Provide a concise explanation of Gaussian elimination, LU decomposition, bisection method, Newton$-$Raphson method, Gauss$-$Seidel, and Simpson's rule in numerical analysis. Share examples to illustrate when each method is applicable.

Compare these methods, highlighting their strengths and weaknesses. Clearly identify where each method is most useful. Support your comparisons with the assistance of examples and present them using MATLAB graphs for better visualization.