 is at the origin
return abs(cross(v2, v3)) / 2.0
def area(self): from scitools.std import array, sqrt from numpy import dot, cross v1 = array(self.v1) v2 = array(self.v2) v3 = array(self.v3) v2 = v2 - v1 v3 = v3 - v1 v1 = v1 - v1 # now v1 is at the origin return abs(cross(v2,v3))/2.0
def _test():
from scitools.std import sin, pi, array
s = Integral(sin, 0)
x = array([pi / 4, pi / 2, 3 * pi / 4, pi])
a = s(x)
print 'The integral of sin(x) for'
for i in range(4):
print '0 to %.2f' % x[i], 'is %.2f' % a[i]
def _test(): from scitools.std import sin, pi, array s = Integral(sin, 0) x = array([pi/4, pi/2, 3*pi/4, pi]) a = s(x) print 'The integral of sin(x) for' for i in range(4): print '0 to %.2f' % x[i], 'is %.2f' % a[i]
def circumference(self): from scitools.std import array, sqrt from numpy import dot v1 = array(self.v1) v2 = array(self.v2) v3 = array(self.v3) v2 = v2 - v1 v3 = v3 - v1 #now v1 is at the origin v1 = v1 - v1 side1 = sqrt(dot(v2,v2)) side2 = sqrt(dot(v3,v3)) v1 = v1 - v2 v3 = v3 - v2 v2 = v2 - v2 #now v2 is at the origin side3 = sqrt(dot(v3, v3)) return side1 + side2 + side3
def circumference(self):
from scitools.std import array, sqrt
from numpy import dot
v1 = array(self.v1)
v2 = array(self.v2)
v3 = array(self.v3)
v2 = v2 - v1
v3 = v3 - v1 #now v1 is at the origin
v1 = v1 - v1
side1 = sqrt(dot(v2, v2))
side2 = sqrt(dot(v3, v3))
v1 = v1 - v2
v3 = v3 - v2
v2 = v2 - v2 #now v2 is at the origin
side3 = sqrt(dot(v3, v3))
return side1 + side2 + side3
def trapezoidal_vectorized(f, a, x, n): h = (x-a)/float(n) I = 0.5*f(a) from scitools.std import array s = f(a + array(range(1,n))*h) I += sum(s) I += 0.5*f(x) I *= h return I
def trapezoidal_vectorized(f, a, x, n):
h = (x - a) / float(n)
I = 0.5 * f(a)
from scitools.std import array
s = f(a + array(range(1, n)) * h)
I += sum(s)
I += 0.5 * f(x)
I *= h
return I
def midpointsum_num(f,a,b,n):
h = float(b - a) / n
x = array([f(a-0.5*h + i * h) for i in range(1,n+1)])
s = h * np.sum(x[i] for i in range(len(x)))
return s
# %timeit results with f(x)=x^2, 0, 10, 100
# midpointint: 10000 loops, best of 3: 125 microseconds per loop
# midpointsum_py: 10000 loops, best of 3: 185 microseconds per loop
# midpointsum_num: 10000 loops, best of 3: 179 microseconds per loop
def trapezoidal(f, a, x, n): from scitools.std import array, iseq, linspace, zeros x = array(x) I = zeros(len(x)) index = 0 new_sum = 0 old_sum = 0 for i in x: new_sum = 0.5*f(a) h = (i-a)/float(n) new_sum += sum(f(a + array(range(1,n))*h)) new_sum += 0.5*f(i) new_sum *= h new_sum += old_sum I[index] = new_sum index += 1 a = i old_sum = new_sum if len(I) == 1: return I[0] else: return I
def trapezoidal(f, a, x, n):
from scitools.std import array, iseq, linspace, zeros
x = array(x)
I = zeros(len(x))
index = 0
new_sum = 0
old_sum = 0
for i in x:
new_sum = 0.5 * f(a)
h = (i - a) / float(n)
new_sum += sum(f(a + array(range(1, n)) * h))
new_sum += 0.5 * f(i)
new_sum *= h
new_sum += old_sum
I[index] = new_sum
index += 1
a = i
old_sum = new_sum
if len(I) == 1:
return I[0]
else:
return I
def main(nx, dt):
# Parameters
T = 1.0
rho = 1.0
# Create mesh and define function space
mesh = UnitIntervalMesh(nx)
V = FunctionSpace(mesh, 'Lagrange', 1)
# Define boundary conditions
u0 = Expression('t * x[0]*x[0]*(0.5 - x[0]/3.)', t=0)
# Initial condition
u_1 = interpolate(u0, V)
#Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression('rho*x[0]*x[0]*(-2*x[0] + 3)/6. - \
(-12*t*x[0] + 3*t*(-2*x[0] + 3))*(pow(x[0],4.)*\
pow((-dt + t),2)*pow((-2*x[0] + 3),2.) + 36)/324.\
- (-6*t*x[0]*x[0] + 6*t*x[0]*(-2*x[0] + 3))*\
(36*pow(x[0],4.)*pow((-dt + t),2.)*(2*x[0] - 3)+\
36*pow(x[0],3.)*pow((-dt + t),2.)*pow((-2*x[0] + 3),2.))/5832.',
