Hosted by CoCalc
 Download
import numpy
import matplotlib

%python
def f(x):
    return numpy.array([x[0]**2 + x[1]**2 - 1, x[0] - x[1]])
def J(x):
    return numpy.matrix([[2*x[0], 2*x[1]],[1,-1]])

x1 = numpy.linspace(-2,2,100)
x2 = numpy.linspace(-2,2,100)
[xx1,xx2] = numpy.meshgrid(x1,x2)
fig = matplotlib.pyplot.figure()
matplotlib.pyplot.axis('equal')
ax = fig.gca()
ax.contour(xx1,xx2,xx1**2 + xx2**2 - 1,colors='k',levels=[0])
ax.contour(xx1,xx2,xx1 - xx2,colors='b',levels=[0])

x = numpy.array([1,0])
converged = False
iter = 0
tol = 1e-6
maxit = 2
while not converged:
    ax.plot(x[0],x[1],marker='o',color='red')
    p = numpy.linalg.solve(J(x),f(x))
    x = x - p
    iter = iter + 1
    converged = (numpy.linalg.norm(p)<tol) or (iter > maxit)
    print(iter,numpy.linalg.norm(p))
matplotlib.pyplot.show()
(-0.059999999999999998, 0.059999999999999998, -0.059999999999999998, 0.059999999999999998) <matplotlib.contour.QuadContourSet object at 0x7fd1b38a3810> <matplotlib.contour.QuadContourSet object at 0x7fd1b37ce650> [<matplotlib.lines.Line2D object at 0x7fd1b37ebcd0>] (1, 1.0) [<matplotlib.lines.Line2D object at 0x7fd1b37ce650>] (2, 0.35355339059327379) [<matplotlib.lines.Line2D object at 0x7fd1b37ce710>] (3, 0.05892556509887896) 
%python
def f(x):
    return numpy.array([x[0]**2 + x[1]**2 - 2, x[0] - x[1], x[0] + x[1] - 2])
def J(x):
    return numpy.matrix([[2*x[0], 2*x[1]],[1,-1],[1,1]])

x1 = linspace(-2,2,100)
x2 = linspace(-2,2,100)
s = linspace(-.1,.1,10)
[xx1,xx2] = numpy.meshgrid(x1,x2)
fig = matplotlib.pyplot.figure()
matplotlib.pyplot.axis('equal')
ax = fig.gca()
ax.contour(xx1,xx2,xx1**2 + xx2**2 - 2,colors='k',levels=[0])
ax.contour(xx1,xx2,xx1 - xx2,colors='b',levels=[0])
ax.contour(xx1,xx2,xx1 + xx2 - 2,colors='g',levels=[0])
x = numpy.array([1,0])
converged = False
iter = 0
tol = 1e-10
maxit = 10
while not converged:
    a = ax.plot(x[0],x[1],marker='o',color='red')
    p = linalg.lstsq(J(x),f(x))[0]
    x = x - p
    iter = iter + 1
    converged = (linalg.norm(p)<tol) or (iter > maxit)
    print(iter,linalg.norm(p))
matplotlib.pyplot.show()
(-0.059999999999999998, 0.059999999999999998, -0.059999999999999998, 0.059999999999999998) <matplotlib.contour.QuadContourSet object at 0x7f9eb7c9b8d0> <matplotlib.contour.QuadContourSet object at 0x7f9eb4a82ad0> <matplotlib.contour.QuadContourSet object at 0x7f9eb777f250> (1, 1.0540925533894594) (2, 0.31119636958012253) (3, 0.028027720048995688) (4, 0.000223025653887907) (5, 1.406916519038287e-08) (6, 0.0) 

%python
def f(x):
    return numpy.array([x[0]**2 + x[1]**2 - 2])
def J(x):
    return numpy.matrix([[2*x[0], 2*x[1]]])

x1 = linspace(-2,2,100)
x2 = linspace(-2,2,100)
[xx1,xx2] = numpy.meshgrid(x1,x2)
fig = matplotlib.pyplot.figure()
ax = fig.gca()
matplotlib.pyplot.axis('equal')
ax.contour(xx1,xx2,xx1**2 + xx2**2 - 2,colors='k',levels=[0])
x = numpy.array([1,0.5])
converged = False
iter = 0
tol = 1e-10
maxit = 10
while not converged:
    a = ax.plot(x[0],x[1],marker='o',color='red')
    p = linalg.lstsq(J(x),f(x))[0]
    x = x - p
    iter = iter + 1
    converged = (linalg.norm(p)<tol) or (iter > maxit)
    print(iter,linalg.norm(p))
matplotlib.pyplot.show()
(-0.059999999999999998, 0.059999999999999998, -0.059999999999999998, 0.059999999999999998) <matplotlib.contour.QuadContourSet object at 0x7f9eb4a82ad0> (1, 0.33541019662496846) (2, 0.038701176533650287) (3, 0.00052934739939640679) (4, 9.9068718271003633e-08) (5, 3.4542034091042956e-15) 
%python
def f(x):
    return numpy.array([x[0]**2 + x[1]**2 - 2, x[0] - x[1], x[0] - 1.2])
def J(x):
    return numpy.matrix([[2*x[0], 2*x[1]],[1,-1],[1,0]])

x1 = linspace(-2,2,100)
x2 = linspace(-2,2,100)
s = linspace(-.1,.1,10)
[xx1,xx2] = numpy.meshgrid(x1,x2)
fig = matplotlib.pyplot.figure()
matplotlib.pyplot.axis('equal')
ax = fig.gca()
ax.contour(xx1,xx2,xx1**2 + xx2**2 - 2,colors='k',levels=[0])
ax.contour(xx1,xx2,xx1 - xx2,colors='b',levels=[0])
ax.contour(xx1,xx2,xx1 - 1.2,colors='g',levels=[0])
x = numpy.array([1,0])
converged = False
iter = 0
tol = 1e-10
maxit = 10
while not converged:
    a = ax.plot(x[0],x[1],marker='o',color='red')
    p = linalg.lstsq(J(x),f(x))[0]
    x = x - p
    iter = iter + 1
    converged = (linalg.norm(p)<tol) or (iter > maxit)
    print(iter,linalg.norm(p))
matplotlib.pyplot.show()
(-0.059999999999999998, 0.059999999999999998, -0.059999999999999998, 0.059999999999999998) <matplotlib.contour.QuadContourSet object at 0x7f9eb6144c50> <matplotlib.contour.QuadContourSet object at 0x7f9eb758fed0> <matplotlib.contour.QuadContourSet object at 0x7f9eb70eb790> (1, 1.5057224179774977) (2, 0.52405083388102425) (3, 0.086759087166500354) (4, 0.001773477548524111) (5, 1.4448587341850866e-05) (6, 1.2640024582108905e-07) (7, 1.1207583193209142e-09) (8, 1.1556340073250288e-11)