def comparison_plot(f, u, Omega, filename='tmp.pdf',
plot_title='', ymin=None, ymax=None,
u_legend='approximation'):
"""Compare f(x) and u(x) for x in Omega in a plot."""
x = sm.Symbol('x')
print 'f:', f
f = sm.lambdify([x], f, modules="numpy")
u = sm.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
exact = f(xcoor)
approx = u(xcoor)
plot(xcoor, approx, '-')
hold('on')
plot(xcoor, exact, '-')
legend([u_legend, 'exact'])
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def comparison_plot(f, u, Omega, plotfile='tmp'):
"""Compare f(x,y) and u(x,y) for x,y in Omega in a plot."""
x, y = sm.symbols('x y')
f = sm.lambdify([x,y], f, modules="numpy")
u = sm.lambdify([x,y], u, modules="numpy")
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
for r in range(2):
for s in range(2):
if not isinstance(Omega[r][s], (int,float)):
Omega[r][s] = float(Omega[r][s].evalf())
resolution = 41 # no of points in plot
xcoor = linspace(Omega[0][0], Omega[0][1], resolution)
ycoor = linspace(Omega[1][0], Omega[1][1], resolution)
xv, yv = ndgrid(xcoor, ycoor)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
exact = f(xv, yv)
approx = u(xv, yv)
figure()
surfc(xv, yv, exact, title='f(x,y)',
colorbar=True, colormap=hot(), shading='flat')
if plotfile:
savefig('%s_f.pdf' % plotfile, color=True)
savefig('%s_f.png' % plotfile)
figure()
surfc(xv, yv, approx, title='f(x,y)',
colorbar=True, colormap=hot(), shading='flat')
if plotfile:
savefig('%s_u.pdf' % plotfile, color=True)
savefig('%s_u.png' % plotfile)
def solver(T, dt, v0, Cd, rho, A, m, Source=None):
"""
This is the main solver function for the program. It takes the problem
specific variables as arguments, and returns two meshes containing the
velocity and time points respectively.
"""
a = Cd * rho * A / (2 * m) # Set up the constant a for compact code
v = zeros(int(T / dt) + 1) # Create the velocity mesh
v[0] = v0
g = 9.81
#Initial velocity and the value of gravity acceleration
# Description of the functions X(t) and Y(t) is given in the PDF file
def X(t):
if (Source == None):
return -g
else:
return -g + Source(t + dt / 2.) / m
def Y(t):
return a
#Calculate the velocity at each meshpoint
for i in range(1, len(v)):
v[i] = skydiving_iterate(v[i - 1], dt * (i - 1), dt, X, Y)
return v, linspace(0, T, T / dt + 1)
def comparison_plot(u, Omega, u_e=None, filename='tmp.eps',
plot_title='', ymin=None, ymax=None):
x = sp.Symbol('x')
u = sp.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
approx = u(xcoor)
plot(xcoor, approx)
legends = ['approximation']
if u_e is not None:
exact = u_e(xcoor)
hold('on')
plot(xcoor, exact)
legends = ['exact']
legend(legends)
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def test_easyviz():
from scitools.std import linspace, ndgrid, plot, contour, peaks, \
quiver, surfc, backend, get_backend
n = 21
x = linspace(-3, 3, n)
xx, yy = ndgrid(x, x, sparse=False)
# a basic plot
plot(x, x**2, 'bx:')
wait()
if backend in ['gnuplot', 'vtk', 'matlab', 'dx', 'visit', 'veusz']:
# a contour plot
contour(peaks(n), title="contour plot")
wait()
# a vector plot
uu = yy
vv = xx
quiver(xx, yy, uu, vv)
wait()
# a surface plot with contours
zz = peaks(xx, yy)
surfc(xx, yy, zz, colorbar=True)
wait()
if backend == 'grace':
g = get_backend()
g('exit')
def test_easyviz():
