Recursive Bit Algorithm for Digital Arithmetic
Introduction:
Vedic Mathematics:
• Vedic Mathematics deals mainly with 16 Sūtras and their applications for carrying out tedious and cumbersome arithmetical operations, and to a very large extent, executing them mentally.
• Nikhilam NavataścaramamDaśatahSūtra (or simply Nikhilam Sūtra) is one of these 16 Sūtras used for multiplication and has been successfully applied to overcome drawbacks of conventional schemes. This is where the fields of modern computing and Vedic Mathematics converge.
Need For Efficient Multiplication Algorithm:
• Use of numerical methods is prevalent in most software algorithms. Such applications demand an efficient code for basic mathematical operations, one of them being multiplication.
• Real Time Systems demand instantaneous response to environmental variables and quick execution of taken decision.
• Multiplication algorithms find applications in Digital Signal Processing (DSP) for discrete Fourier transforms, Fast Fourier transforms, convolution, digital filters, etc. Therefore any new multiplication algorithm opens up a new approach for improving existing schemes.
This calls for a ‘time efficient’ algorithm for ‘multiplication’ to improve processor throughput.
Explanation of Nikhilam Sūtra:
One of the 16 Sūtras of Vedic Mathematics, Nikhilam Sūtra stated algebraically as follows:
Consider two numbers n1 and n2such that n1 = (x – a), a = (x – n1) n2 = (x - b), b = (x – n2)
Where x = base, a,b= differences from base, then n1 x n2 = (x - a)(x - b) = x[(x – a) + (x – b) – x] + a*b
Thus the Nikhilam Sūtra effectively breaks up a large multiplication n1 x n2 into a small multiplication (a x b) and addition [(x – a) + (x – b) – x].
Decimal multiplication using Nikhilam Sūtra :
• 99 x 98 :
Base as power of 10, n1 and n2 less than base:
Consider n1 = 99 and n2 = 98
Number Difference Base x
(x – a) = 99 a = +01
100
(x – b) = 98 b = +02
(x – a) + (x – b) – x = 97 a*b = 02
Product: 9702
Algorithm for Radix 2
Vedic Mathematics:
• Vedic Mathematics deals mainly with 16 Sūtras and their applications for carrying out tedious and cumbersome arithmetical operations, and to a very large extent, executing them mentally.
• Nikhilam NavataścaramamDaśatahSūtra (or simply Nikhilam Sūtra) is one of these 16 Sūtras used for multiplication and has been successfully applied to overcome drawbacks of conventional schemes. This is where the fields of modern computing and Vedic Mathematics converge.
Need For Efficient Multiplication Algorithm:
• Use of numerical methods is prevalent in most software algorithms. Such applications demand an efficient code for basic mathematical operations, one of them being multiplication.
• Real Time Systems demand instantaneous response to environmental variables and quick execution of taken decision.
• Multiplication algorithms find applications in Digital Signal Processing (DSP) for discrete Fourier transforms, Fast Fourier transforms, convolution, digital filters, etc. Therefore any new multiplication algorithm opens up a new approach for improving existing schemes.
This calls for a ‘time efficient’ algorithm for ‘multiplication’ to improve processor throughput.
Explanation of Nikhilam Sūtra:
One of the 16 Sūtras of Vedic Mathematics, Nikhilam Sūtra stated algebraically as follows:
Consider two numbers n1 and n2such that n1 = (x – a), a = (x – n1) n2 = (x - b), b = (x – n2)
Where x = base, a,b= differences from base, then n1 x n2 = (x - a)(x - b) = x[(x – a) + (x – b) – x] + a*b
Thus the Nikhilam Sūtra effectively breaks up a large multiplication n1 x n2 into a small multiplication (a x b) and addition [(x – a) + (x – b) – x].
Decimal multiplication using Nikhilam Sūtra :
• 99 x 98 :
Base as power of 10, n1 and n2 less than base:
Consider n1 = 99 and n2 = 98
Number Difference Base x
(x – a) = 99 a = +01
100
(x – b) = 98 b = +02
(x – a) + (x – b) – x = 97 a*b = 02
Product: 9702
Algorithm for Radix 2
References: • Ajinkya Kale, ShaunakVaidya, AshishJoglekar – “A Generalized Recursive Algorithm for Binary Multiplication based on Vedic Mathematics ” , Proceedings National Conference on Trends in Computing Technologies – 2009 ,Pg – 37-43 • Harpreet Singh Dhillion ,AbhijitMitra – “ A Reduced Bit Multiplication Algorithm for Digital Arithmetic “ , International Journal of Computational and Mathematical Sciences, WASET, Spring 2008. T. K. Ghosh, A. K. Pal, “Computer Organization and Architecture” • Mano M. Morris, Computer Architecture and Design, Pearson Education Inc., South Asia Edition, 2007. • http://vedicmaths.org • http://en.wikipedia.org