## Introduction

For those who are looking for how square root algorithm works and “YES !, Welcome to my article”, but for those who are simply looking for performing square root on your work and I have to tell you, what you have to do is some Googling on “*math.h*” header file to look for square root function that prepared inside it. Well, let’s begin our lesson “How square root algorithm works on C”. It’s not very difficult once you understand how the following equation is calculated.

The left hand side of the diagram shows the square root of 152.2756 and the right hand side shows the square root of 2. For those who understand how this calculation has been done, please skip to the code as I am going to explain how to solve this equation.

### Basic Principle of Square Root

Based on the question given above, we understand that square root of 152.2756 is 12.34 and the square root of 2 is 1.4142. If you do not believe, try calculator to find the answer.

Suppose we are finding the square root of 2 and the answer is 1.4142. We can express it such that:

If we expressed it in an algebra expression as follows:

Suppose a is continued n times, it can be expressed as follows:

Why I am treating it continue as n times but not 5 times? Because we cannot ensure that all positive numbers will give us a 5 digits answer after square root it. So we have to suppose there are n digits after square root. Right-hand-side term can be expanded as follows:

From the equation above, we can understand square root equation can be solved by following equation:

The most important rule of this equation is we pick a number from 1,2,3,4,5,6,7,8,9 and multiple it with 10^m , where m=Integer (exp. 3,2,1,0,-1,-2,...) and substitute into:

that must be smaller than the number we are going to minus or equal to it. It’s the most difficult part on square-root equation so let's understand it by solving square root of 2.

So, from the solution above, we understand that:

And we sum up all the values, we get the answer 1.414 approximate to 1.4142.

### Algorithm

Based on the method above, what kind of work should we apply on algorithm as follows:

- Create equation
- Create function of power of ten
- Find suitable number that is bigger than 0 for variable
**a** from loop
- Find suitable number that is smaller than 0 but positive for variable
**a** from loop
- Sum all variable
**a **and return it as `double`

First, we are going to create the equation as we mentioned above:

We can express it as code as follows:

(( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i))

where **rst** and **j** is declared as `double `

and **i** is `int`

. `powerOfTen `

is a function that return a value of `i`

times multiple of `10`

.

It’s code of function `powerOfTen`

:

1:
2: double powerOfTen(int num){
3: double rst = 1.0;
4: if(num >= 0){
5: for(int i = 0; i < num ; i++){
6: rst *= 10.0;
7: }
8: }else{
9: for(int i = 0; i < (0 - num ); i++){
10: rst *= 0.1;
11: }
12: }
13:
14: return rst;
15: }

We have to judge that variable **a **must be smaller than upper number. It’s how it looks like:

if(z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)

where **z** is declared as `double `

and treated as upper number.

There are two loops, one loop for finding the suitable number from 10000,1000,100,10… and another one loop for find suitable number from 1,2,3,4,5…

1: for(i = max ; i > 0 ; i--){
2: 3: if(z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
4: {
5: while( z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
6: {
7: j++;
8: if(j >= 10) break;
9:
10: }
11: j--; 12: z -= (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)); 13:
14: rst += j * powerOfTen(i); 15:
16: j = 1.0;
17:
18:
19: }
20:
21: }

This is the same loop as above but this time, we are going to find the decimal number.

1: for(i = 0 ; i >= 0 - max ; i--){
2: if(z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
3: {
4: while( z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
5: {
6: j++;
7: if(j >= 10) break;
8: }
9: j--;
10: z -= (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)); 11:
12: rst += j * powerOfTen(i); 13: j = 1.0;
14: }
15: }

This is how the whole `squareRoot `

algorithm looks like:

1: double squareRoot(double a)
2: {
3:
8: double z = a;
9: double rst = 0.0;
10: int max = 8; 11: int i;
12: double j = 1.0;
13: for(i = max ; i > 0 ; i--){
14: 15: if(z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
16: {
17: while( z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
18: {
19: j++;
20: if(j >= 10) break;
21:
22: }
23: j--; 24: z -= (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)); 25:
26: rst += j * powerOfTen(i); 27:
28: j = 1.0;
29:
30:
31: }
32:
33: }
34:
35: for(i = 0 ; i >= 0 - max ; i--){
36: if(z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
37: {
38: while( z - (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)) >= 0)
39: {
40: j++;
41:
42: }
43: j--;
44: z -= (( 2 * rst ) + ( j * powerOfTen(i)))*( j * powerOfTen(i)); 45:
46: rst += j * powerOfTen(i); 47: j = 1.0;
48: }
49: }
50: 51: return rst;
52: }

## Reference

Thank you.