from scitools.std import linspace, ndgrid, plot, contour, peaks, \
quiver, surfc, backend, get_backend
n = 21
x = linspace(-3, 3, n)
xx, yy = ndgrid(x, x, sparse=False)
# a basic plot
plot(x, x**2, 'bx:')
wait()
if backend in ['gnuplot', 'vtk', 'matlab', 'dx', 'visit', 'veusz']:
# a contour plot
contour(peaks(n), title="contour plot")
wait()
# a vector plot
uu = yy
vv = xx
quiver(xx, yy, uu, vv)
wait()
# a surface plot with contours
zz = peaks(xx, yy)
surfc(xx, yy, zz, colorbar=True)
wait()
if backend == 'grace':
g = get_backend()
g('exit')
def _test():
# from InverseFunction import InverseFunction as I
from scitools.std import log, linspace, plot
def f(x):
return log(x)
x = linspace(1, 5, 101)
f_inv = InverseFunction(f, x)
plot(x,f(x),x, f_inv.values,legend=['log(x)', 'inv log(x)'])
raw_input('Press Enter to quit: ')
def _test():
x0 = float(sys.argv[1])
x, info = Newton(_g, x0, _dg, store=True)
print('root: %.16g' % x)
for i in range(len(info)):
print('Iteration %2d: f(%g)=%g' % (i, info[i][0], info[i][1]))
x = linspace(-7, 7, 401)
y = _g(x)
plot(x, y)
def graph(self, resolution=1001): ''' 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)
def graph(self, resolution=1001):
'''
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)
def __call__(self, x):
from scitools.std import linspace, sum, zeros,iseq
f, a, n = self.f, self.a, self.n
h = (x-a)/float(n)
I = 0.5*f(a)
ar = zeros(len(linspace(1,n-1, n-1)))
for i in iseq(1, n-1):
ar[i-1] = f(a + i*h)
I = sum(ar)
I += 0.5*f(x)
I *= h
return I
def __call__(self, x):
from scitools.std import linspace, sum, zeros, iseq
f, a, n = self.f, self.a, self.n
h = (x - a) / float(n)
I = 0.5 * f(a)
ar = zeros(len(linspace(1, n - 1, n - 1)))
for i in iseq(1, n - 1):
ar[i - 1] = f(a + i * h)
I = sum(ar)
I += 0.5 * f(x)
I *= h
return I
def comparison_plot(f, u, Omega, filename='tmp.pdf'):
x = sm.Symbol('x')
f = sm.lambdify([x], f, modules="numpy")
u = sm.lambdify([x], u, modules="numpy")
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
exact = f(xcoor)
approx = u(xcoor)
plot(xcoor, approx)
hold('on')
plot(xcoor, exact)
legend(['approximation', 'exact'])
savefig(filename)
def _test():
from scitools.std import sin, cos, exp, linspace, plot, pi
import sys
x0 = float(sys.argv[1])
x, info = Newton(_g, x0, _dg, store=True)
print("root: %.16g" % x)
for i in range(len(info)):
print("Iteration %2d: f(%g)=%g" % (i, info[i][0], info[i][1]))
x = linspace(-7, 7, 401)
y = _g(x)
plot(x, y)
def _test():
from scitools.std import sin, cos, exp, linspace, plot, pi
import sys
x0 = float(sys.argv[1])
x, info = Newton(_g, x0, _dg, store=True)
print('root: %.16g' % x)
for i in range(len(info)):
print('Iteration %2d: f(%g)=%g' % \
(i, info[i][0], info[i][1]))
x = linspace(-7, 7, 401)
y = _g(x)
plot(x, y)
def forward_leaper(I,a,b,T,dt): """ Solves u'(t) = -a(t)*u(t) + b(t), u(0) = I with a forward difference scheme. """ raise_type(a) raise_type(b) dt = float(dt) N = int(round(T/dt)) # Rounds off to the closest integer T = N*dt # Recalculate the T value based upon the new N value u = zeros(N+1) t = linspace(0,T,N+1) for n in range(0,N): u[n+1] = u[n] - dt*a(dt*n)*u[n] + dt*b(dt*n) return t,u
def forward_leaper(I, a, b, T, dt):
"""
Solves u'(t) = -a(t)*u(t) + b(t), u(0) = I with a forward difference scheme.
