Ansatz
Encyclopedia
Ansatz (ˈanzats, English: "onset"; today, "approach, setup, starting point"; plural: Ansätze) is a German
noun with several meanings in the English language.
It is widely encountered in physics and mathematics literature.
Since ansatz is a noun
, in German texts the initial a of this word is always capitalised.
that is verified later by its results. An ansatz is the establishment of the starting equation(s), theorem(s) or 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). Typically, a word problem
starts by writing down the ansatz with subsequent refinement leading to the solution of the problem.
s 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 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.
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
noun with several meanings in the English language.
It is widely encountered in physics and mathematics literature.
Since ansatz is a noun
Noun
In linguistics, a noun is a member of a large, open lexical category whose members can occur as the main word in the subject of a clause, the object of a verb, or the object of a preposition .Lexical categories are defined in terms of how their members combine with other kinds of...
, in German texts the initial a of this word is always capitalised.
Definition
In physics and mathematics, an ansatz is an educated guessthat is verified later by its results. An ansatz is the establishment of the starting equation(s), theorem(s) or 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). Typically, a word problem
Word problem (mathematics education)
In mathematics education, the term word problem is often used to refer to any math exercise where significant background information on the problem is presented as text rather than in mathematical notation...
starts by writing down the ansatz with subsequent refinement leading to the solution of the problem.
Examples
Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameterParameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s of the line by a least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
curve fit. Variational approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
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 ansatz is to suppose the solutions of a homogeneous linear differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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.
See also
- HypothesisHypothesisA hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...
- Bethe ansatzBethe ansatzIn physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
- ErsatzErsatzErsatz means 'substituting for, and typically inferior in quality to', e.g. 'chicory is ersatz coffee'. It is a German word literally meaning substitute or replacement...
- GuesstimateGuesstimateGuesstimate is an informal English contraction of guess and estimate, first used by American statisticians in 1934 or 1935. It is defined as an estimate made without using adequate or complete information, or, more strongly, as an estimate arrived at by guesswork or conjecture...
- Trial and errorTrial and errorTrial and error, or trial by error, is a general method of problem solving, fixing things, or for obtaining knowledge."Learning doesn't happen from failure itself but rather from analyzing the failure, making a change, and then trying again."...