Numerical Methods Using MATLAB - Part 5

Graeffe's Root Squaring Method:
           This is a direct method and it is used to find the roots of a polynomial equation with real coefficients. Let us consider an equation of the form:

                         ax3 + bx^2 + cx + d = 0

To find the solution of the above equation, we fill up the following table:

		m		2^m		n1		n2		n3		n4
Given Equation		0		1		a		b		c		d
						a^2		b^2		c^2		d^2
						0		-2*a*c		-2*b*d		0
First Squaring		1		2		a^2		b^2 - 2*a*c		c^2 - 2*b*d		d^2

              Thus, the table goes on. The number of squaring done depends on the required accuracy of the solution. The roots are calculated by using the final values of n1, n2, n3 and n4 as follows:

                   a1^(2^m)=n2/n1

               a2^(2^m)=n3/n2

               a3^(2^m)=n4/n3

          From the above equations, the roots a1, a2, a3 can be calculated.The MATLAB code for solving a polynomial by the above method is given below:

Source Code: 

    % a function that uses Graeffe's method 

    % to calculate roots of Algebraic equation 

    function GraeffeFunc(co,gcount)

             % variable b contains the size of co

             % co  coeff/: vector

             [a b]=size(co);

  

             % initialize count to zero

             count = 0;

            % loop where elements of co vector are squared

            % step by step for further processing

            while count<gcount

                    for i=1:b

                         % for other coeff.

                         if i~=1 && i~=b

                              co1(i)=(co(i)^2)-(2*co(i-1)*co(i+1));

                              % for 1st and last coeff.

                        else

                              co1(i)=co(i)^2;

                        end

                    end

                    count=count+1;

                    co=co1;

           end

           % end of while

           % Finding the solution by operating on co1 vector values

           for i=2:b

               sol(i-1)=(co1(i)/co1(i-1))^(1/(2^gcount));

           end

          % displaying calculated values

          disp(co1);

          disp(sol);

    end

    % end of program

Coming up next:

       * Solution of Simultaneous Linear Algebraic Equations

Posted in: Graeffe's method,Matlab,Numerical Methods,Root Squaring method,Seeker1.0