Question

For what value of positive integer does the bisection search code fails?

As explained in the Lecture 3, for bisection method the code is pasted below
```
x = int(raw_input("Enter a Number to find the square root: "))
epsilon = 0.01
numOfGuesses = 0
low = 0
high = x
ans = (low + high)/2.0

while abs(ans**2 - x) > epsilon and ans <= x:
print "LOW: ", low, " HIGH: ", high, " ANS: ", ans
numOfGuesses += 1
if ans**2 < x:
low = ans
else:
high = ans
ans = (low + high)/2.0

print 'numOfGuesses =', numOfGuesses
print ans, 'is close to square root of', x
```

Answer 1

The code does work for all positive

Here it the output for integers for 0 to 9

>>>
numOfGuesses = 0
0.0 is close to square root of 0

LOW: 0 HIGH: 1 ANS: 0.5
LOW: 0.5 HIGH: 1 ANS: 0.75
LOW: 0.75 HIGH: 1 ANS: 0.875
LOW: 0.875 HIGH: 1 ANS: 0.9375
LOW: 0.9375 HIGH: 1 ANS: 0.96875
LOW: 0.96875 HIGH: 1 ANS: 0.984375
LOW: 0.984375 HIGH: 1 ANS: 0.9921875
numOfGuesses = 7
0.99609375 is close to square root of 1

LOW: 0 HIGH: 2 ANS: 1.0
LOW: 1.0 HIGH: 2 ANS: 1.5
LOW: 1.0 HIGH: 1.5 ANS: 1.25
LOW: 1.25 HIGH: 1.5 ANS: 1.375
LOW: 1.375 HIGH: 1.5 ANS: 1.4375
LOW: 1.375 HIGH: 1.4375 ANS: 1.40625
LOW: 1.40625 HIGH: 1.4375 ANS: 1.421875
numOfGuesses = 7
1.4140625 is close to square root of 2

LOW: 0 HIGH: 3 ANS: 1.5
LOW: 1.5 HIGH: 3 ANS: 2.25
LOW: 1.5 HIGH: 2.25 ANS: 1.875
LOW: 1.5 HIGH: 1.875 ANS: 1.6875
LOW: 1.6875 HIGH: 1.875 ANS: 1.78125
numOfGuesses = 5
1.734375 is close to square root of 3

numOfGuesses = 0
2.0 is close to square root of 4

LOW: 0 HIGH: 5 ANS: 2.5
LOW: 0 HIGH: 2.5 ANS: 1.25
LOW: 1.25 HIGH: 2.5 ANS: 1.875
LOW: 1.875 HIGH: 2.5 ANS: 2.1875
LOW: 2.1875 HIGH: 2.5 ANS: 2.34375
LOW: 2.1875 HIGH: 2.34375 ANS: 2.265625
LOW: 2.1875 HIGH: 2.265625 ANS: 2.2265625
LOW: 2.2265625 HIGH: 2.265625 ANS: 2.24609375
numOfGuesses = 8
2.236328125 is close to square root of 5

LOW: 0 HIGH: 6 ANS: 3.0
LOW: 0 HIGH: 3.0 ANS: 1.5
LOW: 1.5 HIGH: 3.0 ANS: 2.25
LOW: 2.25 HIGH: 3.0 ANS: 2.625
LOW: 2.25 HIGH: 2.625 ANS: 2.4375
LOW: 2.4375 HIGH: 2.625 ANS: 2.53125
LOW: 2.4375 HIGH: 2.53125 ANS: 2.484375
LOW: 2.4375 HIGH: 2.484375 ANS: 2.4609375
numOfGuesses = 8
2.44921875 is close to square root of 6

LOW: 0 HIGH: 7 ANS: 3.5
LOW: 0 HIGH: 3.5 ANS: 1.75
LOW: 1.75 HIGH: 3.5 ANS: 2.625
LOW: 2.625 HIGH: 3.5 ANS: 3.0625
LOW: 2.625 HIGH: 3.0625 ANS: 2.84375
LOW: 2.625 HIGH: 2.84375 ANS: 2.734375
LOW: 2.625 HIGH: 2.734375 ANS: 2.6796875
LOW: 2.625 HIGH: 2.6796875 ANS: 2.65234375
LOW: 2.625 HIGH: 2.65234375 ANS: 2.638671875
numOfGuesses = 9
2.6455078125 is close to square root of 7

LOW: 0 HIGH: 8 ANS: 4.0
LOW: 0 HIGH: 4.0 ANS: 2.0
LOW: 2.0 HIGH: 4.0 ANS: 3.0
LOW: 2.0 HIGH: 3.0 ANS: 2.5
LOW: 2.5 HIGH: 3.0 ANS: 2.75
LOW: 2.75 HIGH: 3.0 ANS: 2.875
LOW: 2.75 HIGH: 2.875 ANS: 2.8125
LOW: 2.8125 HIGH: 2.875 ANS: 2.84375
numOfGuesses = 8
2.828125 is close to square root of 8

LOW: 0 HIGH: 9 ANS: 4.5
LOW: 0 HIGH: 4.5 ANS: 2.25
LOW: 2.25 HIGH: 4.5 ANS: 3.375
LOW: 2.25 HIGH: 3.375 ANS: 2.8125
LOW: 2.8125 HIGH: 3.375 ANS: 3.09375
LOW: 2.8125 HIGH: 3.09375 ANS: 2.953125
LOW: 2.953125 HIGH: 3.09375 ANS: 3.0234375
LOW: 2.953125 HIGH: 3.0234375 ANS: 2.98828125
LOW: 2.98828125 HIGH: 3.0234375 ANS: 3.005859375
LOW: 2.98828125 HIGH: 3.005859375 ANS: 2.9970703125
numOfGuesses = 10
3.00146484375 is close to square root of 9

Answer 2

The program can be adapted to run for negative integers, as well for floats, but then there is a problem with the ans <= x: because then you're comparing floating points nrumbers. Here it will be ok, because 1 works out ok. Comparing a float and an integers might also lead to problems
In case of floating points
The other condition abs(ans**2 - x) > epsilon, won't work in the domain
x in (0,1) x²<x and if adapted for negative numbers it is in the domain x in (-1,0) also x²< |x|