Netwon's method (or any other iterative solving technique) with numpy

newton.py

from numpy.random import randint
from numpy import sqrt

# Newton's method:
# `x' is input, `y' is iteratively refined answer
# solve for square root of x: y**2 = x
# f (y) = y^2 - x 
# f'(y) = 2y
# y[1] = y[0] - f(y[0]) / f'(y[0])

xs = randint(1, 100, size=10).astype(float) # 10 numbers on [1,100)
ys = xs / 2                                 # first guess

unsolved = lambda xs, ys, accuracy = 0.01: abs(xs - ys ** 2) > accuracy

print '<{}>'.format(''.join('{:>4.0f}'.format(x) for x in xs))

us = unsolved(xs, ys)
while us.any():
    ys[us] = ys[us] - (ys[us] ** 2 - xs[us])/(2 * ys[us])
    us = unsolved(xs, ys)

    print ' {} '.format(''.join('  {} '.format(' ' if u else '✓') for u in us))

for x, y, z, e in zip(xs, ys, sqrt(xs), abs(sqrt(xs) - ys)):
    print '√{:2.0f} = {:>5.2f} = {:>5.2f} err: {:>6.4f}'.format(x, y, z, e)