Introduction
Modern applied mathematics started in the 17 and
18 century with people like
Simon Stevin,
René Descartes,
Isaac Newton
and
Leonhard Euler.
Numerical aspects were used in analysis in a natural way; the name numerical
mathematics was unknown. Numerical methods developed by Newton, Euler and later
Carl Friedrich Gauss
play an important role in present day numerical mathematics. The order
symbol of
Edmund Georg Hermann Landau
is used to give a short notation of the approximation errors.
Interpolation
In the error estimate of linear interpolation we
use 'Rolle's Theorem' (
Michel Rolle). Thereafter linear interpolation is generalized to
Lagrange interpolation (
Joseph-Louis Lagrange). In Hermite polynomials (
Charles Hermite), not only the function values but also the
derivatives of a function are used. In many applications (CAD/CAM in technical
applications, visualization, animation etc.) smooth curves are very important.
One way to obtain this is to use cubic splines.
Isaac Jacob Schoenberg
initiated work on splines.
Garrett Birkhoff
was quick to recommend the use of cubic splines for the representation of
smooth curves.
Differentiation
The Taylor polynomial (
Brook Taylor) is used to analyze the error in the approximation of a
derivative by a finite difference formula. Richardson' s extrapolation (
Lewis Fry Richardson) is used to obtain an error estimate or a more
accurate formula.
Initial value
problems
Several numerical integration methods for initial
value problems are given and analyzed as there are
Euler methods (Leonhard Euler)
Modified Euler method, Trapezoidal rule
Runge-Kutta method (Martin Wilhelm Kutta, Carle David Tolme Runge)
Adams-Bashforth method
Integration
The value of definite integrals can be computed by
numerical integration methods. Below are some of these methods
Trapezium (trapezoid) rule,
Integration method of Simpson (
Thomas Simpson)
Simpson Integration
A short introduction to some of the pioneers (incomplete) in this field is
provided below.
Gauss
Romberg
Simpson
Euler