"""
raise_type(a)
raise_type(b)
dt = float(dt)
N = int(round(T / dt)) # Rounds off to the closest integer
T = N * dt # Recalculate the T value based upon the new N value
u = zeros(N + 1)
t = linspace(0, T, N + 1)
for n in range(0, N):
u[n + 1] = u[n] - dt * a(dt * n) * u[n] + dt * b(dt * n)
return t, u
def finite_difference(L, Nx, N, dt, C, P_L, P_R):
x = linspace(0, L, Nx+1)
dx = x[1] - x[0]
C = 0.4*C*dt/(dx**2)
print C
Q = zeros(Nx+1)
Q_1 = P_R**2*ones(Nx+1)
for n in range(0, N):
# Compute u at inner mesh points
for i in range(1, Nx):
Q[i] = Q_1[i] + C*(Q_1[i-1]**2.5 - 2*Q_1[i]**2.5 + Q_1[i+1]**2.5)
# Insert boundary conditions
Q[0]=P_L**2; Q[Nx]=P_R**2
# Update u_1 before next step
Q_1[:]= Q
return sqrt(Q), x
def finite_difference(L, Nx, N, dt, C, P_L, P_R):
x = linspace(0, L, Nx+1)
dx = x[1] - x[0]
C = C*dt/(dx**2)
print C
P = zeros(Nx+1)
P_1 = P_R*ones(Nx+1)
for n in range(0, N):
# Compute u at inner mesh points
for i in range(1, Nx):
#print P_1
P[i] = P_1[i] + C*(P_1[i-1]**2 - 2*P_1[i]**2 + P_1[i+1]**2)
# Insert boundary conditions
P[0]=P_L; P[Nx]=P_R
# Update u_1 before next step
P_1[:]= P
return P, x
def solver(T, dt, v0, Cd, rho, A, m, Source=None):
"""
This is the main solver function for the program. It takes the problem
specific variables as arguments, and returns two meshes containing the
velocity and time points respectively.
"""
a = Cd*rho*A/(2*m) # Set up the constant a for compact code
v = zeros(int(T/dt) + 1) # Create the velocity mesh
v[0] = v0; g = 9.81;#Initial velocity and the value of gravity acceleration
# Description of the functions X(t) and Y(t) is given in the PDF file
def X(t):
if(Source == None):
return -g
else:
return -g + Source(t+dt/2.)/m
def Y(t):
return a
#Calculate the velocity at each meshpoint
for i in range(1,len(v)):
v[i] = skydiving_iterate(v[i-1], dt*(i-1), dt, X, Y)
return v, linspace(0, T, T/dt +1)
"""
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 solver(I, V_, f_, c, Lx, Ly, Nx, Ny, dt, T, b,
user_action=None, version='scalar', skip_every_n_frame=10, show_cpu_time=False,display_warnings=True, plotting=False):
order = 'C' # Store arrays in a column-major order (in memory)
x = linspace(0, Lx, Nx+1) # mesh points in x dir
y = linspace(0, Ly, Ny+1) # mesh points in y dir
dx = x[1] - x[0]
dy = y[1] - y[0]
xv = x[:,newaxis] # for vectorized function evaluations
yv = y[newaxis,:]
# Assuming c is a function:
c_ = zeros((Nx+1,Ny+1), order='c')
for i,xx in enumerate(x):
for j,yy in enumerate(y):
c_[i,j] = c(xx,yy) # Loop through x and y with indices i,j at the same time
c_max = c_.max() # Pick out the largest value from c(x,y)
q = c_**2
stability_limit = (1/float(c_max))*(1/sqrt(1/dx**2 + 1/dy**2))
if dt <= 0: # shortcut for max time step is to use i.e. dt = -1
safety_factor = -dt # use negative dt as safety factor
extra_factor = 1 # Easy way to make dt even smaller
dt = safety_factor*stability_limit*extra_factor
elif dt > stability_limit and display_warnings:
print '\nWarning: (Unless you are testing the program), be aware that'
print 'dt: %g is currently exceeding the stability limit: %g\n' %(dt, stability_limit)
Nt = int(round(T/float(dt)))
t = linspace(0, Nt*dt, Nt+1) # mesh points in time
dt2 = dt**2
# Constants for simple calculation
A = (1 + b*dt/2)**(-1)
B = (b*dt/2 - 1)
dtdx2 = dt**2/(2*dx**2)
dtdy2 = dt**2/(2*dy**2)
# Make f(x,y,t) and V(x,y) ready for computation with different schemes
if f_ is None or f_ == 0:
f = (lambda x, y, t: 0) if version == 'scalar' else \
lambda x, y, t: zeros((xv.shape[0], yv.shape[1]))
else:
if version == 'scalar':
f = f_
if V_ is None or V_ == 0:
V = (lambda x, y: 0) if version == 'scalar' else \
