Ansatz
In physics and mathematics, an ansatz (/ˈænsæts/; German: [ˈʔanzats], meaning: "initial placement of a tool at a work piece", plural ansätze /ˈænsɛtsə/; German: [ˈʔanzɛtsə] or ansatzes) is an educated guess^{[1]} that is verified later by its results.
Contents
UseEdit
An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It can take into consideration boundary conditions. After an ansatz has been established (constituting nothing more than an assumption), the equations are solved for the general function of interest (constituting a confirmation of the assumption).
ExamplesEdit
Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters.
Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the ansatz to most basic problems of thermodynamics.
Another example of an ansatz is to suppose the solutions of a homogeneous linear differential equation and difference equation to have, respectively, exponential and power form. More generally, one can guess a particular solution of a system of equations and test such an ansatz by direct substitution of the solution in the system of equations. In many cases assumed form of the solution is general enough that can represent arbitrary functions, thus the set of solutions found this way is a full set of all solutions.
See alsoEdit
Look up ansatz in Wiktionary, the free dictionary. |
Look up Ansatz in Wiktionary, the free dictionary. |
- Bethe ansatz
- Coupled cluster, a technique for solving the many-body problem that is based on an exponential Ansatz
- Demarcation problem
- Guesstimate
- Heuristic
- Hypothesis
- Trial and error
NotesEdit
- ^ In his book on "The Nature of Mathematical Modelling", Neil Gershenfeld introduces ansatz, with interpretation "a trial answer", to be an important technique for solving differential equations. Gershenfeld 1999, p.10.
ReferencesEdit
- Gershenfeld, Neil A. (1999), The Nature of Mathematical Modeling, Cambridge University Press, ISBN 0-521-57095-6
- Weis, Dr. Erich; Dr. Heinrich Mattutat (1968), The New Schöffler-Weis Compact German and English Dictionary, Ernst Klett Verlag, Stuttgart, ISBN 0-245-59813-8
- Karbach, M.; Müller, G. (September 10, 1998), Introduction to the Bethe ansatz I. Computers in Physics 11 (1997), 36-43. (PDF), retrieved 2008-10-25
- Karbach, M.; Hu, K.; Müller, G. (September 10, 1998), Introduction to the Bethe ansatz II. Computers in Physics 12 (1998), 565-573. (PDF), retrieved 2008-10-25
- Karbach, M.; Hu, K.; Müller, G. (August 1, 2000), Introduction to the Bethe ansatz III. (PDF), retrieved 2008-10-25