Over-determination

In sub-areas of mathematics and its applications, overdetermination is typically the problem that a system is described by more equations than unknowns . In the general case, the restrictions on the system can also be given in the form of inequalities and other things. So there is more information available than is necessary to determine the parameters in a model description of the system.

The additional, possibly contradicting information can serve various purposes:

• to control the system, e.g. when combining several operations or when different processing methods are available,
• to increase the accuracy , because each additional observation can reduce the effect of small, unavoidable measurement errors,
• for statements about the certainty and reliability of a system.

The additional equations or measurements often lead to contradictions in the system, which can often be resolved appropriately.

Geodesy, satellite systems

In geodesy , “overdetermination” refers to the presence or the measurement of additional, in particular geometric, variables such as directions or distances that go beyond the necessary determinants of a model. The simplest example is the measurement of a third angle in the triangle , which should complement the other two to 180 °. More complex cases are geometrical bodies or survey networks where there are excess measurements or data. Navigation systems are a current everyday example : with three receivable navigation satellites, the geographic longitude and latitude can be calculated directly, with four satellites also the height above sea level, but with even more satellites the system is overdetermined.

Tools

The mathematical tools for correct processing of overdeterminations are in many cases the adjustment calculation and the analysis of variance . They are based on the statistical distribution of imperceptible influences (see normal distribution ) and minimize the contradictions between excess measurements or information using the least squares method . The result is the most likely values ​​of the unknowns and the so-called residuals (residual deviations) between the final values ​​and the individual parameters. The individually effective error components can be calculated from these residuals and used to refine the mathematical-physical model.

Typically, over-constrained systems do not have an exact solution . For overdetermined systems of linear equations , instead of the original system, a suitably determined linear compensation problem is solved instead , with which a solution vector is determined that makes the error as small as possible. The so-called Gauss-Newton method is often used for over-determined systems of non-linear equations .

Differential equations

The term is also used in systems of differential equations. For example, in two dimensions, let the partial derivatives of a function be given by two different functions (which are continuous in a domain including their partial derivatives). So there are more equations than unknowns, the differential equation system is overdetermined. One then often makes the additional condition that the function f and its derivatives should be continuous, for which the partial derivatives have to be exchanged, which results in an additional integrability condition . ${\displaystyle z(x,y)}$ ${\displaystyle \phi _{1}(x,y,z),\phi _{2}(x,y,z)}$

conditions

In general, with overdetermined systems of differential equations, similar to linear equations, the question to be answered is whether they describe the same system and how it is specified. In the case of linear equations, this leads to the question of whether all equations are linearly independent of one another (determination of the rank of the associated matrix). In systems of partial differential equations this leads to compatibility conditions from the interchangeability of the partial derivatives. One speaks here of complete integrability. An example is Frobenius' theorem in differential geometry, which specifies when a system of partial differential equations in q dimensions belongs to a q -dimensional tangent space of a manifold. For Frobenius, the condition for this is that the commutator of the vector fields of the system lies in this again. ${\displaystyle \mathbb {R} ^{n}}$

Statics

For the term overdetermination (indeterminacy) in statics, see static determinacy (in contrast to the cases considered above, the situation here is that fewer linear equations are available than are necessary to determine the unknowns).

See also

• QR decomposition # application

literature

• Richard L. Branham, Jr .: Scientific Data Analysis . An Introduction to Overdetermined Systems. Springer Verlag, New York 1990, ISBN 0-387-97201-3 ( MR1043632 ).
• Martin Brokate , Norbert Henze , Frank Hettlich, Andreas Meister, Gabriela Schranz-Kirlinger, Thomas Sonar : Basic knowledge of mathematics studies: higher analysis, numerics and stochastics . With the participation of Daniel Rademacher. 1st edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-642-45077-8 , doi : 10.1007 / 978-3-642-45078-5 .
• Josef Stoer : Introduction to Numerical Mathematics . Taking into account lectures by FL Bauer (=  Heidelberger Taschenbücher . Volume 105 ). 4th, improved edition. Springer-Verlag, Berlin / Heidelberg / New York / Tokyo 1983, ISBN 3-540-12536-1 .
• Guido Walz (Red.): Lexicon of Mathematics in six volumes . Second volume. Spectrum Academic Publishing House, Heidelberg / Berlin 2001, ISBN 3-8274-0434-7 .
• Guido Walz (Red.): Lexicon of Mathematics in six volumes . Fifth volume. Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0437-1 .

References and comments

1. ↑ It is then said that the additional equations or measurements “stiffen” the system.
2. ↑ Martin Brokate et al. a .: Basic knowledge of mathematics studies: advanced analysis, numerics and stochastics. 2016, p. 584 ff.
3. ↑ Josef Stoer: Introduction to Numerical Mathematics I. 1983, p. 179 ff.
4. ^ Lexicon of Mathematics in six volumes. Fifth volume. 2002, p. 258.
5. ^ Lexicon of Mathematics in six volumes. Second volume. 2001, pp. 252-253.
6. ↑ Bieberbach: Theory of differential equations. Springer 1930, p. 276.