function [x, fx, exitFlag] = bisection(f,lb,ub,target,options)
% BISECTION Fast and robust root-finding method that handles n-dim arrays.
%
% [x,fVal,ExitFlag] = BISECTION(f,LB,UB,target,options) finds x +/- TolX
% (LB < x < UB) such that f(x) = target +/- TolFun.
%
% x = BISECTION(f,LB,UB) finds the root(s) of function f on the interval
% [LB, UB], i.e. finds x such that f(x) = 0 where LB <= x <= UB. f will
% never be evaluated outside of the interval specified by LB and UB. f
% should have only one root and f(UB) and f(LB) must bound it. Elements
% of x are NaN for instances where a solution could not be found.
%
% x = BISECTION(f,LB,UB,target) finds x such that f(x) = target.
%
% x = BISECTION(f,LB,UB,target,TolX) will terminate the search when the
% search interval is smaller than TolX (TolX must be positive).
%
% x = BISECTION(f,LB,UB,target,options) solves with the default
% parameters replaced by values in the structure OPTIONS, an argument
% created with the OPTIMSET function. Used options are TolX and TolFun.
% Note that OPTIMSET will not allow arrays for tolerances, so set the
% fields of the options structure manually for non-scalar TolX or TolFun.
%
% [x,fVal] = BISECTION(f,...) returns the value of f evaluated at x.
%
% [x,fVal,ExitFlag] = BISECTION(...) returns an ExitFlag that describes
% the exit condition of BISECTION. Possible values of elements of
% ExitFlag and the corresponding exit conditions are
%
% 1 Search interval smaller than TolX.
% 2 Function value within TolFun of target.
% 3 Search interval smaller than TolX AND function value within
% TolFun of target.
% -1 No solution found.
%
% Any or all of f(scalar), f(array), LB, UB, target, TolX, or TolFun may
% be scalar or n-dim arrays. All non-scalar arrays must be the same size.
% All outputs will be this size.
%
% Default values are target = 0, TolX = 1e-6, and TolFun = 0.
%
% There is no iteration limit. This is because BISECTION (with a TolX
% that won't introduce numerical issues) is guaranteed to converge if f
% is a continuous function on the interval [UB, LB] and f(x)-target
% changes sign on the interval.
%
% The <a href="http://en.wikipedia.org/wiki/Bisection_method">bisection method</a> is very robust root-finding method. The absolute
% error is halved at each step so the method converges linearly. However,
% <a href="http://en.wikipedia.org/wiki/Brent%27s_method">Brent's method</a> (such as implemented in FZERO) can converge
% superlinearly and is as robust. FZERO also has more features and input
% checking, so use BISECTION in cases where either the optimization
% toolbox is unavailable or if FZERO would have to be implemented in a
% loop to solve multiple cases, in which case BISECTION will be much
% faster because of vectorization.
%
% Define LB, UB, target, TolX, and TolFun for each specific application
% using great care for the following reasons:
% - There is no iteration limit, so given an unsolvable task, such as
% TolX = TolFun = 0, BISECTION remains in an unending loop.
% - Spacing between very large floating point numbers is likely to be
% greater than TolX.
% - There is no initial check to make sure that f(x) - target changes
% sign between LB and UB.
% - Very large or very small numbers can introduce numerical issues.
%
% Example 1: Find cube root of array 'target' without using NTHROOT and
% compare speed to using FZERO.
% options = optimset('TolX', 1e-9);
% target = [(-100:.1:100)' (-1000:1:1000)'];
%
% tic;
% xfz = zeros(size(target));
% for ii = 1:numel(target)
% xfz(ii) = fzero(@(x) x.^3-target(ii), [-20 20], options);
% end
% fzero_time = toc
%
% tic;
% xbis = bisection(@(x) x.^3, -20, 20, target, options);
% bisection_time = toc
%
% fprintf('FZERO took %0.0f times longer than BISECTION.\n',...
% fzero_time/bisection_time)
%
% Example 2: Find roots by varying the function coefficients.
% [A, B] = meshgrid(linspace(1,2,6), linspace(4,12,10));
% f = @(x) A.*x.^0.2 + B.*x.^0.87 - 15;
% xstar = bisection(f,0,5);
%
% See also FZERO, FMINBND, OPTIMSET, FUNCTION_HANDLE.
%
% [x,fVal,ExitFlag] = BISECTION(f,LB,UB,target,options)
% Copyright 2010-2013 Sky Sartorius
% Author - Sky Sartorius
% Contact - www.mathworks.com/matlabcentral/fileexchange/authors/101715
% --- Process inputs. ---
% Set default values
tolX = 1e-6;
tolFun = 0;
if nargin == 5
if isstruct(options)
if isfield(options,'TolX') && ~isempty(options.TolX)
tolX = options.TolX;
end
if isfield(options,'TolFun') && ~isempty(options.TolFun)
tolFun = options.TolFun;
end
else
tolX = options;
end
end
if nargin<4 || isempty(target); target=0; end
ub_in = ub; lb_in = lb;
f = @(x) f(x) - target;
% --- Flip UB and LB if necessary. ---
isFlipped = lb>ub;
if any(isFlipped(:))
ub(isFlipped) = lb_in(isFlipped);
lb(isFlipped) = ub_in(isFlipped);
ub_in = ub; lb_in = lb;
end
% --- Make sure everything is the same size for a non-scalar problem. ---
if isscalar(lb) && isscalar(ub)
% Test if f returns multiple outputs for scalar input.
if ~isscalar(target)
ub = ub + zeros(size(target));
else
jnk = f(ub);
if ~isscalar(jnk)
ub = ub + zeros(size(jnk));
end
end
end
% Check if lb and/or ub need to be made into arrays.
if isscalar(lb) && ~isscalar(ub)
lb = lb + zeros(size(ub));
elseif ~isscalar(lb) && isscalar(ub)
ub = ub + zeros(size(lb));
end
testconvergence
% --- Iterate ---
while any(stillNotDone(:))
bigger = fx.*f(ub) > 0;
ub(bigger)= x(bigger);
lb(~bigger)= x(~bigger);
testconvergence;
end
function testconvergence
x=(ub+lb)/2;
fx=f(x);
outsideTolFun = abs(fx) > tolFun;
outsideTolX = (ub - lb) > tolX;
stillNotDone = outsideTolX & outsideTolFun;
end
% --- Check that f(x+tolX) and f(x-tolX) have opposite sign. ---
fu = f(min(x+tolX,ub_in));
fl = f(max(x-tolX,lb_in));
unboundedRoot = (fu.*fl) > 0;
% Throw out unbounded results if not meeting TolFun convergence criteria.
x(unboundedRoot & outsideTolFun) = NaN;
% --- Catch NaN elements of UB, LB, target, or other funky stuff. ---
x(isnan(fx)) = NaN;
% --- Characterize results. ---
fx = fx + target;
if nargout > 2
exitFlag = +~outsideTolX;
exitFlag(~outsideTolFun) = 2;
exitFlag(~outsideTolFun & ~outsideTolX) = 3;
exitFlag(isnan(x)) = -1;
end
end
% V2: July 2010
% V3: December 2012
% don't remember when
% typo line 39; added fn handle to see also; made array in example 2
% smaller; changed wording in example 1
% 2013-08-23
% -changed scalar*ones(...) calls to scalar+zeros(...) calls based on
% http://undocumentedmatlab.com/blog/allocation-performance-take-2/
% -rearranged help block and formatted a tiny bit