lambda x, y: zeros((xv.shape[0], yv.shape[1]))
else:
if version == 'scalar':
V = V_
if version == 'vectorized': # Generate and fill matrices for first timestep
f = zeros((Nx+1,Ny+1), order=order)
V = zeros((Nx+1,Ny+1), order=order)
f[:,:] = f_(xv,yv,0)
V[:,:] = V_(xv,yv)
u = zeros((Nx+1,Ny+1), order=order) # solution array
u_1 = zeros((Nx+1,Ny+1), order=order) # solution at t-dt
u_2 = zeros((Nx+1,Ny+1), order=order) # solution at t-2*dt
Ix = range(0, u.shape[0]) # Index set notation
Iy = range(0, u.shape[1])
It = range(0, t.shape[0])
import time; t0 = time.clock() # for measuring CPU time
# Load initial condition into u_1
if version == 'scalar':
for i in Ix:
for j in Iy:
u_1[i,j] = I(x[i], y[j])
else: # use vectorized version
u_1[:,:] = I(xv, yv)
if user_action is not None:
if plotting:
user_action(u_1, x, xv, y, yv, t, 0, skip_every_n_frame)
else:
user_action(u_1, x, xv, y, yv, t, 0)
# Special formula for first time step
n = 0
# First step requires a special formula, use either the scalar
# or vectorized version (the impact of more efficient loops than
# in advance_vectorized is small as this is only one step)
if version == 'scalar':
u,cpu_time = advance_scalar(u, u_1, u_2, q, f, x, y, t, n, A, B,
dt2, dtdx2,dtdy2, V, step1=True)
else:
u,cpu_time = advance_vectorized(u, u_1, u_2, q, f, t, n, A, B,
dt2, dtdx2,dtdy2, V, step1=True)
if user_action is not None:
if plotting:
user_action(u, x, xv, y, yv, t, 1, skip_every_n_frame)
else:
user_action(u_1, x, xv, y, yv, t, 1)
# Update data structures for next step
u_2, u_1, u = u_1, u, u_2
# Time loop for all later steps
for n in It[1:-1]:
if version == 'scalar':
# use f(x,y,t) function
u,cpu_time = advance_scalar(u, u_1, u_2, q, f, x, y, t, n, A, B, dt2, dtdx2,dtdy2)
if show_cpu_time:
percent = (float(n)/It[-2])*100.0
sys.stdout.write("\rLast step took: %.3f sec with [scalar-code]. Computation is %d%% " %(cpu_time,percent))
sys.stdout.flush()
else: # Use vectorized code
f[:,:] = f_(xv, yv, t[n]) # must precompute the matrix f
u,cpu_time = advance_vectorized(u, u_1, u_2, q, f, t, n, A, B,
dt2, dtdx2,dtdy2)
if show_cpu_time:
percent = (float(n)/It[-2])*100.0
sys.stdout.write("\rLast step took: %.5f sec with [vec-code]. Computation is %d%% " %(cpu_time,percent))
sys.stdout.flush()
if user_action is not None:
if plotting:
if user_action(u, x, xv, y, yv, t, n+1, skip_every_n_frame):
break
else:
if user_action(u, x, xv, y, yv, t, n+1):
break
# Update data structures for next step
#u_2[:] = u_1; u_1[:] = u # safe, but slower
u_2, u_1, u = u_1, u, u_2
# Important to set u = u_1 if u is to be returned!
t1 = time.clock()
# dt might be computed in this function so return the value
return dt, t1 - t0
def graph(self, resolution=1001):
'''
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')
def visualize(self, r_start, r_stop, n=100):
from scitools.std import plot, linspace
r = linspace(r_start, r_stop, n)
g = self.force(r)
title = 'Gravity force: m=%g, M=%g' % (self.m, self.M)
plot(r, g, title=title)
def table(self, n, l, r):
from scitools.std import linspace, array
x = linspace(l, r, n)
print '%10s %10s' % ('x', 'f(x)')
for i in x:
print '%10f %10f' % (i, value(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')
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)
from scitools.std import plot, linspace, vectorize
def v_all(x, mu=1E-6, exp=math.exp):
x = np.float128(x)
mu = np.float128(mu)
num = (1 - exp(float(x) / mu))
denom = (1 - exp(1.0 / mu))
return num, denom, num / denom
def v(x, mu=1E-6, exp=math.exp):
return v_all(x, mu, exp)[2]
for x in linspace(0, 1, 25):
for f in [math.exp, np.exp]:
try:
print v_all(x, 1E-3, f)
except:
print('Could not compute an instance of v(x).')
