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Recursion based Derivation of Duplex Square MethodVitthal Jadhav

Pune, Maharashtra, India.

Email: Jadhavvitthal1989@gmail.com

Abstract

Duplex square method, introduced by Vedic mathematics, is well known faster method for

squaring of any number. This paper derives duplex method using recursive approach.

1. Method Derivation

i

i

S ..................n n n

S ..................n n

S ( .................. ) Sn n n n n

nLet a a a a a a

in

Then a a a a ai

a a a a a a

= + + + + + =

= + + + + =

= + + + + + = +

==

S Sn n n

a = +

. (1)

Squaring on both side, we get(S ) ( S ) ( ) * *S (S )

n n n n n n na a a= + = + +

i( S ) ( ) * * (S )

n n n n

na a a

i= + +

=

(2)

i( S = )

n

na

i

=

Similarly,

i

i

i

( S ) ( ) * * (S )n n n n

( S ) ( ) * * (S )n n n n

( S ) ( ) * *

na a a

i

na a ai

a a ai

= + +

= + +

= +

=

=

=

i

i

(S )+

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Thus, general recurrance relation given as

i( S ) ( ) * * (S ) 0 i n

n i n i n i n i

n ia a a

i= + +

=

.(3)

Also ( S ) ( )a=

(4)

Now using equation (3) and (4), we will expand equation (2) as below

i

i i

( S ) ( ) * * (S )n n n n

( S ) ( ) * * ( ) * * (S )n n n n n n

na a a

in n

a a a a a ai i

= + +

= + + + +

= = =

i i i

( S ) ( ) * * ( ) * * ( ) * * (S )

n n n n n n n n

n n na a a a a a a a a

i i i

= + + + + + +

= = =

i

i i

i i

( S ) ( ) * * ( ) * * ( )n n n n n n

* * ..... ( ) * * (a )n

n na a a a a a a

i in

a a a a ai i

= + + + +

+ + + + +

= =

= =

i

Rearranging, we get

i i

i i

i i i i

( S ) ( ) ( ) ( ) ..... ( ) (a ) * * * *n n n n n n

* * ..... * *n

( S ) ) ( * * * .....n n n n

n na a a a a a a a

i in

a a a ai i

n n n n(a a a a a a a

i i i i

= + + + + + + +

+ + +

= + + + +

= = = =

= = = =

i

* )a ai

n( S ) (a ) a a for 2 j ni jn i

i i j

+ =

= + =