ABSTRACT
The analytical techniques are more informative and efficient to use, but much less general in their range of applicability, i.e., it is common to have equations with no known explicit analytic solution. A sparse matrix is one in which there are few non-zero coefficients in comparison to the zero coefficients. Unless the non-zero coefficients are banded relatively closely to the diagonal so that one of the special banding algorithms can be used effectively, the number of arithmetic operations to solve a sparse matrix is the same as for a non-sparse matrix using direct methods. One of the most useful of the direct methods is know as Gaussian elimination. It is usually effective, although not the most efficient method. In particular, Gaussian elimination does not take advantage of any particular structure in the matrix. Linear equations are easy to set up, solve, and analyze compared to nonlinear equations.
