Finite Difference Equation

This chapter deals with two main topics. The first topic is how to solve linear systems of
equations numerically. We start with Gauss elimination, which may be familiar to some
readers, but this time in an algorithmic setting with partial pivoting. Variants of this method
(Doolittle, Crout, Cholesky, Gauss–Jordan) are discussed in Sec. 20.2. All these methods
are direct methods, that is, methods of numerics where we know in advance how many
steps they will take until they arrive at a solution. However, small pivots and roundoff
error magnification may produce nonsensical results, such as in the Gauss method. A shift
occurs in Sec. 20.3, where we discuss numeric iteration methods or indirect methods to
address our first topic. Here we cannot be totally sure how many steps will be needed to
arrive at a good answer. Several factors—such as how far is the starting value from our
initial solution, how is the problem structure influencing speed of convergence, how
accurate would we like our result to be—determine the outcome of these methods.
Moreover, our computation cycle may not converge. Gauss–Seidel iteration and Jacobi
iteration are discussed in Sec. 20.3. Section 20.4 is at the heart of addressing the pitfalls
of numeric linear algebra. It is concerned with problems that are ill-conditioned. We learn
to estimate how “bad” such a problem is by calculating the condition number of its matrix.
The second topic (Secs. 20.6–20.9) is how to solve eigenvalue problems numerically.
Eigenvalue problems appear throughout engineering, physics, mathematics, economics,
and many areas. For large or very large matrices, determining the eigenvalues is difficult
as it involves finding the roots of the characteristic equations, which are high-degree
polynomials. As such, there are different approaches to tackling this problem. Some
methods, such as Gerschgorin’s method and Collatz’s method only provide a range in
which eigenvalues lie and thus are known as inclusion methods. Others such as
tridiagonalization and QR-factorization actually find all the eigenvalues. The area is quite
ingeneous and should be fascinating to the reader