Problem 2: Compare performance of Newton\'s method and Muller\'s method on the p

Question

Problem 2: Compare performance of Newton's method and Muller's method on the problem of finding roots of a polynomial with real co- efficients by the method of deflation. Write a code implementing deflation method for finding all roots of a polynomial using (a) Newton's method, b) Muller's method ·On the example of P(x) = 2.3 +4x2 +4x+3, show that Newton's ·On the example of P(r) = 2.3 + 4x2 + 4x + 3, show that Muller's ·Show that Newton's method gives correct roots of P(x) = 2.3 + method can not produce complex roots when starts from real initial approximation. method is not sensitive to a real initial approximation 4r2 +4x +3, if the initial approximation has a nonzero complex component. Write a report. Include results of the calculations done in previ ous items, add minor comments explaining the results

Explanation / Answer

SOL

p(x) = x^3+4x^2+4x+3;

actual roots of p(x) is   -3, -0.5-0.866j , -0.5+0.866j

newton method

x(n+1)=x(n)-f(x(n)) /f'(x(n))

as if we choose x(n) as initial guess to real number then f(xn) also real and f'(xn) ia also real

therfore xn+1 is always real till the iteration convergence to real number in this case -3

thus with the initial guess of real number we never going to find out the complex roots of polynomial

%%%% Matlab code %%%%

% Newton Raphson method
syms x
f= x^3+4*x^2+4*x+3;
tol=10^(-6);

x0=[-5,1,4]; %%% initial guess
for k=1:length(x0)
    s(1)=x0(k);
for n=1:100
    l1=subs(f,s(n));
    l2=subs(diff(f),s(n));
    s(n+1)=s(n)-(l1/l2);
    e=abs(s(n+1)-s(n));
    if (e < tol)
        break;
    end
  
end
fprintf('Initial guess x0= %f ', x0(k));
fprintf(' roots of equation is = %f ', s(end));
fprintf('Number of iteration required = %d ', n);
end

OUTPUT:

Initial guess x0= -5.000000
roots of equation is = -3.000000
Number of iteration required = 7
Initial guess x0= 1.000000
roots of equation is = -3.000000
Number of iteration required = 7
Initial guess x0= 4.000000
roots of equation is = -3.000000
Number of iteration required = 13

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.