def __apply_newton(self,guess,i): #not going to scale well. ans = newton.solve(guess, self.max_iter, self.tol, i, self.f) if np.isnan(ans): c = 0 rev = np.log(self.number) #traverse log as new guess while np.isnan(rev)==False and np.isnan(ans) and c < 10: ans = newton.solve(int(rev), self.max_iter, self.tol, i, self.f) c += 1 rev = np.log(rev) return ans
def test1(debug_solve=True):
"""
Test Newton iteration for the square root with different initial
conditions.
"""
# points to evaluate polynomial
x = np.linspace(-5., 5, 1000)
g1 = x*np.cos(np.pi*x)
g2 = 1-.6*x**2
plt.figure(1) # open plot figure window
plt.clf() # clear figure
ax = plt.subplot(111)
ax.plot(x, g1, 'b-', label='g1') # connect points with a blue line
ax.plot(x, g2, 'r-', label='g2')
ax.legend(loc='lower center')
for x0 in [-2., -1.5, -1., 1.5]:
print " " # blank line
x, iters = solve(fvals_sqrt, x0, debug=debug_solve)
print "Initial guess: x = %22.15e" % x0
print "solve returns x = %22.15e after %i iterations " % (x,iters)
fx = 1-.6*x**2
print "the value of f(x) is %22.15e" % fx
# Add data points (polynomial should go through these points!)
ax.plot(x, fx, 'ko') # plot as red circles
plt.title("Intersections")
plt.savefig('intersections.png') # save figure as .png file
def test1(debug_solve=False):
"""
Find 4 intersections.
"""
xs = []
ys = []
for x0 in [-2.5, -1.7, -1., 1.3]:
print " " # blank line
x,iters = solve(fvals, x0, debug=debug_solve)
print "solve returns x = %22.15e after %i iterations " % (x,iters)
fx,fpx = fvals(x)
print "the value of f(x) is %22.15e" % fx
xs.append(x)
ys.append(gvals_g1(x)[0])
#assert abs(x-2.) < 1e-14, "*** Unexpected result: x = %22.15e" % x
x = np.linspace(-5, 5, 1001)
y1, y1p = gvals_g1(x)
y2, y2p = gvals_g2(x)
plt.figure(1) # open plot figure window
plt.clf() # clear figure
plt.plot(x,y1,'b-') # connect points with a blue line
plt.plot(x,y2,'r-') # connect points with a blue line
plt.legend(["g1", "g2"])
# Add data points (polynomial should go through these points!)
plt.plot(xs,ys,'ko') # plot as red circles
#plt.ylim(-2,8) # set limits in y for plot
plt.title("functions and thier intersections")
plt.savefig('hw3.png') # save figure as .png file
def intersect(g1vals,g2vals,x0):
'''
given two functions and initial guess this function returns their intersections
'''
x, iterations = solve( lambda x: ( g1vals(x)[0] - g2vals(x)[0], g1vals(x)[1] - g2vals(x)[1] ) , x0)
return x
def test1(debug_solve=False): """ Test Newton iteration for the square root with different initial conditions. """ is_values = [] for x0 in [-2.5,-1.5,-0.5,1.5]: print " " x, iters = solve(g_fcns, x0, debug_solve) print "solve returns x = %22.15e after %d iterations " % (x,iters) fx,fpx = g_fcns(x) print "the value of f(x) is %22.15e" % fx is_values.append(x) #assert abs(x-2.) < 1e-14, "*** Unexpected result: x = %22.15e" % x plot_fcns(asarray(is_values))
def test1(debug_solve=False):
"""
Test Newton iteration for the square root with different initial
conditions.
