History of mathematicians
In this document we give some information of mathematicians which
work or names are used in the course wi2023 "Numerieke Wiskunde voor
technici". November 1947 can be seen as
the
birthday of modern numerical analysis.
1. Ordinary differential equations
Ordinary differential equations are splitted into two classes: initial
value problems and boundary value problems. In Chapter 1 initial value
problems are considered. Several numerical integration methods are given
and analysed as there are
As an application of the theory given in Chapter 1 of this course a
simulation (using a Java-applet) of a double pendulum is possible.
The integration
is done by a Runge-Kutta method.
To analyse the discretisation error of these methods
the Taylor
(
Brook Taylor (1685-1731))
polynomial is used together with numerical
integration methods: midpoint rule, trapezium rule and the
integration method of Simpson
(
Thomas Simpson (1710-1761)).
The order symbol of Landau (
Edmund Georg Hermann Landau (1877-1938))
is used to give a short notation of the
approximation error when a finite difference is used.
To estimate the difference between the solution of an exact and
perturbed system of equations we the Euclid norm
(
Euclid (365 BC-300 BC ).
2. The solution of a system of linear equations
A system of linear equations can be solved in several ways. Which method
is chosen depends on the dimensions of the coefficient matrix.
When the dimension is small Cramers's
(
Gabriel Cramer (1704-1752)) rule can be used. However for large
dimensions this method becomes too expensive with respect to the amount
of work. So for large problems the inverse of the matrix can be used,
but in many problems (especially if the coefficient matrix is a band
matrix) the Gauss elimination method
(
Carl Friedrich Gauss (1777-1855))
is preferred. This method is only robust with respect to rounding errors
when complete or partial pivotting is used. In general it is difficult
to get a good analysis of rounding errors
due to floating point arithmetic done by computers. To motivate the
study of rounding errors we note that several
disasters are originated by rounding errors. Recent examples are:
Patriot Missile Failure and the
Explosion of the Ariane 5.
Some historical notes on matrices and determinants are given
here.
3. Eigenvalue problems
Eigenvalue problems are solved using the Power method. This method gives
the largest eigenvalue and its corresponding eigenvector.
To compute smaller eigenvalues, deflation can be used. When the smallest
eigenvalue is necessary it is better to use the inverse Power method.
Biographies index
Contact information:
Kees Vuik
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Last modified on 02-11-2001 by Kees Vuik
