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Bairstow

The document contains MATLAB code implementing Bairstow's method for finding roots of a polynomial. It initializes polynomial coefficients, iteratively applies synthetic division, and updates guesses for roots until convergence is achieved or the maximum number of iterations is reached. The results include the quadratic factor and its roots, along with the remaining polynomial coefficients after deflation.

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wonwoong5
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22 views2 pages

Bairstow

The document contains MATLAB code implementing Bairstow's method for finding roots of a polynomial. It initializes polynomial coefficients, iteratively applies synthetic division, and updates guesses for roots until convergence is achieved or the maximum number of iterations is reached. The results include the quadratic factor and its roots, along with the remaining polynomial coefficients after deflation.

Uploaded by

wonwoong5
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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[Bairstow]

% MATLAB Code for Bairstow's Method


clear all;

% Define the coefficients of the polynomial f(x)


% Coefficients of f(x) Order: constant to high n
coeff = [1.25, -3.875, 2.125, 2.75, -3.5, 1];

% Initial guesses for r and s


r = -1;
s = -1;

% Error tolerance
epsilon_s = 0.01; % 1%

% Maximum iterations to prevent infinite loops


max_iter = 100;

% Number of iterations counter


iter = 0;

while true
iter = iter + 1;

% Synthetic division to calculate b and c arrays


n = length(coeff) - 1; % Degree of the polynomial
b = zeros(size(coeff));
c = zeros(size(coeff));

% Compute b array using synthetic division


b(n+1) = coeff(n+1);
b(n) = coeff(n) + r * b(n+1);
for i = n-1:-1:1
b(i) = coeff(i) + r * b(i+1) + s * b(i+2);
end
fprintf('b values are: [');
fprintf('%f ', b);
fprintf(']\n');

% Compute c array using synthetic division on b array


c(n+1) = b(n+1);
c(n) = b(n) + r * c(n+1);
for i = n-1:-1:2
c(i) = b(i) + r * c(i+1) + s * c(i+2);
end
fprintf('c values are: [');
fprintf('%f ', c);
fprintf(']\n');

% Update r and s using Newton-Raphson method


det = c(3)*c(3) - c(2)*c(4); % Determinant of Jacobian matrix

if abs(det) < eps % Prevent division by zero


error('Determinant too small, method failed.');
end

dr = (-b(2)*c(3) + b(1)*c(4)) / det;


ds = (-b(1)*c(3) + b(2)*c(2)) / det;

r_new = r + dr;
s_new = s + ds;

% Check for convergence


err_r = abs(dr / r_new) * 100; % Relative error in r
err_s = abs(ds / s_new) * 100; % Relative error in s

if err_r < epsilon_s && err_s < epsilon_s


break;
end

if iter >= max_iter


error('Maximum iterations reached without convergence.');
end

% Update r and s for next iteration


r = r_new;
s = s_new;
end

% Display results for quadratic factor found (x^2 + rx + s)


fprintf('Converged after %d iterations.\n', iter);
fprintf('Quadratic factor: x^2 + (%.4f)x + (%.4f)\n', r, s);

% Use quadratic formula to find roots of x^2 + rx + s


root_1 = (r + sqrt(r^2 + 4*s)) / 2;
root_2 = (r - sqrt(r^2 + 4*s)) / 2;

fprintf('1st Root of quadratic factor: %.4f + %.4f i\n', real(root_1), imag(root_1));


fprintf('2nd Root of quadratic factor: %.4f + %.4f i\n', real(root_2), imag(root_2));

% Deflate the polynomial by dividing by (x^2 + rx + s)


remaining_poly_coeffs = b(3:end);

fprintf('Remaining polynomial coefficients after deflation:\n');


disp(remaining_poly_coeffs);

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