Matlab Ridder's Method Code - matlab

Question

I need to write a proper implementation of the Ridder's method in Matlab. I must define the function as
function [x sol, f at x sol, N iterations] = Ridders(f, x1, x2, eps f, eps x)
The explanation I was given is:
bracket the roots (x1, x2)
evaluate the midpoint (x1 + x2)/2
find new approximation for the root
x4 = x3 + sign(f1 - f2) [f3/((f3)^2 - f1f2)^1/2)](x3 - x1)
check if x4 satisfies the convergence condition. if yes, stop. if not...
rebracket the root using x4 and whichever of x1, x2, or x3 is closer to the root
loop back to 1
I have no idea how to implement this in matlab. Help?

Answer 1

In Matlab, you would define your function as:
function [list of outputs] = myfunc(list of input variables)
%function definition to compute outputs using the input variables.
In your case if you would like x4 (i.e root) to be your output, you would do:
function root = riddler(func, x1, x2, xaccuracy, N)
xl = x1;
xh = x2;
fl=func(x1)
fh=func(x2)
for i = 1:N
xm = 0.5*(xl+xh);
fm = func(xm);
s = sqrt(fm*fm - fl*fh)
if s == 0
return;
end
xnew = xm + (xm - xl)*sign(fl - fh)*fm/s %update formula
.
.
. % extra code to check convergence and assign final answer for root
. % code to update xl, xh, fl, fh, etc. (i.e rebracket)
.
end % end for

Answer 2

I wrote a Matlab implementation of Ridder's method on the Matlab file exchange: submission 54458. I've copied the code below for reference:
function xZero = rootSolve(func,xLow,xUpp)
% XZERO = ROOTSOLVE(FUNC, XLOW, XUPP)
%
% FUNCTION: This function uses Ridder's Method to return a root, xZero,
% of func on the interval [xLow,xUpp]
%
% INPUTS:
% func = a function for a SISO function: y = f(x)
% xLow = the lower search bound
% xUpp = the upper search bound
%
% OUTPUTS:
% xZero = the root of the function on the domain [xLow, xUpp]
%
% NOTES:
% 1) The function must be smooth
% 2) sign(f(xLow)) ~= sign(f(xUpp))
% 3) This function will return a root if one exists, and the function is
% not crazy. If there are multiple roots, it will return the first one
% that it finds.
maxIter = 50;
fLow = feval(func,xLow);
fUpp = feval(func,xUpp);
xZero = [];
tol = 10*eps;
if (fLow > 0.0 && fUpp < 0.0) || (fLow < 0.0 && fUpp > 0.0)
for i=1:maxIter
xMid = 0.5*(xLow+xUpp);
fMid = feval(func,xMid);
s = sqrt(fMid*fMid - fLow*fUpp);
if s==0.0, break; end
xTmp = (xMid-xLow)*fMid/s;
if fLow >= fUpp
xNew = xMid + xTmp;
else
xNew = xMid - xTmp;
end
xZero = xNew;
fNew = feval(func,xZero);
if abs(fNew)<tol, break; end
%Update
if sign(fMid) ~= sign(fNew)
xLow = xMid;
fLow = fMid;
xUpp = xZero;
fUpp = fNew;
elseif sign(fLow) ~= sign(fNew)
xUpp = xZero;
fUpp = fNew;
elseif sign(fUpp) ~= sign(fNew)
xLow = xZero;
fLow = fNew;
else
error('Something bad happened in riddersMethod!');
end
end
else
if fLow == 0.0
xZero = xLow;
elseif fUpp == 0.0
xZero = xUpp;
else
error('Root must be bracketed in Ridder''s Method!');
end
end

Answer 3

Some principal concepts that might help:
function handles (you need to provide f in this format)
valid variable names (many variable names in your definition are not valid)