find the least positive root of the equation y=cos(sqrt​(abs(polyv​al(p,x))))​-sin(x)/2 with the use of fzero

what function should i create and what i should do after

Answer 1

[Star Strider]

Similarly to your earlier Question:

p = polyfit(x,y,7);                                     % Fit Polynomial
P = @(x,p) polyval(p,x);

f = @(x,p) cos(sqrt(abs(P(x,p)))) - sin(x)/2;           % Function

f_root = fzero(@(x) f(x,p), 1);                         % Find Single Root

xv0 = linspace(0, 5, 100);                              % Vector Of Initial Estimates For Multiple Roots
for k1 = 1:length(xv0)                                  % Find Multiple Roots
    f_root_v(k1) = fzero(@(x) f(x,p), xv0(k1));         % Find Vector Of Roots
end

roundn = @(x,n) round(x .* 10.^n)./10.^n;               % Round ‘x’ To ‘n’ Digits, Emulates Latest ‘round’ Function
function_roots = unique(roundn(f_root_v, 4));           % Find Unique Roots

EDIT — To find the least positive root, add this line:

LeastPositiveRoot = min(function_roots(function_roots > 0))

Answer 2

[Walter Roberson]

A difficulty with the linspace approach is that when you ask fzero for a root and supply an initial positive guess, there is no guarantee that the resulting root will be positive even if there are positive roots. For any fixed spacing like linspace(0,5,100) you can come up with a function which will project all of those values to negative roots even though positive roots exist.

It is tempting to use fzero with an interval of eps(realmin) to realmax. The difficulty with that is that fzero requires that the endpoints have opposite signs, which we can certainly not guarantee in general.

You can fzero adjacent intervals over a range hoping for some root, if you check the exitflag. But if the function is sufficiently wiggly then you might still run into the sign-change problem.

Some other methods of finding roots:

http://www.mathworks.com/matlabcentral/fileexchange/15122-real-roots-on-interval

http://www.mathworks.com/matlabcentral/answers/89577-finding-the-zeros-of-a-function