Question

1) (80pts) Consider the following function f(x)--x5-4x4 + 2x3 + x2-3x + 5 Develop a simple program which will give an iterative solution to the problem f(x)=0 by Newton's algorithm. The solution should display the results on an Excel spreadsheet such as the one given below (the example below is given for a different function). Choose the programming language which suits you most, however the program should be able to read data from an Excel spreadsheet and write the successive results (for all iterations) on the same spreadsheet just as given in the example below. Please use x 2 as your initial solution. What is the value of x which solves this problem? 0.0 0.4444 -0.5063 0.5110 -0.5110 0.5110 4.0 0.4048 0.0265 0.0002 0.0000 9.0 6.5418 5.6475 5.5698 5.5693 2) (20pts) For the program obtained in Q.1 try different initial points to determine a different value of x which solves the problem f(x)0 While writing the program and answering these questions please do not make use of any built-in functions of Excel or the Solver Tool.

Answer 1

Matlab code for Newton Raphson method clear all close all syms function for which root have to find f(x)--x 5-4*x* 4+2*x* 3+x 2-3*x+5; fprintf function for which d value have to find\n') disp(f) 1st Derivative of this function g1 (x,-- 5+x"4-16*x^3 + 6#x"2 + 2*x - 3; fprintf(1st Derivative of this functionn) disp gl) max it-100; xx-0 fprint f ("Newton iteration for initial guess %f\n",xx) fprintf(t klt x\t f(x) tdf (x)/dxln) %Loop for all intial guesses %Maximum number of iterations tinitial guess n= 10^_6; %error for i= 1: max it limit for close itte ration fprint f ("It%d \t%2.5f \t%2 . 5 f \t%2.5f\n',1-1 , xx,f(xx ) , gl (xx)) x2=double(xx-(f(xx). /gl (xx ))); %Newton Raphson Formula cc=abs (xx-x2 ) ; xx=x2 ; %Error break end fprintf(nThe function is f (x)-n) disp(f) fprint f ('value of x for the function is %f for total iteration count %d.\n\n',xx,1) max it-100; xx--20; fprint f (.Newton iteration for initial guess %f\n.xx) fprintf(t klt xlt f(x) tdf (x)/dxln) %Loop for all intial guesses SMaximum number of iterations Binitial guess n-10^-6; %error limit for close itte ration for i-1:max it fprint f ("It%d \t%2.5f \t%2 . 5 f \t%2.5f\n',ì-1 , xx, f(xx ) , gl (xx)) x2-double(xx-(f(xx). /g 1(xx))); %Newton Raphson Formula cc-abs (xx-x2); xx x2; %Error break end end fprintf nThe function is f (x)n') disp(f) fprint f ('value of x for the function is %f for total iteration %d.\n\n', xx,i) count 888888888888888888 End of Code 8888888888888888%

function for which d value have to find symbolic function inputs: x 1st Derivative of this function -5*x"4-16#x^3 + 6*x^2 + 2*x- symbolic function inputs: x 3 Newton iteration for initial guess 0.000000 k x f(x) df (x)/dx 0 0.00000 5.00000 -3.00000 I1.66667 -31.68724 -95.65432 2 1.33540 -9.42731 -43.63248 3 1.11934-2. 33654 -23.53175 4 1. 02004-0.33174 -17.11158 5 1. 00066 -0. 01052 -16.03550 6 1.00000 -0. 00001 -16.00004 The function is f(x)= symbolic function inputs: x Value of x for the function is 1.000000 for total iteration count 7 Newton iteration for initial guess -20.000000 k x f(x) df (x)/dx 0 -20.00000 2544465.00000 -669643.00000 I16. 20027 832158.19280 -274829.32628 2 -13.17236 271790.53640 -112949.83113 3 -10.76606 88557.42496 -46536.55323 4-8.86310 28726.93175 -19263.67318 5 -7.37185 9238.26788 -8048.24705 6 -6.22399 2917.59509 3428.51445 7-5.37301 884. 069051525.85800 8 -4. 79362 241. 10581 752. 41099 9 -4.47317 48.86386 -461.66742 104.36733 4.20993 -383. 49042 11 4.35635 0.04203 -375.84794 12-4.35624 0.00000 -375.77053 The function is f(x)- symbolic function inputs: x Value of x for the function is-4. 356241 for total iteration count 13. Published with MATLAB R2018a

%%Matlab code for Newton Raphson method
clear all
close all
syms x
% function for which root have to find
f(x)=-x^5-4*x^4+2*x^3+x^2-3*x+5;
fprintf('function for which d value have to find\n')
disp(f)
%1st Derivative of this function
g1(x)=- 5*x^4 - 16*x^3 + 6*x^2 + 2*x - 3;
fprintf('1st Derivative of this function\n')
disp(g1)
max_it=100;         %Maximum number of iterations
xx=0;               %initial guess
fprintf('Newton iteration for initial guess %f\n',xx)
fprintf('\t k\t x\t f(x) \tdf(x)/dx\n')
%Loop for all intial guesses
    n=10^-6; %error limit for close itteration
    for i=1:max_it
        fprintf('\t%d \t%2.5f \t%2.5f \t%2.5f\n',i-1,xx,f(xx),g1(xx))
        x2=double(xx-(f(xx)./g1(xx))); %Newton Raphson Formula
        cc=abs(xx-x2);                 %Error
        xx=x2;
        if cc<=n
            break
        end
    end
    fprintf('\nThe function is f(x)=\n')
    disp(f)
    fprintf('Value of x for the function is %f for total iteration count %d.\n\n',xx,i)
max_it=100;         %Maximum number of iterations
xx=-20;               %initial guess
fprintf('Newton iteration for initial guess %f\n',xx)
fprintf('\t k\t x\t f(x) \tdf(x)/dx\n')
%Loop for all intial guesses
    n=10^-6; %error limit for close itteration
    for i=1:max_it
        fprintf('\t%d \t%2.5f \t%2.5f \t%2.5f\n',i-1,xx,f(xx),g1(xx))
        x2=double(xx-(f(xx)./g1(xx))); %Newton Raphson Formula
        cc=abs(xx-x2);                 %Error
        xx=x2;
        if cc<=n
            break
        end
    end
    fprintf('\nThe function is f(x)=\n')
    disp(f)
    fprintf('Value of x for the function is %f for total iteration count %d.\n\n',xx,i)
  
    %%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%