Householder's method for solving the nonlinear equation f(x) = 0 is In+1 = In +d (1/f)(d-)...

 

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Householder's method for solving the nonlinear equation f(x) = 0 is In+1 = In +d (1/f)(d-) (zn) (1/f)(x) starting from an initial guess ro. Various values of d give methods of order d+ 1 but require calculation of the dth derivative. Note: The notation (1/f) (rn) means computing the dth derivative of 1/f(r) and evaluate it at z = n When d = 1 the method is just Newton's method, while if d = 2 the method becomes Halley's method which has an order of convergence of 3 for simple roots: Xn+1 = n 1 (Kn)/ [F"(x) 1(2n)" (zn) - 2f'(x) (i) Write a short MATLAB function implementing Halley's method called Halley.m. Using this function repeat question (a) parts (i) and (ii) above. Comment on the number of iterations taken for Newton's Method and Halley's Method. (ii) In your ODEs course MS4403 you will have studied Sturm-Liouville problems. In some instances the solution requires you to numerically solve an equation to calculate the eigenvalues. One such problem is to find all positive solutions (or the first n at least) to = -tan(X). Use the Halley.m function you wrote to find the first two positive zeros of A=tan(A) to 12 decimal places. In order to identify good initial guesses for the roots first plot the function in MATLAB. Label your plot neatly. Householder's method for solving the nonlinear equation f(x) = 0 is In+1 = In +d (1/f)(d-) (zn) (1/f)(x) starting from an initial guess ro. Various values of d give methods of order d+ 1 but require calculation of the dth derivative. Note: The notation (1/f) (rn) means computing the dth derivative of 1/f(r) and evaluate it at z = n When d = 1 the method is just Newton's method, while if d = 2 the method becomes Halley's method which has an order of convergence of 3 for simple roots: Xn+1 = n 1 (Kn)/ [F"(x) 1(2n)" (zn) - 2f'(x) (i) Write a short MATLAB function implementing Halley's method called Halley.m. Using this function repeat question (a) parts (i) and (ii) above. Comment on the number of iterations taken for Newton's Method and Halley's Method. (ii) In your ODEs course MS4403 you will have studied Sturm-Liouville problems. In some instances the solution requires you to numerically solve an equation to calculate the eigenvalues. One such problem is to find all positive solutions (or the first n at least) to = -tan(X). Use the Halley.m function you wrote to find the first two positive zeros of A=tan(A) to 12 decimal places. In order to identify good initial guesses for the roots first plot the function in MATLAB. Label your plot neatly.