Upon your opinion and expert...What is the best Nonlinear Solution Algorithm?
I do not know the meaning of your word "best". However, good and fruitful algorithms can be developed on the existing tools as follows. For ODE, you may apply RK-IV; for PDE, Finite Element Method (FEM); for complex polynomial equations, Decomposition Method (DM); and so on.
Some suggested readings are:
1. J.C. Tannehill et al., COMPUTATIONAL FLUID MECHANICS AND HEAT TRANSFER (Taylor & Francis, USA, 1994).
2. Lindfield, G.R., Penny, J.E.T.: NUMERICAL METHODS USING MATLAB. Elsevier, Amsterdam (2012).
For ODE, it is better to use high order methods with variable step, for example, Dormand-Prince method 8, or extrapolation algorithms.
this depends on the problem to be studied,type of loading,material,etc.
Most nonlinear solution algorithms are best suited for a certain class of problems.
Newtons method: if the hessian is easy to compute and evaluate
Quasi Newton Methods e.g Gauss Newton Algorithm: for cases where you do not have an hessian, you pay with convergence
conjugate gradient methods: good for large sparse matrices
Trust region methods: better for quadratic like problems
For ODES
this depends on the order of error you're willing to tolerate
ODE45, Dormand Prince
or if youre dealing with a stiff differential equation then you need ODE23s or ODE15s in MATLAB.
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