#This function is subject to overflow.
#Many more instances succeed with float128 than default float64.
x = linspace(0, 1, 10000)
v = vectorize(v)
plot(x,
v(x, 1.0, np.exp),
'-r',
""" Exercise 5.7: Demonstrate array slicing Author: Weiyun Lu """ from scitools.std import linspace w = linspace(0, 3, 31) print w print w[:] #should just be the whole thing print w[:-2] #up to but not including 2nd last, so should be up to 2.8 print w[::5] #every fifth element starting with 0 print w[2:-2:6] #start with 0.2 and up to 2.8; every 6th element
if __name__ == '__main__':
c = Calculator(10, 5)
#d = Calculator(10, 5)
#print c + d
print 'Adding: ', c.add()
print 'Subtracting: ', c.subtract()
print 'Multiplying: ', c.multiply()
print 'Divifing: ', c.divide()
print 'Changeing variables...'
c.change_variables(6, 3)
print 'Now adding: ', c.add()
print 'Now trying functions...'
print 'f(x) = x^2, x = 2: ', c.function('x**2', 2)
print 'Trying with array...'
c.change_variables(linspace(-5, -3, 3), linspace(-1, 1, 3))
print 'Adding linspace(-5, -3, 5), linspace(-1, 1, 5):'
print c.add()
print 'f(x) = x^2, x = linspace(-1, 1, 11): '
print c.function('x**2', linspace(-1, 1, 21))
x = linspace(-5, 5)
plot(x, c.function('x**2+x**4-3', x))
raw_input()
'''
Bergem$ python Calculator.py
Adding: 15
Subtracting: 5
Multiplying: 50
Divifing: 2.0
Changeing variables...
omega1 = 2 * pi / P1
a1 = sqrt(omega1 / (2 * k))
A2 = 7 # amplitude of yearly temperature variations (in C)
P2 = 24 * 60 * 60 * 365. # oscillation period of 1 yr (in seconds)
# angular frequency of yearly temperature variations (in rad/s)
omega2 = 2 * pi / P2
a2 = sqrt(omega2 / (2 * k))
dt = P2 / 30 # time lag: 0.1 yr
tmax = 3 * P2 # 3 year simulation
T0 = 10 # mean surface temperature in Celsius
D = -(1 / a1) * log(0.001) # max depth
n = 501 # no of points in the z direction
z = linspace(0, D, n)
def z_scaled(x, s):
a = x[0]
b = x[-1]
return a + (b - a) * ((x - a) / (b - a))**s
zs = z_scaled(z, 3)
animate(tmax, dt, zs, T, T0 - A2 - A1, T0 + A2 + A1, 0, 'z', 'T')
movie('tmp_*.png', encoder='convert', fps=6)
import glob
import os
# from __future__ import unicode_literals
# import matplotlib
# matplotlib.rcParams['text.usetex'] = True
# matplotlib.rcParams['text.latex.unicode'] = True
from matplotlib import rc
import matplotlib.pyplot as plt
from scitools.std import sqrt, pi, exp, linspace
import os
def f(x, m, s):
return (1.0/(sqrt(2*pi)*s))*exp(-0.5*((x-m)/s)**2)
m=0
s_min = 0.2
s_max = 2
x = linspace(m -3*s_max, m + 3*s_max, 100)
s_values = linspace(s_max, s_min, 5)
# f is max for x=m; smaller s gives larger max value
max_f = f(m, m, s_min)
os.system("rm ./tmp*.png")
#plt.figure(1, figsize=(6, 4))
#
#
plt.ion()
k = 1E-6 # thermal diffusivity (in m**2/s)
A1 = 15 # amplitude of the daily temperature variations (in C)
P1 = 24 * 60 * 60. # oscillation period of 24 h (in seconds)
omega1 = 2 * pi / P1 # angular freq of daily temp variations (in rad/s)
a1 = sqrt(omega1 / (2 * k))
A2 = 7 # amplitude of yearly temperature variations (in C)
P2 = 24 * 60 * 60 * 365. # oscillation period of 1 yr (in seconds)
