Nonlinear algebraic or transcendental scalar equations are usually solved by using iteration methods. Because some of them are so intricate to solve analytically or an approximate numerical solution is required. Nonlinear algebraic equations can occur during several modeling processes or when solving nonlinear BVPs using FDMs......
I cover all of the best methods in this book, which will be free on Thursday (7/15) https://www.amazon.com/dp/B07FL7JR1J The software examples are free here http://dudleybenton.altervista.org/software/Nonlinear_Equations_examples.zip In answer to your question: It depends. Some methods are better than others. Some work well for one class of problems but not another class. In the literature you often find problems carefully selected to prove a point or make one method stand out above all the rest, which might make for an interesting paper but is not really helpful. I try to present a balanced approach.
Dear Dudley J. Benton, thanks for attaching links, and for suggestions!
Of course, it is not simple to judge the best one amongst the others.The basics and bases of comparisons have to be rationally and carefully decided. Basic root bracketing derivative free methods such as Bisection method & Regula-falsi method guarantee convergence but may be slow. Fixed point (open methods) such as Newton's method, Halley's method, Chebyshev's method, and Super-Halley’s method are popularly used with better speeds but involve derivatives and may diverge when the initial guess is chosen randomly. There are many other fast converging methods (having different properties) in the existing literature....
I think the short answer is to run several experiments and evaluate the outcomes based on your set metric for comparison.
Dear Tekle Gemechu,
You can use finite element method for solving non-linear problems. For this method, you can do computation in matlab or in FEniCS like software where you can find some demos which will be very helpful for you.
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