"""
is_values = []
for x0 in [-2.5, -1.5, -0.5, 1.5]:
print " "
x, iters = solve(g_fcns, x0, debug_solve)
print "solve returns x = %22.15e after %d iterations " % (x, iters)
fx, fpx = g_fcns(x)
print "the value of f(x) is %22.15e" % fx
is_values.append(x)
#assert abs(x-2.) < 1e-14, "*** Unexpected result: x = %22.15e" % x
plot_fcns(asarray(is_values))
def plot_intersections(fvals_func):
x = np.linspace(-5, 5, 300)
xx = np.empty(4)
for i, x0 in enumerate([-2.20, -1.5, -0.7, 1.5]):
xx[i], iters = solve(fvals_func, x0)
yy = 1 - 0.6 * xx**2
fig, ax = plt.subplots()
ax.plot(x, x * np.cos(np.pi * x), 'r', label='$g_{1}(x)=sin(x)$')
ax.plot(x, 1 - 0.6 * x**2, 'b', label='$g_{2}(x)=1-x^2$')
ax.plot(xx, yy, 'ko', label='intersections')
ax.set_xlim(-5, 5)
ax.legend(loc='lower left')
fig.savefig('intersections.png')
def plot_intersections(fvals_func):
x = np.linspace(-5, 5, 300)
xx = np.empty(4)
for i, x0 in enumerate([-2.20, -1.5, -0.7, 1.5]):
xx[i], iters = solve(fvals_func, x0)
yy = 1 - 0.6 * xx**2
fig, ax = plt.subplots()
ax.plot(x, x * np.cos(np.pi * x), 'r', label='$g_{1}(x)=sin(x)$')
ax.plot(x, 1 - 0.6 * x**2, 'b', label='$g_{2}(x)=1-x^2$')
ax.plot(xx, yy, 'ko', label='intersections')
ax.set_xlim(-5, 5)
ax.legend(loc='lower left')
fig.savefig('intersections.png')
def intersectionG(x0input):
intersec = []
xinter = []
for x0 in x0input:
x, iters = solve(f_val, x0, debug=False)
print "solve returns x= %22.15e after %i iterations" % (x, iters)
fx, fpx = f_val(x)
g1vx, g1vx_p = g1val(x)
print "The value of intersection is %22.15e" % fx
if fx not in intersec:
intersec.append(g1vx)
xinter.append(x)
xplot = linspace(-5., 5., 100.)
g1 = xplot * cos(pi * xplot)
g2 = 1. - 0.6 * xplot**2
plt.plot(xplot, g1, 'b', xplot, g2, 'g', xinter, intersec, 'ro')
plt.show()
def main(debug = False):
"""
Main function to show Newton Method to calculate intersections.
"""
x = np.linspace(-5,5,100)
y1 = [x1 * math.cos(x1 * math.pi) for x1 in x]
y2 = 1 - 0.6*x**2
plt.figure(1)
plt.clf()
plt.plot(x,y1,label='g1')
plt.plot(x,y2,label='g2')
guesses = [-2,-1.5,-0.8,1.5]
for x0 in guesses:
x,iter = solve(fvals,x0,debug)
print 'With initial guess x0 = %22.15e' %x0
print '\tsolve return x = %22.15e after %d iterations' %(x,iter)
plt.plot(x,1 - 0.6*x**2,'k.')
plt.legend(('g1','g2'))
plt.show()
plt.savefig('intersections.png')
def p4(debug_solve=False):
"""
Problem 4 from Week 3 HW, find intersects of two functions.
"""
from numpy import sqrt
a_x = []
a_fx = []
for x0 in [-2.2, -1.6, -.75, 1.4]:
print " " # blank line
# print "for init of the funcial guess %22.15e" % x0
x,iters = solve(fvals_inter, x0, debug=debug_solve)
print "solve returns x = %22.15e after %i iterations " % (x,iters)
fx = f1(x)
print "the value of f(x) is %22.15e" % fx
# store values to plot
a_x = np.append(a_x,x)
a_fx = np.append(a_fx,fx)
x_lin = np.linspace(-5,5,1001)
pi = math.pi
g1 = x_lin*np.cos(pi*x_lin)
g2 = 1 - 0.6*x_lin**2
plt.clf(); plt.plot(x_lin,g1,'r-',label='$g_1(x)$');
plt.plot(x_lin,g2,label='$g_2(x)$')
plt.legend()
plt.plot(a_x, a_fx,'ko')
plt.title("Intersections of $g_1(x)$ and $g_2(x)$")
plt.legend()
plt.savefig('intersections.png')
if debug_solve:
print "Intersection points (x,y)"
