A description of numerical analysis
The subject of numerical analysis is concerned with devising methods
for approximating, in an efficient way, the solutions to
mathematically expressed problems
(Trefethen gives a more elaborate
definition for numerical analysis). The efficiency of the method
depends both upon the accuracy required and the
ease with which the method can be implemented. In a practical situation,
the mathematical problem is derived from a physical
(chemical, biological, economical) phenomenon
where some simplifying assumptions have been made to allow
the mathematical representation to develop. Generally a relaxation
on the physical assumptions leads to a more appropriate
mathematical model, but at the same time one that is more
difficult or impossible to solve explicitly. Since the mathematical
problem ordinarily does not solve the physical problem exactly
in any case, it is often more
appropriate to find an approximate solution to a more complicated
mathematical model of a physical problem than to find an exact
solution of a simplified model. To obtain such an approximation
a method called an algorithm is devised. The algorithm consists
of a sequence of algebraic and logical operations that produces
the approximation to the mathematical problem, and, it is hoped,
to the physical problem as well, within a prescribed tolerance or
accuracy.
Since the efficiency of a method depends upon its
of implementation, the choice of the appropriate method for
approximation the solution to a problem is influenced significantly
by changes in calculator and computer technology. Twenty-five
years ago, before the widespread use of digital computing equipment,
methods requiring a large amount of computational effort could not
be reasonably applied. Since that time, however, the advances
in computing equipment have made some of these methods increasingly
attractive. At present, the limiting factor generally involves the
amount of computer storage requirements of the method, although the
cost factor associated with a large amount of computation time is,
of course, also important. The availability of personal computers and
low cost programmable calculators is also an influencing factor in the
choice of an approximation method, since these can be used to solve
many relatively simple problems.
The basic ideas that underlie most
current numerical techniques have been known for some time,
as, have the methods used in predicting bounds for the maximum error
that can be produced in an application of the methods. It is of
primary interest, then, to determine the way in which these methods have
developed and how their error can be estimated, since variations of
these techniques will undoubtedly be used to develop and apply numerical
procedures in the future, irrespective of the technology.
Contact information:
Kees
Vuik
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