Step by Step solution about How to resolve the algorithm Ethiopian multiplication step by step in the JavaScript programming language

Objective:

The provided JavaScript code defines two functions, eth.mult and ethMult, that perform fast integer multiplication using the "Ethiopian multiplication" method. This method was traditionally used in ancient Ethiopia and is known for its efficiency and simplicity.

Implementation Details:

A. eth.mult Function:

• The eth.mult function takes two integer arguments, a and b, and returns their product.
• It starts by initializing sum to 0 and converting a and b to arrays with a single element each.
• It then iteratively halves a and doubles b until a[0] becomes 1 (indicating that it's reached the limit).
• For each iteration, it checks if a[i] is odd. If it is, it adds b[i] to the sum.
• Finally, it adds b[0] to sum and returns the result.

B. ethMult Function:

• The ethMult function is a more concise implementation of the Ethiopian multiplication method.
• It handles both integers and strings (for cases like concatenating strings).
• It initializes o to an empty string if the input is a string, or 0 if it's an integer.
• It then iteratively performs the following steps:
  • If n is odd (determined by a bitwise AND operation with 1), it adds m to o.
  • It right-shifts n by 1, which is equivalent to integer division by 2.
  • It adds m to itself, which is equivalent to integer multiplication by 2.
• It repeats this process until n is less than or equal to 1.
• Finally, it returns the concatenation of o with m (for strings) or o + m (for integers).

Example Usage:

eth.mult(17, 34) // returns 578

-  ```javascript
ethMult(17, 34) // returns 578

ethMult('Ethiopian', 34) // returns 'EthiopianEthiopianEthiopian...' (34 times)

<div id="sourcecode"/>

## Source code in the javascript programming language

```javascript  
var eth = {
	
	halve : function ( n ){  return Math.floor(n/2);  },
	double: function ( n ){  return 2*n;              },
	isEven: function ( n ){  return n%2 === 0);       },
	
	mult: function ( a , b ){
		var sum = 0, a = [a], b = [b];
		
		while ( a[0] !== 1 ){
			a.unshift( eth.halve( a[0] ) );
			b.unshift( eth.double( b[0] ) );
		}
		
		for( var i = a.length - 1; i > 0 ; i -= 1 ){
			
			if( !eth.isEven( a[i] ) ){
				sum += b[i];
			}
		}		
		return sum + b[0];
	}
}
// eth.mult(17,34) returns 578


function ethMult(m, n) {
  var o = !isNaN(m) ? 0 : ''; // same technique works with strings
  if (n < 1) return o;
  while (n > 1) {
    if (n & 1) o += m;  // 3. integer odd/even? (bit-wise and 1)
    n >>= 1;            // 1. integer halved (by right-shift)
    m += m;             // 2. integer doubled (addition to self)
  }
  return o + m;
}

ethMult(17, 34)


ethMult('Ethiopian', 34)