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8/12/2019 8 - Derivacin Basada Recursividad de Duplex Mtodo Square
1/3
Recursion based Derivation of Duplex Square MethodVitthal Jadhav
Pune, Maharashtra, India.
Email: [email protected]
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 )+
8/12/2019 8 - Derivacin Basada Recursividad de Duplex Mtodo Square
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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
+ =
= + =