omega2 = 2 * pi / P2 # angular freq of yearly temp variations (in rad/s)
a2 = sqrt(omega2 / (2 * k))
dt = P2 / 20 # time lag: 0.1 yr
tmax = 3 * P2 # 3 year simulation
T0 = 10 # mean surface temperature in Celsius
D = -(1 / a1) * log(0.001) # max depth
n = 501 # no of points in the z direction
# set T0, A, k, omega, D, n, tmax, dt
z = linspace(0, D, n)
animate(tmax, dt, z, T, T0 - A2 - A1, T0 + A2 + A1, 0, 'z', 'T')
movie('tmp_*.png', encoder='convert', fps=6, outputfile='tmp_heatwave.gif')
import glob
import os
# Remove frames
for filename in glob.glob('tmp_*.png'):
os.remove(filename)
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)
plt.figure()
plt.plot(t,nsy,'r-')
plt.title('Diffusion D = %.1f ' % D)
plt.xlabel('time [MD]')
#plt.xlabel('time [fs]')
plt.ylabel('msq/t [MD]')
#plt.ylabel('msq/t [m^2/t]')
plt.legend(('Diffusion constant'), loc='lower right')
#######################################3
print timeiterations
l_binz = len(binz)
print l_binz
radius = linspace(0,(l_binz+1)*0.1,l_binz)
#plt.figure()
#plt.plot(radius,binz)
volumeR = zeros(l_binz)
for i in range(l_binz-1):
volumeR[i] = (4./3)*pi*(radius[i+1]**3 - radius[i]**3) # volume of shell in range [r,r+dr]
binz[1:] = binz[1:]/(timeiterations*N*rho*volumeR[i]) # mean number of particles in volume
plt.figure()
plt.plot(radius,binz,'r-*')
plt.title('Average number of particles in distance r from atom')
"""
Exercise 5.41: Plot functions from the command line
Author: Weiyun Lu
"""
import sys
from scitools.std import StringFunction, plot, linspace
from numpy import *
try:
f = StringFunction(sys.argv[1])
f = vectorize(f)
min = eval(sys.argv[2])
max = eval(sys.argv[3])
except IndexError:
print('You must supply a function, a minimum value, and a maximum value!')
sys.exit(1)
except ValueError or TypeError:
print('You must supply a function (as a string), and valid min and max',
'numerical values!')
sys.exit(1)
if sys.argv[4] != None:
n = eval(sys.argv[4])
else:
n = 501
x = linspace(min, max, n)
plot(x, f(x), xlabel='x', ylabel='f(x)', title='f(x)=%s' % sys.argv[1])
"""
Exercise 5.28: Plot the viscosity of water
Author: Weiyun Lu
"""
from scitools.std import plot, linspace
A = 2.414e-5
B = 247.8
C = 140.0
mu = lambda T: A * 10**(B / (T - C)) #viscosity in terms of Kelvin
mu_celcius = lambda T: mu(T + 273) #calculate mu in terms of celcius
x = linspace(0, 100, 100)
y = mu_celcius(x)
plot(x, y, '-b', xlabel='Temperature (C)', ylabel='Viscosity (Pa s)', \
title='Viscocity of Water')
def table(self, n, l, r):
from scitools.std import linspace, array
x = linspace(l, r, n)
print '%10s %10s' % ('x', 'f(x)')
for i in x:
print '%10f %10f' % (i, value(i))
""" Exercise 5.26: Plot a wave packet Author: Weiyun Lu """ from scitools.std import exp, sin, pi, linspace, plot f = lambda x, t : exp(-(x-3*t)**2) * sin(3*pi*(x-t)) xp = linspace(-4,4,100) yp = f(xp, 0) plot(xp, yp , '-b', title='A wave localized in space')
def graph(self, resolution=1001):
'''
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')