rho=rho,
dt=dt,
t=0)
a = u * v * dx + (dt / rho) * (1 - u_1**2) * inner(nabla_grad(u),
nabla_grad(v)) * dx
L = u_1 * v * dx + (dt / rho) * f * v * dx
u = Function(V)
# Initial condition already stored in u_1, so start iteration at t=dt
t = dt
counter = 1
E = []
while t <= T:
u0.t = t
f.t = t
solve(a == L, u)
u_e = interpolate(u0, V)
e = u_e.vector().array() - u.vector().array()
E.append(numpy.sqrt(numpy.sum(e**2) / u.vector().array().size))
if (counter % 10 == 0):
print e
u_1.assign(u) # Copy solution to u_1 to prepare for next time-step
t += dt
counter += 1
return array(E).max()
def main(nx, dt):
# Parameters
T = 1.0
rho = 1.0
# Create mesh and define function space
mesh = UnitIntervalMesh(nx)
V = FunctionSpace(mesh, 'Lagrange', 1)
# Define boundary conditions
u0 = Expression('t * x[0]*x[0]*(0.5 - x[0]/3.)', t=0)
# Initial condition
u_1 = interpolate(u0, V)
#Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression('rho*x[0]*x[0]*(-2*x[0] + 3)/6. - \
(-12*t*x[0] + 3*t*(-2*x[0] + 3))*(pow(x[0],4.)*\
pow((-dt + t),2)*pow((-2*x[0] + 3),2.) + 36)/324.\
- (-6*t*x[0]*x[0] + 6*t*x[0]*(-2*x[0] + 3))*\
(36*pow(x[0],4.)*pow((-dt + t),2.)*(2*x[0] - 3)+\
36*pow(x[0],3.)*pow((-dt + t),2.)*pow((-2*x[0] + 3),2.))/5832.',
rho=rho, dt=dt, t=0)
a = u*v*dx + (dt/rho)*(1-u_1**2)*inner(nabla_grad(u), nabla_grad(v))*dx
L = u_1*v*dx + (dt/rho)*f*v*dx
u = Function(V)
# Initial condition already stored in u_1, so start iteration at t=dt
t = dt
counter = 1
E=[]
while t<=T:
u0.t = t; f.t = t
solve(a==L, u)
u_e = interpolate(u0, V)
e = u_e.vector().array() - u.vector().array()
E.append(numpy.sqrt(numpy.sum(e**2)/u.vector().array().size))
if(counter%10 == 0):
print e
u_1.assign(u) # Copy solution to u_1 to prepare for next time-step
t += dt
counter += 1
return array(E).max()
'''
graphs the Lagrange polynomial over self.points
'''
#from scitools.std import *
from scitools.std import zeros, linspace, plot, array
points = self.points
xlist = linspace(points[0, 0], points[-1, 0], resolution)
ylist = self.__call__(xlist)
plot(xlist, ylist)
if __name__ == '__main__':
from scitools.std import *
x = linspace(0, pi, 5)
y = sin(x)
points = array(zip(x, y))
l = Lagrange(points)
l.verify()
figure()
hold('on')
axis([-.1, pi + .1, -.1, 1.1])
for i in [5, 10, 20, 55, 70, 100]:
x = linspace(0, pi, i)
y = sin(x)
points = array(zip(x, y))
l = Lagrange(points)
l.graph()
xlabel('x')
ylabel('y')
legend('n = %d' % i)
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')
def __init__(self, coefficients):
from scitools.std import array
self.coeff = array(coefficients)
infile = open('lnoutput.txt', 'r')
eps = [] #intialize three empty lists
exa = []
enn = []
for line in infile:
if line.find('epsilon:') == -1: #if we can't find epsilon, skip this line
continue
else: #otherwise, process the line
words = line.split(',')
epsilon = float(words[0].split(':')[1].strip())
exact = float(words[1].split(':')[1].strip())
n = float(words[2].split('=')[1].strip())
eps.append(epsilon)
exa.append(exact)
enn.append(n)
eps = array(eps) #convert lists to arrays
exa = array(exa)
enn = array(enn)
infile.close()
semilogy(enn,
eps,
'-b',
enn,
exa,
'-r',
xlabel='n',
ylabel='Log of eps/error',
legend=['epsilon', 'exact error'],
title='Ln sum errors')
"""
Exercise 5.14: Plot data in a two-column file
Author: Weiyun Lu
"""
from scitools.std import plot, array, average
myfile = open('xydat.txt', 'r')
x = []
y = []
for line in myfile:
w1 = float(line.split()[0])
w2 = float(line.split()[1])
x.append(w1)
y.append(w2)
plot(x, y)
y = array(y)
avg = average(y)
maxi = max(y)
mini = min(y)
myfile.close()
print 'The mean is', avg, 'the min value is', mini, 'and the max value is', maxi, '.'
