%[text] # Solving a Transcendental Equation by Recursion
%[text] 
%[text] We want to solve the equation
%[text]{"align":"center"} $x = \\cos \\big(x\\big)$
%[text] The key idea is transforming the equation by recursion, obtaining a succession that converges to the solution:
%[text]{"align":"center"} $x\_{n} = \\cos \\big(x\_{n-1} \\big) \\; \\Longrightarrow \\; \\big\\{ x\_1\\,,\\; x\_2\\,,\\; \\ldots x\_n\\,,\\; \\ldots \\big\\}$
%[text] where
%[text]{"align":"center"} $\\displaystyle lim\_{n \\to \\infty} \\,x\_n = \\bar x \\;:\\quad \\bar x = \\cos \\big( \\bar x \\big)$
%[text]{"align":"center"} 
% 1st approach
tic % <-- what does this command? try "help tic"
x = zeros(1,20); x(1) = pi/4; % initial guess
n = 1; d = 1;
 while d > 0.001 & n < 20
    n = n+1; x(n) = cos(x(n-1));
    d = abs( x(n) - x(n-1) );
end
[n,x(n)] %[output:7aa2d985]
toc % <-- what does this command? try "help toc" %[output:790c2803]
%%
%[text] A more efficient approach (allocating less RAM memory to store the intermediate results)
% 2nd approach
tic
 xold = pi/4; n = 1; d = 1;
 while d > 0.001 & n < 20
    n = n+1; xnew = cos(xold);
    d = abs( xnew - xold );
    xold = xnew;
end
[n, xnew, d] %[output:8dc10699]
toc %[output:347b6c49]

%[appendix]{"version":"1.0"}
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%   data: {"dataType":"text","outputData":{"text":"Elapsed time is 0.081278 seconds.\n","truncated":false}}
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%   data: {"dataType":"text","outputData":{"text":"Elapsed time is 0.018065 seconds.\n","truncated":false}}
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