Numeric mathematics

numeric mathematics, briefly Numerik mentioned, is occupied as subsection of mathematics with the construction and analysis of algorithms for continuous mathematical problems.

overview

interest in such algorithmsconsists usually of one of the two following reasons:

1. There is no explicit solution representation to the problem (so for example with the Navier Stokes equations or the Dreikörperproblem) or
2. the solution representation existed, is not suitable however to calculate around the solution fast and/or it lies in a formforwards, in which calculation errors become strongly apparent (for example with many power series).

Differences become two types of procedures. Once direct, those after finite time with infinite computer accuracy the accurate solution of a problem supply and on the other side approximation method, which - like thatName says - only approximations supply. An example of first is the Gauss Eliminationsverfahren, which supplies the solution of a linear set of equations. Approximation methods are among other things quadrature formulas, those the value of an integral approach compute or also the Newton procedure, that iterative better approximations ona zero of a function supplies.

Different procedures are compared after running time , stability and robustness.

history

the desire to be able to solve mathematical equations in terms of figures (also approach) exists since the antique one. The ancient Greeks already knew problems, which solve them only approachcould, like the computation of surfaces (integration) or the circle number π. In this sense can Archimedes, which supplied for both problems algorithms, when the first important Numeriker are designated. In the age of the computer engineering the numeric procedure gains against it dramatically significance.

ThoseNames of classical procedures show clearly that the algorithmic and approximate entrance to mathematical problems was always important, in order to be able to use purely theoretical statements fruitfully. Concepts such as convergence speed or stability were very important also when counting by hand. Thus for example a high convergence speed leaves on ithope to become fast with the computation finished. And already Gauss noticed that sometimes its calculation errors affected with the Gauss Eliminationsverfahren desaströs the solution and made them so completely useless. It preferred therefore the Gauss Seidel procedure, with which one errors by implementing onefurther iteration step easily to adjust could.

In order to facilitate the monotonous accomplishing from algorithms to, 19 became in. Century mechanical calculating machines develops and finally into the 1930ern the first computer of Konrad Zuse. The Second World War accelerated the development dramatically and in particular John von Neumann floatedin the context of the Manhattan Projects both mathematically and technically the Numerik in front. The time of the cold war was particularly coined/shaped of military applications such as reentry problems, but the explosion of the computer achievement since the 1980ern civilian applications into the foreground let step. Furthermore hasthe need according to fast algorithms with the speed increase strengthens accordingly. For many problems the research could carry this out and the speed of the algorithms in such a way in the last 20 years around approximately the same order of magnitude improved as the CCU achievements. Nowadays are numeric procedureswithin each technical or scientific range present and everyday life tool.

error analysis

an aspect with the analysis of the algorithms in the Numerik is the error analysis. With a numeric computation different types from errors come to carrying: When counting on floating-point numbers inevitably rounding error arises.These errors can be made smaller for example by an increase of the number of digits. To completely eliminate one cannot do it however, since each computer can count only in principle on finally many places.

How the problem reacts to disturbances in the initial data is measured, with the condition.If a problem has a large condition, then the solution of the problem depends rounding errors sensitively on the initial data, which makes a numeric solution more difficult, as disturbance of the initial data to be understood in particular there to be able. One speaks of a problem badly placed, which after possibility by a reformulatingto be gone around should.

Furthermore the numeric procedure replaces the continuous mathematical problem by a discrete, thus finite problem. The discretisation error in such a way specified already arises , which is measured and evaluated in the context of the consistency analysis. This is necessarily, there a numeric procedure as a rule notthe accurate solution supplies.

How such errors become larger with further calculating is evaluated, with the help of the stability analysis.

Consistency and stability of the algorithm lead as a rule to convergence.

subsections

of subsections of the Numerik are among other things:

• Optimization
• approximation
• numeric solution of nonlinear equations
• Numerik of Differential equations
• Numerik numeric procedures
• selected by integral equations
• numeric linear

algebra numeric number theory a commentated composition of is here: List of numeric procedures.

literature

• Martin Hermann: Numeric mathematics. Oldenbourg publishing house, Munich and Vienna 2001 ISBN 3-486-25558-4
• Gerhard victims: Numeric mathematics for beginners. OneIntroduction for mathematicians, engineers and computer scientists. 4. Aufl. Vieweg publishing house, Braunschweig 2002 ISBN 3-528-37265-6
• Robert Plato: Numeric mathematics compactly. Basic knowledge for study and practice. Vieweg publishing house, Braunschweig 2000 ISBN 3-528-03153-0
• Thomas Huckle, Stefan cutter: Numerik for computer scientists. Springer publishing house, Berlin 2002 ISBN 3-540-42387-7