def broyden (func, x1, x2, tol = 1e-8):
"""
Implementation of the Broyden method finding roots to a multivariable func-
tion.
Inputs:
func: The vector function you wish to find a root for
x1: One initial guess vector
x2: A second initial guess vector
[tol]: optional. Tolerance level
Returns:
x such that func(x) = 0
"""
if abs(func(x1)) < tol:
return x1
if abs(func(x2)) < tol:
return x2
x1Array = sp.array(x1)
x2Array = sp.array(x2)
jaco = Jacobian(func)
f1 = func(x1)
f2 = func(x2)
deltax = (x2-x1)
Jacob = jaco.jacobian(deltax)
Jnew = Jacob + (f2- f1 - sp.dot(sp.dot(Jacob,deltax),deltax.T))\
/(la.norm(deltax)**2)
xnew = la.inv(Jnew) * (-f2) + x2
x1 = x2
x2 = xnew
Jacob = Jnew
i=0
while abs(la.norm(func(xnew))) > tol:
f1 = func(x1)
f2 = func(x2)
deltax = (x2-x1)
Jnew = Jacob + (f2- f1 - sp.dot(sp.dot(Jacob,deltax),deltax.T))\
/(la.norm(deltax)**2)
xnew = la.inv(Jnew) * (-f2) + x2
x1 = x2
x2 = xnew
Jacob = Jnew
i += 1
print i
return xnew
def systemNewton(function, x0, fPrime = None, tol = 0.0001):
epsilon = 1e-5
xArray = sp.array(x0, dtype = float)
funArray = sp.array(function)
numEqns = funArray.size
numVars = xArray.size
if numVars==1:
if fPrime:
return myNewNewton(function, x0, fPrime)
else:
return myNewNewton(function, x0)
else:
xold = xArray
jacfun = Jacobian(function)
jac0 = sp.mat(jacfun.jacobian(xold))
xnew = xold - la.solve(jac0,function(xold))
while max(abs(xnew-xold))> tol:
xold = xnew
jacNew = sp.mat(jacfun.jacobian(xold))
xnew = xold - la.solve(jacNew,function(xold))
return xnew