# minte9 reminder

Last update:   19-11-2021

### Approximate

p 126 $\sqrt{4} = 2$ 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.
# 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

p 128 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


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