33 Section Summary

We only had a brief overview of numerical methods. Again, the author emphasizes there are a lot that could not be included in this summary, and thus awaiting to be explored by the reader. Following is the list of a few “missing” items by sections:

• In root-finding algorithm, the extensions of Newton’s method such as the secant and Steffensen’s method, applicable to situations where obtaining the derivative or computation involving derivative is complicated;
• In polynomial interpolation, the method to facilitate fitting a cubic spline with the use of matrix, and other interpolating methods such as Hermite interpolation were not discussed;
• In numerical calculus, in-depth error analysis, for example, through the use of Richardson’s extrapolation, was not conducted, and the method of undetermined coefficients was not discussed;
• Iterative techniques in matrix algebra, that primarily deals with methods for faster} computation of the solutions of linear systems, were not even included in this summary. In linear algebra, one way of obtaining solutions was by converting a given matrix into a reduced row echelon form, and most often Gaussian elimination} algorithm is used. In numerical analysis, the focus is on minimizing operation counts, namely replacement, interchange and scaling, so that computation burden is minimized and thus solutions be obtained in the most efficient manner.

Indeed, above list is not comprehensive. The world of mathematics to be trodden is as good as infinite.