for i in range(0,len(a_x)):
print a_x[i], a_fx[i]
def part(s, a):
g = lambda x: arc(x) - s * arc(1)
return solve(g, f, a, 0.5 * 10**-6)
"""
use four initial guess to solve the problem and then
plot the figure as required
"""
from newton import solve
import matplotlib.pyplot as plt
import numpy as np
# initial guess by eyeballs
guess = (-2.2, -1.5, -0.6, 1.5)
x = np.ones(4)
for i in range(4):
print "\n*********************************************"
x[i], iters = solve(fvals, guess[i], False)
print "With initial guess x0 = %22.15e" % i
print "solve returns x = %22.15e after %i iterations" % (x[i], iters)
# plot figure
xvals = np.linspace(-5, 5, 1000)
g1 = xvals * np.cos(np.pi * xvals)
g2 = 1 - 0.6 * xvals ** 2
# for intersection point
fx = x * np.cos(np.pi * x)
plt.figure(1) # open plot figure window
plt.clf() # clear figure
plt.plot(xvals, g1) # connect points with a blue line
plt.plot(xvals, g2)
plt.plot(x, fx, "ko") # highlight intersection with black dots
def print_intersections(fvals_func):
for x0 in [-2.20, -1.5, -0.7, 1.5]: # x0 taken looking at the plot
x, iters = solve(fvals_func, x0)
print('With initial guess x0 %20.15e,\n solve returns x = %20.15e after %i iterations' % (x0, x, iters))
Example use:
>>> run intersections
"""
import newton
import matplotlib.pyplot as plt
import numpy as np
guesses = np.array([-2.2, -1.6, -0.8, 1.4])
xarray = np.zeros(4)
yarray = np.zeros(4)
i = 0
for xin in guesses:
(xout, iters) = newton.solve(xin)
xarray[i] = xout
yarray[i] = (1 - (0.6 * xout**2))
i = i + 1
x = np.linspace(-5, 5, 1000)
g1 = x * np.cos(np.pi * x)
g2 = (1 - (0.6 * x**2))
plt.figure(1)
plt.clf()
plt.xlim(-5, 5)
plt.xlabel('x (-)')
plt.ylabel('y (-)')
def fvals(x):
"""
f and f prime for this homework
"""
#from numpy import pi,cos, sin
f = (x * cos(pi * x)) - (1. - 0.6 * x * x)
fp = (cos(pi * x) - pi * x * sin(pi * x)) - (-1.2 * x)
return f, fp
xlist = []
for x0 in [-2.2, -1.5, -0.75, 1.5]:
print " " # blank line
x, iters = solve(fvals, x0, debug=True)
xlist.append(x)
print "solve returns x = %22.15e after %i iterations " % (x, iters)
fx, fpx = fvals(x)
print "the value of f(x) is %22.15e" % fx
x0s = array(xlist)
x1 = linspace(-5., 5., 1000)
f1 = x1 * cos(pi * x1)
g1 = 1. - 0.6 * x1 * x1
plt.figure(1) # open plot figure window
plt.clf() # clear figure
plt.plot(x1, f1, 'r-') # connect points with a blue line
plt.plot(x1, g1, 'b-') # connect points with a blue line
# add intersections
Example use:
>>> run intersections
"""
import newton
import matplotlib.pyplot as plt
import numpy as np
guesses = np.array([-2.2, -1.6, -0.8, 1.4])
xarray = np.zeros(4)
yarray = np.zeros(4)
i = 0
for xin in guesses:
(xout, iters) = newton.solve(xin)
xarray[i] = xout
yarray[i] = (1-(0.6*xout**2))
i = i+1
x = np.linspace(-5,5,1000)
g1 = x*np.cos(np.pi*x)
g2 = (1-(0.6*x**2))
plt.figure(1)
plt.clf()
plt.xlim(-5,5)
plt.xlabel('x (-)')
plt.ylabel('y (-)')
g2x,g2px = g2(x)
fx = g1x - g2x
fpx = g1px - g2px
return fx, fpx
# Points
xi = linspace(-5,5,1000)
g1xi, g1pxi = g1(xi)
g2xi, g2pxi = g2(xi)
fxi, fpxi = f(xi)
# Functions
plt.figure(1)
plt.clf()
plt.plot(xi, g1xi, 'b')
plt.plot(xi, g2xi, 'r')
plt.legend(['g1(x)=x*cos(pi*x)','g2(x)=1-0.6*x**2'])
plt.figure(1)
