The "fzero" function does not reuse or recycle the function values calculated during the initial interval search. To improve the speed of the interval search, you can mention the interval you identified, i.e. 
as the initial interval when calling "fzero" again.
as the initial interval when calling "fzero" again. To reuse the function values during the initial interval search, use a "Map" and define a custom initial root interval finder. However, if the function to be evaluated is complex, the "Map" may serve more harm than good by running into memory issues as the algorithm requires more iterations depending upon the initial guess and the step size.
% Code for finding the initial root interval using Map
function [interval, map] = rootIntervalMap(func, x0, step, maxIter)
% Initialize variables
a = x0;
b = x0 + step;
% Creating the Map
map = containers.Map('KeyType', 'double', 'ValueType', 'any');
% Function to get key value or compute if not cached
fa = getMapValue(func, map, a);
fb = getMapValue(func, map, b);
% Expand interval until a sign change is found or max iterations reached
iter = 0;
while iter < maxIter
if sign(fa) ~= sign(fb)
interval = [a, b];
return;
end
% Expand the interval
a = b;
fa = fb;
b = b + step;
fb = getMapValue(func, map, b);
iter = iter + 1;
end
% Optional: If no interval is found
interval = [NaN, NaN];
warning('Failed to find a valid interval after %d iterations.', maxIter);
end
% Helper function for getting the Map value
function y = getMapValue(func, map, x)
if isKey(map, x)
y = map(x);
else
y = func(x);
map(x) = y;
end
end
% Define the function
myFunction = @(x) 5*(x.^10) - x.^4 - x.^2 - 4;
% Configuring the initialGuess, stepSize and maxIterations
initialGuess = -2;
stepSize = 0.1;
maxIterations = 10000;
% Using the map based interval search for the function f(x) = 5x^10 - x^4 - x^2 - 4
[interval, map] = rootIntervalMap(myFunction, initialGuess, stepSize, maxIterations);
% Finding the root using the "fzero" function
root = fzero(myFunction, interval);
fprintf('Interval found: [%.4f, %.4f]\n', interval(1), interval(2));
fprintf('Root found using fzero and Map: %.4f\n', root);
% Running fzero without finding out the interval first
rootAlt = fzero(myFunction, initialGuess);
fprintf('Root found using fzero: %.4f\n', rootAlt);
For more information regarding "containers.Map", refer to this documentation: https://www.mathworks.com/help/matlab/ref/containers.map.html
