Contractions - algorithm and example for nested fixed points

Contractions takes an objective function and uses its two-argument output to find a fixed point. This algorithm in particular can
1) handle the routine in the background while parameters are easily manipulated.
2) can accelerate and dampen stepsizes while it recognizes when to dampen it
3) it respects box constraints.
4) functional arguments from the objective functions can be passed on to carry over "good" starting values for problems in the objective.
5) it can switch to a levenberg-marquardt algorithm. The LM algorithm can use the Broyden method to update the Jacobian which reduces function evaluations.
The LevenbergMarquardt needs to be downloaded for full functionality:
http://www.mathworks.com/matlabcentral/fileexchange/53449-levenberg-marquardt-toolbox
The Jacobian toolbox is also required for the LevenbergMarquardt algorithm:
http://www.mathworks.com/matlabcentral/fileexchange/53440-jacobian-toolbox