: Variational definition: A surface M ⊂ R < sup > 3 </ sup > is minimal if and only if it is a critical point of the area functional for all compactly supported variations.

There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently by Variational method.

* S K Adhikari 1998 " Variational Principles for the Numerical Solution of Scattering Problems ".

* Kiyohisa Tokunaga, " Variational Principle for Electromagnetic Field ".

Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

Variational Bayesian methods are primarily used for two purposes:

* Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc ( 1997 ) Unified Notation for Data Assimilation: Operational, Sequential and Variational Journal of the Meteorologcial Society of Japan, vol.

Variational message passing ( VMP ) is an approximate inference technique for continuous-or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn.

* Variational free energy, a construct from Information theory that is used in Variational Bayesian methods

Variational Bayes can be seen as an extension of the EM ( expectation-maximization ) algorithm from maximum a posteriori estimation ( MAP estimation ) of the single most probable value of each parameter to fully Bayesian estimation which computes ( an approximation to ) the entire posterior distribution of the parameters and latent variables.

# REDIRECT Variational Bayesian methods

* M. Ferraris, M. Francaviglia, C. Reina, Variational Formulation of General Relativity from 1915 to 1925 ' Palatini's Method ' Discovered by Einstein in 1925, Gen. Rel.

The optimal solution may also be obtained by Gauss elimination using other sparse-matrix techniques or iterative methods based e. g. on Variational Calculus.

* David Bleecker, Gauge Theory and Variational Principles, ( 1981 ), Addison-Wesley Publishing, ISBN 0-201-10096-7.

The main mathematical tools to study critical points are renormalization group, which takes advantage of the Russian dolls picture to explain universality and predict numerically the critical exponents, and Variational perturbation theory, which converts divergent perturbation expansions into convergent strong-coupling expansions relevant to critical phenomena.

* Yibei Ling, Jie Mi, Xiaola Lin: A Variational Calculus Approach to Optimal Checkpoint Placement.

Gray, G. Karl, and V. A. Novikov, " Progress in Classical and Quantum Variational Principles ".

* Komkov, Vadim ( 1986 ) Variational principles of continuum mechanics with engineering applications.

In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior.

Variational Bayes will then construct an approximation to the posterior probability.

* Dym, C. L. and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973.

Variational Bayes ( VB ) is often compared with expectation maximization ( EM ).

* David Bleecker, Gauge Theory and Variational Principles, Addison-Wesley publishing, Reading, Mass ( 1981 ).

He is the head of the research unit Quantitative Lexicology and Variational Linguistics ( QLVL ).

Algorithms are used for calculation, data processing, and automated reasoning.

Algorithms for calculating variance play a major role in statistical computing.

# Donald E. Knuth, Selected Papers on Analysis of Algorithms ( Stanford, California: Center for the Study of Language and Information — CSLI Lecture Notes, no.

# Donald E. Knuth, Selected Papers on Design of Algorithms ( Stanford, California: Center for the Study of Language and Information — CSLI Lecture Notes, no.

* Paul M. Embree, Damon Danieli: C ++ Algorithms for Digital Signal Processing, Prentice Hall, ISBN 0-13-179144-3

* Richard P. Brent, " Recent Progress and Prospects for Integer Factorisation Algorithms ", Computing and Combinatorics ", 2000, pp. 3-22. download

Algorithms are generally quite specifically tuned to a particular type of file: for example, lossless audio compression programs do not work well on text files, and vice versa.

* Algorithms for the K-server problem

* Algorithms for calculating variance

Algorithms for searching virtual spaces are used in constraint satisfaction problem, where the goal is to find a set of value assignments to certain variables that will satisfy specific mathematical equations and inequations.

Algorithms for these problems include the basic brute-force search ( also called " naïve " or " uninformed " search ), and a variety of heuristics that try to exploit partial knowledge about structure of the space, such as linear relaxation, constraint generation, and constraint propagation.

Algorithms have been developed to ensure the same SMILES is generated for a molecule regardless of the order of atoms in the structure.

* Algorithms for calculating variance

* Michel Raynal: Algorithms for Mutual Exclusion, MIT Press, ISBN 0-262-18119-3

* Goldberg, David E ( 2002 ), The Design of Innovation: Lessons from and for Competent Genetic Algorithms, Addison-Wesley, Reading, MA.

* Schmitt, Lothar M ( 2004 ), Theory of Genetic Algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling, Theoretical Computer Science 310: 181 – 231

* James J. Nutaro, Building Software for Simulation: Theory and Algorithms, with Applications in C ++.

Algorithms for cladograms include least squares, neighbor-joining, parsimony, maximum likelihood, and Bayesian inference.

Algorithms based on row or column reduction, that is, Gaussian elimination, presented in introductory linear algebra textbooks and in the preceding sections of this article are not suitable for a practical computation of the null space because of numerical accuracy problems in the presence of rounding errors.

One potential method for determining underpricing is through the use of IPO Underpricing Algorithms.

* Huang, Te-Ming ; Kecman, Vojislav ; and Kopriva, Ivica ( 2006 ); Kernel Based Algorithms for Mining Huge Data Sets, in Supervised, Semi-supervised, and Unsupervised Learning, Springer-Verlag, Berlin, Heidelberg, 260 pp. 96 illus., Hardcover, ISBN 3-540-31681-7

The following EBNF-like grammar ( for Niklaus Wirth's PL / 0 programming language, from Algorithms + Data Structures = Programs ) is in LL ( 1 ) form:

* Algorithms for finding the solution to a NLLSQ problem require initial values for the parameters, LLSQ does not.