# Intersection points
for x0 in [-2.2, -1.6, -0.8, 1.45]:
x,iters = solve(f, x0)
print "\nWith initial guess x0 = %22.15e," % x0
print " solve returns x = %22.15e after %i iterations " % (x,iters)
g1xi,g1pxi = g1(x)
fxi,fpxi = f(x)
print " f(x) = %22.15e" % fxi
plt.plot(x,g1xi,'ko')
plt.savefig('intersections.png')
"""
from newton import solve
import numpy as np
from numpy import pi, cos, sin
import pylab
guesses = np.array([-2.15, -1.7, -.9, 1.3])
solns = np.zeros(4)
def fvals_intersect(x):
"""
Return f(x) and f'(x) for f(x) = x*cos(pi*x)
"""
f = x * cos(pi * x) - 1 + 0.6 * x**2
fp = -x * sin(pi * x) * pi + cos(pi * x) + 1.2 * x
return f, fp
for j in range(len(guesses)):
soln, i = solve(fvals_intersect, guesses[j])
solns[j] = soln
print 'With initial guess x0 = %22.15e,' % guesses[j]
print '\tsolve returns x = %22.15e after %i iterations' % (soln, i)
#Begin Plotting
x = pylab.linspace(-5, 5, 1000)
fsolns = solns * cos(pi * solns)
pylab.plot(x, x * cos(pi * x), x, 1 - .6 * x**2, solns, fsolns, 'ko')
pylab.savefig('intersections.png')
xx = linspace(-10,10,1000)
g1xx,g1pxx = g1vals(xx)
g2xx,g2pxx = g2vals(xx)
plt.figure(1)
plt.clf()
plt.plot(xx,g1xx,'b')
plt.plot(xx,g2xx,'r')
plt.legend(['g1','g2'])
plt.figure(2)
plt.clf()
fxx,fpxx=fvals(xx)
plt.plot(xx,fxx,'b')
plt.plot(xx,0*xx,'k') # x axis
plt.title("plot of f(x)=g2(x)-g1(x)")
plt.figure(1)
for x0 in [-2.2,-1.6,-0.8,1.45]:
x,iters = solve(fvals,x0)
print "\nWith initial guess x0 = %22.15e, " % x0
print " solve returns x = %22.15e after %i iterations " % (x,iters)
g1x,g1px = g1vals(x)
fx,fpx = fvals(x)
print " f(x)=%22.15e" % fx
plt.plot(x,g1x,'ko')
plt.axis((-5,5,-4,4))
plt.savefig('intersections.png')
xx = linspace(-10, 10, 1000)
g1xx, g1pxx = g1vals(xx)
g2xx, g2pxx = g2vals(xx)
plt.figure(1)
plt.clf()
plt.plot(xx, g1xx, 'b')
plt.plot(xx, g2xx, 'r')
plt.legend(['g1', 'g2'])
plt.figure(2)
plt.clf()
fxx, fpxx = fvals(xx)
plt.plot(xx, fxx, 'b')
plt.plot(xx, 0 * xx, 'k') # x-axis
plt.title("plot of f(x) = g2(x) - g1(x)")
plt.figure(1)
for x0 in [-2.2, -1.6, -0.8, 1.45]:
x, iters = solve(fvals, x0)
print "\nWith initial guess x0 = %22.15e," % x0
print " solve returns x = %22.15e after %i iterations " % (x, iters)
g1x, g1px = g1vals(x)
fx, fpx = fvals(x)
print " f(x) = %22.15e" % fx
plt.plot(x, g1x, 'ko')
plt.axis((-5, 5, -4, 4))
plt.savefig('intersections.png')
f = g1(x) - g2(x)
fp = np.cos(np.pi * x) + -1 * np.pi * x * np.sin(np.pi * x) + 1.2 * (x)
return f, fp
if __name__ == "__main__":
print "Running test..."
x_guess = [-2, -1.6, -0.8, 1.4]
xfinal = []
plt.figure(1)
plt.clf()
x = np.linspace(-5, 5, 1000)
y1 = g1(x)
y2 = g2(x)
g1plot, = plt.plot(x, y1, label=r"$xcos(\pi x)$")
g2plot, = plt.plot(x, y2, label=r"$1-.6x^2$")
for x0 in x_guess:
print "With initial guess x0 = %22.16e " % x0
(xf, i) = solve(fval_g1_g2, x0)
print " \t solve returns x = %22.16e after %i iterations \n" % (xf, i)
xfinal.append(xf)
plt.plot(xfinal, map(g1, xfinal), 'ro', label='intersection')
plt.grid()
plt.legend()
plt.savefig('intersections.png')
def g1(x):
return x * cos(pi*x)
def g2(x):
return 1 - .6*x**2
def g1g2(x):
f = g1(x) - g2(x)
fp = cos(pi*x) - x*pi*sin(pi*x) + 1.2*x
return f, fp
##calculating the 4 zeros from the 4 initial guesses.
