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Derivative Matrices 149
Only in a very few large-scale computations n is bigger than 100 (over 200 in extreme
cases). In most of the applications n does not exceed 30. In this case it is convenient
to use a direct method, such as Gauss elimination with maximal pivots, rather than
an iterative method. Direct factorization (such as LU decomposition) is recommended
when the same system has to be solved several times with different data sets. It is well
known that the global cost of these procedures is proportional to n 3 .
As we shall see in chapter thirteen, for partial differential equations, the derivative
matrices in each variable have basically the same structure as those described here. This
is because spectral methods are usually applied to boundary value problems, defined
on domains obtained by direct product of intervals (rectangles, cubes, and so forth). In
this context, to save both computer memory and CPU time, an iterative procedure is
often preferable to a direct approach. A survey of the most popular iterative algorithms
in spectral methods is given in canuto, hussaini, quarteroni and zang(1988),
chapter 5, and boyd(1989), chapter 12. The development of these techniques is based
on the analysis of problems in one variable. Therefore, a detailed study of the simplest
situations will be helpful in finding new strategies.
A lot of problems are defined in the physical space (see section 7.4). The major
drawback, however, is that the corresponding matrices are full and non symmetric. With
the exception of the Chebyshev case, where the FFT algorithm considerably reduces
the amount of operations (see sections 4.3 and 7.2), the cost of a matrix-vector multi-
plication is proportional to n 2 . To be competitive, an iterative method has to reach an
accurate solution in very few iterations. To achieve this goal, it is imperative to work
with preconditioned matrices. How to construct and implement this kind of matrix is
discussed in the next chapter. For the moment, let us briefly recall some basic facts.
Let D be a n×n matrix (n ≥ 1) and ¯q a given vector in R n . We want to determine
the solution ¯p ∈ R n of the linear system
(7.6.1) D¯p = ¯q, where det(D) = 0.
A one-step iterative method to evaluate ¯p is obtained by introducing a new n × n non
singular matrix R and defining an initial guess ¯p (0) ∈ R n . Successively, one constructs