from scitools.std import array, plot
infile = open("../output/TEST3.txt", 'r')
n = []
t = [[],[],[]]
i = [[],[],[]]
for line in infile:
if line.startswith("n="):
n.append(int(line.split("=")[1]))
else:
tmp = line.split(":")
j = int(tmp[0])
t[j].append(float(tmp[2]))
i[j].append(int(tmp[3]))
t0 = array(t[0])/array(i[0])
t1 = array(t[1])/array(i[1])
t2 = array(t[2])/array(i[2])
T = array([t0, t1, t2])
styles = ["k--","k:","k-."]
methods = ["QR", "HessenbergQR", "shiftedQR"]
for i in range(1, len(T)):
plot(n, T[i,:], styles[i], hold="on",legend=methods[i], title="Comparison of runtime pr. iteration. n = matrix dim.", ylabel="n", xlabel="runtime/iteration [s]")
raw_input()
def midpointsum_py(f,a,b,n):
h = float(b - a) / n
x = array([f(a-0.5*h + i * h) for i in range(1,n+1)])
s = h * sum(x[i] for i in range(len(x)))
return s
def __init__(self, coefficients):
from scitools.std import array
self.coeff = array(coefficients)
"""
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
graphs the Lagrange polynomial over self.points
'''
#from scitools.std import *
from scitools.std import zeros, linspace, plot, array
points = self.points
xlist = linspace(points[0,0], points[-1,0], resolution)
ylist = self.__call__(xlist)
plot(xlist, ylist)
if __name__ == '__main__':
from scitools.std import *
x = linspace(0, pi, 5)
y = sin(x)
points = array(zip(x,y))
l = Lagrange(points)
l.verify()
figure()
hold('on')
axis([-.1, pi+.1, -.1, 1.1])
for i in [5,10,20,55,70, 100]:
x = linspace(0, pi, i)
y = sin(x)
points = array(zip(x,y))
l = Lagrange(points)
l.graph()
xlabel('x')
ylabel('y')
legend('n = %d' % i)
elif 30 <= i <= 39:
hat[i] = 'purple'
from scitools.std import array, append
import copy
N = 10**4
successes = 0
total = 0
for i in range(N):
temp = copy.deepcopy(hat)
selection = []
for j in range(10):
pick = random.choice(temp)
temp.remove(pick)
selection = append(selection, pick)
n_blue = sum(array(selection) == 'blue')
n_purple = sum(array(selection) == 'purple')
if n_blue == 2 and n_purple == 2:
successes += 1
total += 1
print 'Our probability of picking two blue balls and two'
print 'purple balls from ten picks without repalcement is'
print '%.6f' % (float(successes) / total)
'''
python 4balls_from10.py
Our probability of picking two blue balls and two
purple balls from ten picks without repalcement is
0.090600
'''
elif 30 <= i <= 39: hat[i] = 'purple' from scitools.std import array, append import copy N = 10**4 successes = 0 total = 0 for i in range(N): temp = copy.deepcopy(hat) selection = [] for j in range(10): pick = random.choice(temp) temp.remove(pick) selection = append(selection, pick) n_blue = sum(array(selection) == 'blue') n_purple = sum(array(selection) == 'purple') if n_blue == 2 and n_purple == 2: successes += 1 total += 1 print 'Our probability of picking two blue balls and two' print 'purple balls from ten picks without repalcement is' print '%.6f' % (float(successes) / total) ''' python 4balls_from10.py Our probability of picking two blue balls and two purple balls from ten picks without repalcement is 0.090600 '''
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',
legend=(['Instaneous velocity=%6f' % velo, 'present location']),
savefig='pix/planet%4d.png' % counter)
counter += 1
def __call__(self, x):
"""Evaluate the polynomial."""
from scitools.std import array
x_values = array([x**i for i in range(len(self.coeff))])
s = sum(self.coeff * x_values)
return s
