Task 2 (3 points) The following equation, expl_ (x-1)21+ expl_ (x-291+ expl_ (x-

Question

Task 2 (3 points) The following equation, expl_ (x-1)21+ expl_ (x-291+ expl_ (x-4%]-09-0, has four solutions within the interval of 0 sxs5. Suppose that Newton's method is used to seek the solutions, different choices of the initial guess would potentially lead to different solutions. Consider the 51 initial guesses, xo-(0, 0.1, 0.2, 0.3, , 49, 5). Use Newton's method to perform 100 iterations for each of those initial guesses. List the solutions (as obtained at the end of 100 iterations) and their corresponding initial guesses in the following format: Xp=0 solution= xo= 0.1 solution x0=0.2 solution If for a given xo the process of iteration diverges such that Matlab returns Inf or NaN at the end of 100 iterations, please list the "Inf" or "NaN" as the solution to indicate that Newton's method fails to converge. Briefly discuss the result. Focus on the following questions: Is there a general predictive relation between the given initial guess and the final solution? What happens when the initial guess is close to wheref(x) -0? (Here, "/(x)-0" describes the original equation and f'(x) is the derivative of j(x).) What happens when the initial guess is very close to a solution?

Explanation / Answer

clc;
clear all;
format long
f=@(x)exp(-(x-1).^2)+exp(-(x-2).^2)+exp(-(x-4).^2)-0.9;%function

f1=@(x)-2*x.*(exp(-(x-1).^2)+exp(-(x-2).^2)+exp(-(x-4).^2)); %derivative of function
y=0:0.1:5;
for i=1:length(y)
x0=y(i);%initial guess
n=1;
erorr=0.1;
del=1e-8;
x(1)=x0;
m=2;
while (abs(erorr>del)& (n<=100))
y1=x0-(f(x0)/f1(x0));% Newton method

erorr(n+1)=abs((y1-x0)); %erorr

% if abs(erorr<1e-3)
% break
%end
n=1+n;
x0=y1; % update xold
x(n+1)=x0;
if (n>100)
      fprintf('x0=%0.3f   solution = NaN ',x(1))
end
end
   fprintf('x0=%0.3f   solution =%f ',x(1),x0)
end

x0=0.000   solution =NaN
x0=0.100   solution =NaN
x0=0.200   solution =NaN
x0=0.300   solution = NaN
x0=0.300   solution =2.623384
x0=0.400   solution =NaN
x0=0.500   solution =NaN
x0=0.600   solution = NaN
x0=0.600   solution =2.623383
x0=0.700   solution = NaN
x0=0.700   solution =2.623383
x0=0.800   solution = NaN
x0=0.800   solution =2.623383
x0=0.900   solution = NaN
x0=0.900   solution =2.623383
x0=1.000   solution = NaN
x0=1.000   solution =2.623383
x0=1.100   solution = NaN
x0=1.100   solution =2.623383
x0=1.200   solution = NaN
x0=1.200   solution =2.623383
x0=1.300   solution = NaN
x0=1.300   solution =2.623383
x0=1.400   solution = NaN
x0=1.400   solution =2.623383
x0=1.500   solution = NaN
x0=1.500   solution =2.623384
x0=1.600   solution = NaN
x0=1.600   solution =2.623384
x0=1.700   solution = NaN
x0=1.700   solution =2.623384
x0=1.800   solution = NaN
x0=1.800   solution =2.623384
x0=1.900   solution = NaN
x0=1.900   solution =2.623384
x0=2.000   solution = NaN
x0=2.000   solution =2.623384
x0=2.100   solution = NaN
x0=2.100   solution =2.623384
x0=2.200   solution = NaN
x0=2.200   solution =2.623384
x0=2.300   solution = NaN
x0=2.300   solution =2.623384
x0=2.400   solution = NaN
x0=2.400   solution =2.623384
x0=2.500   solution =2.623384
x0=2.600   solution =2.623384
x0=2.700   solution =2.623384
x0=2.800   solution = NaN
x0=2.800   solution =2.623384
x0=2.900   solution = NaN
x0=2.900   solution =2.623384
x0=3.000   solution = NaN
x0=3.000   solution =2.623384
x0=3.100   solution = NaN
x0=3.100   solution =2.623384
x0=3.200   solution = NaN
x0=3.200   solution =2.623384
x0=3.300   solution = NaN
x0=3.300   solution =2.623385
x0=3.400   solution = NaN
x0=3.400   solution =2.623386
x0=3.500   solution = NaN
x0=3.500   solution =2.623440
x0=3.600   solution = NaN
x0=3.600   solution =4.328213
x0=3.700   solution = NaN
x0=3.700   solution =4.330858
x0=3.800   solution = NaN
x0=3.800   solution =4.331442
x0=3.900   solution = NaN
x0=3.900   solution =4.331686
x0=4.000   solution = NaN
x0=4.000   solution =4.331817
x0=4.100   solution = NaN
x0=4.100   solution =4.331896
x0=4.200   solution = NaN
x0=4.200   solution =4.331948
x0=4.300   solution = NaN
x0=4.300   solution =4.331984
x0=4.400   solution = NaN
x0=4.400   solution =4.332010
x0=4.500   solution = NaN
x0=4.500   solution =4.332028
x0=4.600   solution = NaN
x0=4.600   solution =4.332042
x0=4.700   solution = NaN
x0=4.700   solution =4.332052
x0=4.800   solution = NaN
x0=4.800   solution =4.332060
x0=4.900   solution = NaN
x0=4.900   solution =4.332065
x0=5.000   solution = NaN
x0=5.000   solution =4.332069
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