In today's post we look at some more difficulties with numerical algorithms and their computations.
We know the roots of a quadratic with well known formulas.
The first problem is how do we compute the square root. There is a very neat algorithm by Newton which is iterative and converges pretty rapidly,
x0 = (1 +  D ) / 2 and x n+1 = (xn + D / xn ) / 2
 where D > 1 . In 4 iterations, error is within 10 ^ - 15.
Stopping  when we have reasonable values.
For eg: when we have sqrt(2)
we compute x0 = ( 1 + 2 ) / 2 = 1.5
x1  = (1.5 + 2 / 1.5 ) / 2 = 1.416667
x2 = ( x1 + 2 / x1 ) / 2 = 1.414216
and in 4th iteration, there is litttle or no change.
The roots of a quadratic as given by (-b + - Sqrt( b ^ 2 - 4.a.c ) ) / 2 . a
Here another problem is given by the fact that the numerator has a negative component and there is a cancellation possible. This is the substractive cancellation in the x2 numerator.
Take for example, x ^2  - 10 ^ 5 x + 1 = 0
x1 * = 999999.999990
x2  * = 0.000000000001

In this case, we work around it by rewriting the formula as the same multiplied by ( -b - sqrt( b ^2 - 4.a. c) / ( -b - sqrt( b ^2 - 4.a. c)  where the numerators on multiplication gets multiplied and squared and the denominator becomess both components negative and we have eliminated substractive cancelation in favor of addition.

If b < 0 we take x1 as before and x2 as 2c/ ( -b + sqrt( b ^ 2 - 4.a.c))
o summarize, numerical algorithms may suffer from
Truncation errors ( we use Finite Approach)
Round off erors ( we use Bounded Representation )
Overflow, underflow,
substractive cancellation,
ill conditional problems
Therefore they require careful analysis of errors.