# plt.plot(x, B1energyVec)
# plt.xlabel('$x_1$')
# plt.ylabel('$u_3$')
# plt.title('Deflection of a cantilever beam under self weight')
# plt.show()
def f(x,m,s):
return (1.0/(sqrt(2*pi)*s))*exp(-0.5*((x-m)/s)**2)
m=0
s_min=0.2
s_max=2
x=linspace(m-3*s_max, m+3*s_max,10)
s_values =linspace(s_max, s_min, 3)
max_f=f(m,m,s_min)
#
plt.ion()
y=f(x,m, s_max)
print len(x), "\n", len(y)
print x, \
"\n", \
y
plt.plot(x,y)
plt.axis([x[0], x[-1], -0.1, max_f])
# plt.xlabel('x')
# plt.ylabel('f')
from scitools.std import sqrt, pi, exp, linspace, plot, movie
import time, glob, os, sys
# Clean up old frames
for name in glob.glob('tmp_*.pdf'):
os.remove(name)
def f(x, m, s):
return (1.0/(sqrt(2*pi)*s))*exp(-0.5*((x-m)/s)**2)
m = 0
s_max = 2
s_min = 0.2
x = linspace(m -3*s_max, m + 3*s_max, 1000)
s_values = linspace(s_max, s_min, 30)
# f is max for x=m; smaller s gives larger max value
max_f = f(m, m, s_min)
# Show the movie, and make hardcopies of frames simulatenously
counter = 0
for s in s_values:
y = f(x, m, s)
plot(x, y, '-', axis=[x[0], x[-1], -0.1, max_f],
xlabel='x', ylabel='f', legend='s=%4.2f' % s,
savefig='tmp_%04d.png' % counter)
counter += 1
#time.sleep(0.2) # can insert a pause to control movie speed
if '--no-moviefile' in sys.argv:
# Drop making movie files
import sys; sys.exit(0)
"""
Exercise 5.12: Plot exact and inexact Fahrenheit-Celcius conversion formulas
Author: Weiyun Lu
"""
from scitools.std import linspace, plot
F = linspace(-20,120,20)
C_approx = (F-30) / 2.0
C_actual = (F-32) * 5.0/9
plot(F, C_approx, '-r', F, C_actual, '-b', legend=('approx C', 'actual C'),\
xlabel='Fahrenheit', ylabel='Celcius')
plt.figure()
plt.plot(t, nsy, 'r-')
plt.title('Diffusion D = %.1f ' % D)
plt.xlabel('time [MD]')
#plt.xlabel('time [fs]')
plt.ylabel('msq/t [MD]')
#plt.ylabel('msq/t [m^2/t]')
plt.legend(('Diffusion constant'), loc='lower right')
#######################################3
print timeiterations
l_binz = len(binz)
print l_binz
radius = linspace(0, (l_binz + 1) * 0.1, l_binz)
#plt.figure()
#plt.plot(radius,binz)
volumeR = zeros(l_binz)
for i in range(l_binz - 1):
volumeR[i] = (4. / 3) * pi * (radius[i + 1]**3 - radius[i]**3
) # volume of shell in range [r,r+dr]
binz[1:] = binz[1:] / (timeiterations * N * rho * volumeR[i]
) # mean number of particles in volume
plt.figure()
plt.plot(radius, binz, 'r-*')
"""
Exercise 5.3: Fill arrays; vectorized (plot)
Author: Weiyun Lu
"""
from scitools.std import sqrt, pi, exp, linspace, plot
h = lambda x : (1/(sqrt(2*pi))) * exp(-0.5*x**2)
xlist = linspace(-4,4,41)
hlist = h(xlist)
pairs = zip(xlist,hlist)
plot(xlist, hlist, axis=[xlist[0], xlist[-1], 0, 0.5], xlabel='x', \
ylabel='h(x)', title='Standard Gaussian')
def visualize(self, r_start, r_stop, n=100):
from scitools.std import plot, linspace
r = linspace(r_start, r_stop, n)
g = self.force(r)
title='Gravity force: m=%g, M=%g' % (self.m, self.M)
plot(r, g, title=title)
def differentiate(self):
"""Differentiate this polynomial in-place."""
from scitools.std import linspace
n = len(self.coeff)
self.coeff[:-1] = linspace(1, n-1, n-1)*self.coeff[1:]
self.coeff = self.coeff[:-1]