x_guesses = [-2.2,-1.6,-0.77,1.43]
zeros = []
for x0 in x_guesses:
x, i = solve(g1g2, x0, False)
zeros.append(x)
print "With initial guess x0 = %22.15e," %x0
print " solve returns x = %22.15e after %i iterations" %(x,i)
print
##plot the results
x = linspace(-5,5,500)
figure(1)
clf()
plot(x,g1(x))
plot(x,g2(x))
for i in range(4):
plot(zeros[i],g1(zeros[i]),'ko')
savefig('intersections.png')
Returns g1(x) - g2(x) and g1'(x) - g2'(x) """ f = fvals_g1(x)[0] - fvals_g2(x)[0] fp = fvals_g1(x)[1] - fvals_g2(x)[1] return f, fp ### Set parameters debug_solve = True init_guess = [-2.5, -1.7, -1., 1.5] pts_x = [] pts_y = [] ### Solving for intersections for x0 in init_guess: print " " x, iters = solve(fvals_f, x0, debug = debug_solve) print "With intial guess x0 = %22.15e," % x0 print " solve returns x = %22.15e after %i iterations " % (x,iters) fx = fvals_g1(x)[0] pts_x.append(x) pts_y.append(fx) ### Plotting two curves and intersection points x = linspace(-5,5,1000) y1 = fvals_g1(x)[0] y2 = fvals_g2(x)[0] figure(0) clf() plot(x, y1, 'b-', label = 'g1(x)') plot(x, y2, 'r-', label = 'g2(x)')
def print_intersections(fvals_func):
for x0 in [-2.20, -1.5, -0.7, 1.5]: # x0 taken looking at the plot
x, iters = solve(fvals_func, x0)
print(
'With initial guess x0 %20.15e,\n solve returns x = %20.15e after %i iterations'
% (x0, x, iters))
import numpy as np
from newton import solve
x = np.linspace(-10, 10, 1000)
y1 = x * np.cos(np.pi * x)
y2 = 1 - 0.6 * (x * x)
plt.figure(1)
plt.clf()
plt.plot(x, y1, 'b-')
plt.plot(x, y2, 'r-')
def fun(x):
f = x * np.cos(np.pi * x) - 1 + 0.6 * x**2
fp = np.cos(np.pi * x) - x * np.sin(np.pi * x) * np.pi + 1.2 * x
return f, fp
guesses = [-2.2, -1.6, -0.8, 1.4]
for x0 in guesses:
print " " # blank line
x, iters = solve(fun, x0)
print "solve returns x = %22.15e after %i iterations " % (x,iters)
fx1 = x * np.cos(np.pi * x)
fx2 = 1 - 0.6 * x**2
print "the value of f(x) is %22.15e" % (fx1 - fx2)
assert abs(fx1 - fx2) < 1e-14, "*** Unexpected result: x = %22.15e" % x
plt.plot(x, fx1, 'ko')
plt.savefig('intersections.png')
def gdiffvals(x):
"""
Differences betwen g1(x) and g2(x)
"""
return g1vals(x)[0] - g2vals(x)[0], g1vals(x)[1] - g2vals(x)[1]
x0 = [-3.0, -2.0, -1.0, -0.5, 1.0, 2.0, 1.6]
xvals = np.zeros(len(x0))
yvals = np.zeros(len(x0))
for i in range(0, len(x0)):
print " "
x, iters = newton.solve(gdiffvals, x0[i], debug=False)
xvals[i] = x
yvals[i] = g1vals(x)[0]
print "With initial guess x0 = %22.15e" % x0[i]
print " solve returns x = %22.15e after %s iterations" % (x, iters)
print " g1(x) - g2(x) = %22.15e" % gdiffvals(x)[0]
x = np.linspace(-5.0, 5.0, 1000)
y1 = g1vals(x)[0]
y2 = g2vals(x)[0]
plt.clf()
plt.plot(x, y1, "r-", x, y2, "b-", xvals, yvals, "ko")
plt.show()
plt.title("Intersections between g1(x) and g2(x)")
import matplotlib.pyplot as plt
def fvals(x):
"""
f and f prime for this homework
"""
#from numpy import pi,cos, sin
f = (x*cos(pi*x)) - (1. - 0.6*x*x)
fp = (cos(pi*x) - pi*x*sin(pi*x)) - (-1.2*x)
return f, fp
xlist =[]
for x0 in [-2.2, -1.5, -0.75, 1.5]:
print " " # blank line
x,iters = solve(fvals, x0, debug=True)
xlist.append(x)
print "solve returns x = %22.15e after %i iterations " % (x,iters)
fx,fpx = fvals(x)
print "the value of f(x) is %22.15e" % fx
x0s = array(xlist)
x1 = linspace(-5., 5., 1000)
f1 = x1*cos(pi*x1)
g1 = 1. - 0.6*x1*x1
plt.figure(1) # open plot figure window
plt.clf() # clear figure
plt.plot(x1,f1,'r-') # connect points with a blue line
plt.plot(x1,g1,'b-') # connect points with a blue line
# add intersections
of two known functions.
