DEMSLV03

Compute fixedpoint of $f(x) = x^{0.5}$

Compute fixedpoint of $f(x) = x^{0.5}$ using Newton, Broyden, and function iteration methods.

Initial values generated randomly. Some alrorithms may fail to converge, depending on the initial value.

True fixedpoint is $x=1$.

from demos.setup import np, tic, toc
from compecon import NLP

Randomly generate starting point

xinit = np.random.rand(1) + 0.5

Set up the problem

def g(x):
    return np.sqrt(x)

problem_as_fixpoint = NLP(g, xinit)

Equivalent Rootfinding Formulation

def f(x):
    fval = x - np.sqrt(x)
    fjac = 1-0.5 / np.sqrt(x)
    return fval, fjac

problem_as_zero = NLP(f, xinit)

Compute fixed-point using Newton method

t0 = tic()
x1 = problem_as_zero.newton()
t1 = 100 * toc(t0)
n1 = problem_as_zero.fnorm

Compute fixed-point using Broyden method

t0 = tic()
x2 = problem_as_zero.broyden()
t2 = 100 * toc(t0)
n2 = problem_as_zero.fnorm

Compute fixed-point using function iteration

t0 = tic()
x3 = problem_as_fixpoint.fixpoint()
t3 = 100 * toc(t0)
n3 = np.linalg.norm(problem_as_fixpoint.fx - x3)

Print results

print('Hundredths of seconds required to compute fixed-point of g(x)=sqrt(x)')
print('using Newton, Broyden, and function iteration methods, starting at')
print('x = %4.2f\n' % xinit)
print(' Method       Time    Norm of f       x\n', '-' * 40)
ff = '%9s %8.2f    %8.0e     %5.2f'
print(ff % ('Newton', t1, n1, x1))
print(ff % ('Broyden', t2, n2, x2))
print(ff % ('Function', t3, n3, x3))

Hundredths of seconds required to compute fixed-point of g(x)=sqrt(x)
using Newton, Broyden, and function iteration methods, starting at
x = 1.03

 Method       Time    Norm of f       x
 ----------------------------------------
   Newton     0.53       7e-09      1.00
  Broyden     0.40       5e-11      1.00
 Function     0.40       7e-09      1.00