Question

a) What is the strength and limitation od the four methods for finding the roots of equations: (1) the graphical method; (2) the bisection method; (3) the simple (one-point) iteration method and (4) the Newton Raphson method. You can use graphs or example to enhance your explanation.

b) Discuss the strength and limitation of (1)naive Gauss elimination; (2)Gauss elimination with partial pivoting, (3) Jacobi iteration and (4) Gauss-Siedel method in solving the set of linear equations, in terms of error and convergence.

Answer 1

Graphical method

The major strength of graphical method is that it provides a convenient way to visually observe the root as this method involves drawing of the function f whose root needs to be determined. Major limitation is that it is not always convenient to draw the graph of the function.

Bisection method

Major strength is that this methods leads to convergence in most cases, when when the function in context is not smooth. Major limitation is that in this method, the convergence is linear, il.e. it provides very slow convergence.

Simple (one-point) iteration method

In this method, the major strength is that root may be obtained by suceesive substitution of the initial guess. Major limitation is that the convergence is dependent on the initial guess as well as the function under consideration

Newton Raphson method

The strength of this method is that it provides a faster (quadratic) convergence for functions where derivative exists.

Limitation is that for non-smooth functions, or for initial guesses far away from the actual root, the solution diverges.