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# Numerical methods challenge: Day 3

During October (2017) I will write a program per day for some well-known numerical methods in both Python and Julia. It is intended to be an exercise then don't expect the code to be good enough for real use. Also, I should mention that I have almost no experience with Julia, so it probably won't be idiomatic Julia but more Python-like Julia.

## Newton's method

Today's method is Newton's method. This method is used to solve the equation $f(x) = 0$ for $x$ real, and $f$ is a differentiable funciton. It starts with an initial guess $x_0$ and it succesively refine it by finding the intercept of the tangent line to the function with zero. The new approximation is computed from the old one using

\begin{equation*} x_k = x_{k-1} - \frac{f(x)}{f'(x)} \end{equation*}

Convergence of this method is generally faster than bisection method. Nevertheless, the convergence is not guaranteed. Another drawback is the need of the derivative of the function.

We will use the function $f(x) = \cos(x) - x$ to test the codes, and the initial guess is 1.0.

Following are the codes.

### Python

from __future__ import division, print_function
from numpy import abs, cos, sin

def newton(fun, grad, x, niter=50, ftol=1e-12, verbose=False):
msg = "Maximum number of iterations reached."
for cont in range(niter):
x = None
msg = "Derivative near to zero."
break
if verbose:
print("n: {}, x: {}".format(cont, x))
if abs(fun(x)) < ftol:
msg = "Root found with desired accuracy."
break
return x, msg

def fun(x):
return cos(x) - x

return -sin(x) - 1.0



### Julia

function newton(fun, grad, x, niter=50, ftol=1e-12, verbose=false)
msg = "Maximum number of iterations reached."
for cont = 1:niter
x = nothing
msg = "Derivative near to zero."
break
end
if verbose
println("n: $(cont), x:$(x)")
end
if abs(fun(x)) < ftol
msg = "Root found with desired accuracy."
break
end
end
return x, msg
end

function fun(x)
return cos(x) - x
end

return -sin(x) - 1.0
end



### Comparison

Regarding number of lines we have: 28 in Python and 32 in Julia. The comparison in execution time is done with %timeit magic command in IPython and @benchmark in Julia.

For Python:

%timeit newton(fun, grad, 1.0)


with result

10000 loops, best of 3: 27.3 µs per loop


For Julia:

@benchmark newton(fun, grad, 1.0)


with result

BenchmarkTools.Trial:
memory estimate:  48 bytes
allocs estimate:  2
--------------
minimum time:     327.925 ns (0.00% GC)
median time:      337.956 ns (0.00% GC)
mean time:        351.064 ns (0.80% GC)
maximum time:     8.118 μs (92.60% GC)
--------------
samples:          10000
evals/sample:     226