"""
from newton import solve
import numpy as np
from numpy import pi, cos, sin
import pylab
guesses = np.array([-2.15, -1.7,-.9,1.3])
solns = np.zeros(4)
def fvals_intersect(x):
"""
Return f(x) and f'(x) for f(x) = x*cos(pi*x)
"""
f = x*cos(pi*x) - 1 + 0.6*x**2
fp = -x*sin(pi*x)*pi + cos(pi*x) +1.2*x
return f, fp
for j in range(len(guesses)):
soln, i = solve(fvals_intersect, guesses[j])
solns[j]=soln
print 'With initial guess x0 = %22.15e,' % guesses[j]
print '\tsolve returns x = %22.15e after %i iterations' % (soln, i)
#Begin Plotting
x = pylab.linspace(-5,5,1000)
fsolns = solns*cos(pi*solns)
pylab.plot(x,x*cos(pi*x),x,1-.6*x**2,solns,fsolns,'ko')
pylab.savefig('intersections.png')
xspace = linspace(-10,10,1000)
g1x,g1px = g1vals(xspace)
g2x,g2px = g2vals(xspace)
#plot curves
plt.figure(1)
plt.clf()
plt.plot(xspace,g1x,'b')
plt.plot(xspace,g2x,'r')
plt.legend(['g1','g2'])
plt.figure(2)
plt.clf()
fx,fpx = fvals(xspace)
plt.plot(xspace,fx,'b')
plt.savefig('g2minusg1.png')
#plotting root points
plt.figure(1)
for x0 in [-2.2,-1.6,-0.8,1.45]:
(x,i) = newton.solve(fvals,x0)
print "\n Initial Guess: %22.15e" % x0
print " solve returned x = %22.15e after %i iterations " % (x,i)
g1xx,g1pxx = g1vals(x) # we could use g2 too
fxx,fpxx = fvals(x)
print " f(x) = %22.15e " %fxx
plt.plot(x,g1xx,'ko')
plt.axis( (-5,5,-4,4) )
plt.savefig('inter.png')
xspace = linspace(-10, 10, 1000)
g1x, g1px = g1vals(xspace)
g2x, g2px = g2vals(xspace)
#plot curves
plt.figure(1)
plt.clf()
plt.plot(xspace, g1x, 'b')
plt.plot(xspace, g2x, 'r')
plt.legend(['g1', 'g2'])
plt.figure(2)
plt.clf()
fx, fpx = fvals(xspace)
plt.plot(xspace, fx, 'b')
plt.savefig('g2minusg1.png')
#plotting root points
plt.figure(1)
for x0 in [-2.2, -1.6, -0.8, 1.45]:
(x, i) = newton.solve(fvals, x0)
print "\n Initial Guess: %22.15e" % x0
print " solve returned x = %22.15e after %i iterations " % (x, i)
g1xx, g1pxx = g1vals(x) # we could use g2 too
fxx, fpxx = fvals(x)
print " f(x) = %22.15e " % fxx
plt.plot(x, g1xx, 'ko')
plt.axis((-5, 5, -4, 4))
plt.savefig('inter.png')
