Write MATLAB codes to find the roots of the following equations. Submit your cod

Question

Write MATLAB codes to find the roots of the following equations. Submit your codes and outputs as a single pdf file to sakai. Be carefull There may be multiple roots. In all problems above, set a tolerance of 10 for convergence. 1. sin(x2+x2 -2x-0.09-0 using Newton's Method, and using MATLAB's fzero function cos4w-x2)-8x2+8x4+1=0 using Bisection Method to find the brackets, and 2. then Secant Method to find all the roots. So this is like a hybrid method. 3. x-3x +2.25-0 using Newton's Method. How many iterations you need to get a converged result here? Compare this with the number of iterations you got in HW 3 for this problem using Fixed Point iterations (using the same initial guess).

Explanation / Answer

1.

fzero function is different method altogether. It is combination of secant, bisection and backward interpolation.

We solve using newton's method.

function [ x, err_est ] = newton( func, diff_func, x0, tolerance, nmaxiter )

%

% Input:

% func - input funtion

% diff_func - derived input function

% x0 - inicial aproximation

% tolerance - tolerance

% nmaxiter - maximum number of iterations

%

% Output:

% x - aproximation to root

% err_est - error estimate

%

% Example:

% [ x, ex ] = newton( 'sin(x.^2)+x.^2-2.*x-0.09', '2.*x.*cos(x.^2)+2.*x-2', 0, 10^-8, 100)

%

% Result:

%x =

% -0.0450 -0.0431 -0.0431 -0.0431

%ex =

% 0.0450 0.0019 0.0000 0.0000

%Answer:-0.0431

if nargin == 3

tolerance = 1e-4;

nmaxiter = 1e1;

elseif nargin == 4

nmaxiter = 1e1;

elseif nargin ~= 5

error('newton: invalid input parameters');

end

  

func = inline(func);

diff_func = inline(diff_func);

x(1) = x0 - (func(x0)/diff_func(x0));

err_est(1) = abs(x(1)-x0);

k = 2;

while (err_est(k-1) >= tolerance) && (k <= nmaxiter)

x(k) = x(k-1) - (func(x(k-1))/diff_func(x(k-1)));

err_est(k) = abs(x(k)-x(k-1));

k = k+1;

end

end

2.

function b_out = hybrid(func,a_in,b_out)

fa = func(a_in);
fb = func(b_out);
s = '%5.0f %8s %19.15f %23.15e ';
fprintf(s,1,'initial',a_in,fa)
fprintf(s,2,'initial',b_out,fb)
k = 2;

% a_in is the previous value of b_out and [b_out, c] always contains the zero.
c = a_in; fc = fa;
while true
if sign(fb) == sign(fc)
c = a_in; fc = fa;
end

% Swap to insure f(b) is the smallest value so far.
if abs(fc) < abs(fb)
a_in = b_out; fa = fb;
b_out = c; fb = fc;
c = a_in; fc = fa;
end

% Midpoint.
m = (b_out + c)/2;
if abs(m - b_out) <= eps(abs(b_out))
return % Exit from the loop and the function here.
end

% p/q is the the secant step.
p = (b_out - a_in)*fb;
if p >= 0
q = fa - fb;
else
q = fb - fa;
p = -p;
end
% Save this point.
a_in = b_out; fa = fb;
k = k+1;

% Choose next point.
if p <= eps(q)
% Minimal step.
b_out = b_out + sign(c-b_out)*eps(b_out);
fb = func(b_out);
fprintf(s,k,'minimal',b_out,fb)

elseif p <= (m - b_out)*q
% Secant.
b_out = b_out + p/q;
fb = func(b_out);
fprintf(s,k,'secant ',b_out,fb)

else
% Bisection.
b_out = m;
fb = func(b_out);
fprintf(s,k,'bisect ',b_out,fb)
end
end
end

%Result

hybrid(f,-10,10)
1 initial -10.000000000000000 3.515534819284370e+172
2 initial 10.000000000000000 3.515534819284370e+172
3 bisect 0.000000000000000 2.000000000000000e+00
4 minimal 0.000000000000000 2.000000000000000e+00

ans =

4.9407e-324

3.

[ x, ex ] = newton( 'x.^6-3.*x.^3+2.25', '6.*x.^5-9.*x.^2', -100, 10^-8, 40)

x =

Columns 1 through 12

-83.3333 -69.4444 -57.8703 -48.2252 -40.1875 -33.4894 -27.9076 -23.2561 -19.3796 -16.1490 -13.4565 -11.2124

Columns 13 through 24

-9.3417 -7.7819 -6.4808 -5.3947 -4.4870 -3.7267 -3.0876 -2.5468 -2.0838 -1.6789 -1.3104 -0.9464

Columns 25 through 36

-0.5095 0.5382 1.3115 1.2383 1.1949 1.1709 1.1581 1.1515 1.1481 1.1464 1.1456 1.1451

Columns 37 through 40

1.1449 1.1448 1.1448 1.1447

ex =

Columns 1 through 12

16.6667 13.8889 11.5741 9.6451 8.0376 6.6981 5.5818 4.6516 3.8765 3.2306 2.6925 2.2441

Columns 13 through 24

1.8707 1.5598 1.3011 1.0861 0.9077 0.7602 0.6391 0.5408 0.4630 0.4049 0.3685 0.3640

Columns 25 through 36

0.4369 1.0478 0.7732 0.0732 0.0433 0.0241 0.0128 0.0066 0.0034 0.0017 0.0009 0.0004

Columns 37 through 40

0.0002 0.0001 0.0001 0.0000

Converges at 40th iteration

Fixed Point iteration

format long;
%% set initial guess %%
z(1) = -100;
for n = 1:40
z(n+1) = z(1).^6-3.*z(1).^3+2.25;
x(n) = z(n+1)
end

%Result

1.0e+12 *

Columns 1 through 6

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 7 through 12

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 13 through 18

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 19 through 24

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 25 through 30

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 31 through 36

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

Columns 37 through 40

1.000003000002250 1.000003000002250 1.000003000002250 1.000003000002250

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