Dear Dr. Courtney,Dr. Courtney said:I can't think of any general ones, but applications of the least action principles should be able to search around an approximate solution to find an exact solution more quickly.
I've also written code for searching out periodic orbits of chaotic systems. Having approximations to begin with would make the process faster.
Some common numerical methods that require a guess or approximate solution include the bisection method, Newton's method, and the secant method.
These methods rely on iteration to approach the true solution, so an initial guess is needed to start the process. Without a starting point, the methods would not be able to converge to the correct solution.
The accuracy of the solutions obtained from numerical methods that require a guess or approximate solution depends on the initial guess and the number of iterations performed. With a good initial guess and enough iterations, the solutions can be very accurate.
No, these methods are typically used for solving equations that cannot be solved analytically, such as transcendental equations or systems of nonlinear equations. They may not be suitable for all types of problems and may require modification for specific cases.
The choice of numerical method depends on the specific problem and the structure of the equation. It is important to consider factors such as the availability of initial guesses, the convergence rate of the method, and the complexity of the problem before selecting a method.