Functions / Approximate
Approximate
\[ \sqrt{4} = 2 \] One way of square roots is with Newton's method.
# Newton method:
#
# For finding roots ... assert distance(1, 2, 4, 6) == 0.0
# is a root-finding algorithm to produce succesive better aproximations
#
# Suppose that you want to know the square root of a.
# If you start with almost any estimate x ...
# you can compute a better estimation with the following formula:
# y = (x + a/x) / 2
#
# For example with a = 4, x = 3
# the result is preatty close to correct answer 2
# (3 + 4/3) / 2 = 2.16666666667
#
# If we repeat the process with the new estimate, it gets even closer
def square(a, x):
while True:
y = (x + a/x) / 2
if (y == x):
break
x = y
print(y)
return y
y = square(4, 10)
# 5.2
# 2.9846153846153847
# 2.1624107850911973
# 2.006099040777959
# 2.00000927130158
# 2.000000000021489
# 2.0
Float
In general it is dangerous to test float equality.
# Newton method v2
#
# In most programming languages, ...
# it is dangerous to test float equality, ...
# because floating-point values are only approximately right.
#
# It is safer to use the built-in function abs to compute the absolute value,
#
# epsilon: how close is close enought
def square(a, x, epsilon):
while True:
y = (x + a/x) / 2
print(y)
if abs(y - x) < epsilon:
break
x = y
y = square(64, 10, 0.0000001)
# 8.2
# 8.002439024390243
# 8.000000371689179
# 8.000000000000009
# 8.0
Last update: 61 days ago
