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Eucledian algorithm for gcd of integers and polynomials

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Eucledian Algorithm for GCD of Integers and Polynomials

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Eucledian algorithm for gcd of integers and polynomials

  2. 2. CONTENTS 2  Introduction  GCD  Euclidean GCD  Applications of Euclidean Algorithm  Flowchart  MATlab code for integers  GCD for Polynomials.  MATlab Functions  Future work
  3. 3. GCD 3  In mathematics, the greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder.  The GCD is also known as the greatest common factor (gcf), highest common factor (hcf), greatest common measure (gcm), or highest common divisor.
  4. 4. Euclidean GCD 4  It is named after the ancient Greek mathematician Euclid, who first described it in Euclid’s Elements (c. 300 BC)  Efficient algorithm to calculate GCD is Euclidean GCD.  Uses a division Algorithm.  The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number.
  5. 5. CONTD… 5  The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.
  6. 6. Applications of Euclidean Algorithm 6  The Euclidean algorithm has many theoretical and practical applications.  It is used for reducing fractions to their simplest form and for performing division in modular arithmetic.  cryptographic protocols that are used to secure internet communications.  The Euclidean algorithm may be used to solve Diophantine equations.  It is a basic tool for proving theorems in number theory such as Lagrange’s four-square theorem and the uniqueness of prime factorizations.
  7. 7. Euclidean Algorithm: 7  GCD(a,b) where a and b are integers.  For all a , b with a > b there is a q (quotient) and r (remainder) such that a = qb + r with r < b or r = 0  This is calculated repeatedly by making a=b and b=r until r=0.  Finally, GCD=b.
  8. 8. Properties of GCD: 8  Gcd(a,0)=a  Gcd(a,a)=a  Gcd(a,b)=gcd(b,a mod b)  a mod b=a-b [floor(a/b)]
  9. 9. FLOWCHART 9 Fig.1: Flowchart of Euclidean GCD calculation
  10. 10. MATLAB CODE 10 clear; clc a = input('First number: '); b = input('Second number: '); a = abs(a); b = abs(b); r = a - b*floor(a/b); while r ~= 0 a = b; b = r; r = a - b*floor(a/b); end GCD = b
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  13. 13. 13 GCD OF POLYNOMIALS Definition: The greatest common divisor of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the original polynomials. • In of univariate polynomials , the polynomial GCD can be computed, like that for the integer GCD, by Euclid's algorithm using long division. • The similarity between the integer GCD and the polynomial GCD allows us to extend some properties of integer GCD to polynomial GCD. Property: The roots of the GCD of two polynomials are the common roots of the two polynomials. This allows to get information on the roots without computing them.
  14. 14. 14 FLOWCHART:
  15. 15. 15 Example:
  16. 16. 1) De-convolution(p1,p2)=polynomial division(p1/p2) 16 Polynomial Division: Q P2 P1 R
  17. 17. 17 MATLAB deconv command execution:
  18. 18. 18 function b = polystab(a) r = roots(a); v = find(abs(r)>1); //finds indices of the vector “r” whose absolute value is greater than 1 r(v) = 1./conj(r(v)); b = a(1) * poly ( r ); if isreal(a), b = real(b); endif endfunction 2) Polystab(p1): Stabilizing the polynomial transfer function
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  21. 21. 21 FUTURE WORK: • Rectify errors in MATLAB code for GCD of polynomials. • Write Verilog code for GCD of two integers and dump the same into FPGA. • Write Verilog code for GCD of two Polynomials.
  22. 22. 22 Thank you!!