Question

Write a MATLAB function (IncNR) combining the capabilities of both the Incremental Search Method and the Newton-Raphson Method to find all the roots in a given function [xR, err, n] = IncNR(AF, xb, ed) AF = anonymous function xb = initial guess x bracket = [xL xU], where xL = lower x and xU = upper x ed = desired error Outputs: xR = vector of roots err = vector of errors corresponding to the roots n = vector of the number of iterations Suggested Steps: Use the Incremental Search Method to identify the number of roots and the brackets Use the Newton-Raphson Method in each of the brackets starting at either the lower domain or the upper domain

Answer 1

Ans: The matlab code for the question is :

function [xR,err,n,xRV,errV,AFD1,AFD2] = NewtonRaphsonNM(AF,xi,ed)

% AF = anonymous function equation:

% ed = desired approximate relative error = |(current - previous)/current|: ed = 0.01;

% xR = x root

% xRV = x root vector

% errV = approximate relative error vector

% AFD1 = anonymous function 1st derivative

% AFD2 = anonymous function 2nd derivative

%% Computations

% Derivatives

AFSYM = sym(AF); % symbolic math function

AFD1SYM = diff(AFSYM); % symbolic math function 1st derivative

AFD2SYM = diff(AFD1SYM); % symbolic math function 2nd derivative

AFD1 = matlabFunction(AFD1SYM); % anonymous function 1st derivative

AFD2 = matlabFunction(AFD2SYM); % anonymous function 2nd derivative

% Newton-Raphson Method

err = 1; % initial approximate relative error = 100%

k = 1; % initial counter

while ed < err % compares desired versus calculated error

xR = xi-(AF(xi)/AFD1(xi)); % x root using Newton-Raphson method

xRV(k+1) = xR; % stores the x root per iteration in a vector

err = abs((xRV(k+1) - xRV(k))/xRV(k+1)); % approximate relative error

errV(k+1) = err; % stores the error into a vector

xi = xR; % new guess x = xR, where xR = x root

k = k+1; % increase counter

end

n = k - 1; % number of iterations

xRV = xRV(2:end); % readjust x root vector

errV = errV(2:end); % readjust approximate relative error vector