The Division Algorithm: How to Divide Numbers

A division algorithm is one where two integers N and D, are given; and used to calculate the quotient and the remainder. A computation involving a division algorithm can be done by hand, and others are applied in digital circuit designs. The algorithm falls into two major groups: fast division and slow division. Fast division begins with a close estimation till the last quotient. On each iteration of the fast division, twice as many digits of the final quotient are produced. Examples of the fast division include Newton-Raphson and Goldschmidt algorithms. Contrary to fast division, the slow division has one number of the last quotient on each iteration. Examples of slow division methods include; restoring, non-restoring, non-performing restoring, and SRT algorithm (Pantankar et al., 2021).

The division algorithm is the N/D= (Q/R), where N/D is the input and N is the Numerator, and D is the Denominator. Q/R is the output, where Q is the quotient and R is the remainder. Also, the division algorithm formula is: Dividend = (Divisor X Quotient) + Remainder, p(x) = q(x) × g(x) + r(x)

The following are examples of division algorithm:

Example 1

For a polynomial division,

The divisor g(x) = 3x-2

The quotient q(x) = 6x²+4

The remainder r(x) = 5

Find the dividend p(x).

Solution

We know that; p(x) = q(x) × g(x) + r(x)

By substituting the above values,

P(x) = (6x²+4) × (3x-2) + (5)

And by multiplying q(x) and g(x), the result is,

(18x³+12x-12x²-8) +5

18x³+12x-12x²-8+5

Hence, the dividend p(x) is 18x³+12x-12x²-8+5

Example 2

Let us solve cos x = 2x to 5 decimal places.

This is equivalent to solving f(x) = 0 where f(x) = cos x − 2x.

[NB: make sure your calculator is in radian mode].

The recursion formula (1) becomes xn+1 = xn − (cos xn − 2xn) (− sin xn − 2) With an initial guess of x0 = 0.5, we obtain: x0 = 0.5 x1 = 0.45063 x2 = 0.45018 x3 = 0.45018 . . . with no further changes in the digits, to five decimal places.

Therefore, to this degree of accuracy, the root is x = 0.45018.

I chose the above examples because they illustrate the computation of the division algorithm in different formats. Moreover, the second example is the Newton Raphson algorithm which is in the category of fast division.

References

Patankar, U. S., & Koel, A. (2021). Review of Basic Classes of Dividers Based on Division Algorithm.  IEEE Access ,  9 , 23035-23069.