...been constructed, based on certain intuitive principles, i.e. families of algorithms for solving classification problems. The most widely used are: models based
...ication is a priori known. In developing the general theory of recognition algorithms the most complete results have been obtained within an algebraic framework.
8 KB (1,189 words) - 13:43, 17 April 2014
...<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
2 KB (216 words) - 17:08, 7 February 2011
One of the basic concepts in the [[Algorithms, theory of|theory of algorithms]] and its applications. Its background was the fact that solvability (and u
...so-called priority method, which is extensively employed in the theory of algorithms.
6 KB (862 words) - 19:26, 22 January 2016
...ecial method for constructing algorithms based on the requirement that the algorithms should be exact or have an error of prescribed order of accuracy on some se
4 KB (621 words) - 11:59, 12 August 2014
...[[Automatic programming|automatic programming]] and for the publication of algorithms (it is an abbreviation of "ALGOrithmic Language" ).
...lly refers to "Algol-60" . It is especially suited for the description of algorithms of numerical analysis. The language is machine-independent and does not pro
4 KB (632 words) - 17:21, 7 February 2011
...]. The Scarf algorithm and other related ones rapidly evolved into various algorithms for solving equations, for finding extrema and for tracing bifurcating solu
...as a very extensive bibliography. A substantial number of previously known algorithms for solving equations can be obtained as particular cases of the homotopy c
9 KB (1,325 words) - 17:26, 7 February 2011
...nald E. Knuth, ''The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1'', Addison-Wesley (2014) {{ISBN|0133488853}}
1 KB (199 words) - 14:15, 12 November 2023
there are algorithms which are particularly convenient in practice. There is a considerable numb
...>[a4]</TD> <TD valign="top"> D.F. Elliot, K.R. Rao, "Fast transforms: algorithms, analysis, applications" , Acad. Press (1982)</TD></TR>
4 KB (602 words) - 14:32, 17 March 2023
...computable functions and theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]; [[Computable function|Computable function]]) is any mapping o
6 KB (919 words) - 08:26, 6 June 2020
* Rajeev Motwani, Prabhakar Raghavan, "Randomized Algorithms", Cambridge University Press (1995) {{ISBN|978-0-521-47465-8}} {{ZBL|0849.6
1 KB (226 words) - 07:28, 14 November 2023
...o [[Optimization of computational algorithms|Optimization of computational algorithms]]; [[Index transform|Information-based complexity]]; [[Curse of dimension|C
...gn="top"> G.W. Wasilkowski, H. Woźniakowski, "Weighted tensor-product algorithms for linear multivariate problems" ''Preprint'' (1998)</td></tr><tr><td va
8 KB (1,086 words) - 17:43, 1 July 2020
...first such algorithm was proposed by H. Scarf, [[#References|[a3]]]. Such algorithms later developed in the so-called homotopy or continuation methods for calcu
...n="top">[a4]</TD> <TD valign="top"> S. Karamadian (ed.) , ''Fixed points. Algorithms and applications'' , Acad. Press (1977)</TD></TR>
4 KB (586 words) - 11:27, 17 March 2023
The accuracy of the algorithms based on the interpolation formulas of Lagrange, Newton and others strongly
...tly smooth and if the number of nodal points is large [[#References|[1]]]. Algorithms of numerical differentiation involving spline-interpolation [[#References|[
5 KB (779 words) - 19:35, 5 June 2020
...</TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987)</TD></TR><TR><TD valign="top"
2 KB (247 words) - 17:32, 5 June 2020
...tape scanned by the head of the machine. Time and storage costs for normal algorithms (cf. [[Normal algorithm|Normal algorithm]]), iterative arrays, multi-head a
better than other algorithms" . However, as exemplified by the speed-up theorem (see below), such a form
16 KB (2,413 words) - 16:10, 1 April 2020
1) Provide algorithms turning $ R [ {\mathcal X} ] $
By keeping track of intermediate results in the algorithms, it is possible to express the Gröbner basis $ C $
10 KB (1,471 words) - 14:52, 30 May 2020
...theory of calculi is the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]). This is clear from the fact that the notion of a calculus is
...<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
8 KB (1,204 words) - 18:51, 13 November 2014
...to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. [[Iteration algorithm|Iteration algorithm]]).
...m state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [[#References|[a3]]].
4 KB (649 words) - 05:25, 19 March 2022
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
2 KB (250 words) - 20:46, 23 November 2023
...m the previous one, usually by a small amount. The convergence analysis of algorithms like (a7) is treated in e.g. [[#References|[a6]]], [[#References|[a1]]], [[
...</td> <td valign="top"> V. Solo, X. Kong, "Adaptive signal processing algorithms" , Prentice-Hall (1995)</td></tr></table>
7 KB (1,131 words) - 14:45, 16 December 2023
...t in real problems the computations may be too laborious. If so, heuristic algorithms are used to generate better patterns: as a rule, it is enough to consider c
...isted. For linear and rectangular parts applicable heuristic approximation algorithms have been programmed (see [[#References|[2]]]).
8 KB (1,179 words) - 17:25, 27 October 2014
...algorithms to test primality of numbers and, if composite, to factor them. Algorithms of this kind which can handle numbers of 100 or 200 digits have recently at
6 KB (955 words) - 19:40, 1 November 2023
...rocedures that solve every instance of the problem? Which problems possess algorithms for their solution?
...ion is of definite practical significance: One should not try to construct algorithms for an unsolvable problem. (There are some notorious examples of such attem
13 KB (2,014 words) - 20:35, 31 December 2018
This approach makes it possible to construct numerical algorithms for a wide class of differential equations, including partial differential
...polation of operators and functionals in the construction of computational algorithms for solving concrete problems is based on the use of interpolation formulas
26 KB (3,882 words) - 22:13, 5 June 2020
...heorem, one asks to find these factors algorithmically, and to devise such algorithms with low cost.
...lled polynomial representation of $\sigma$ as its basic tool. The last two algorithms are based on Gauss' theorem that $x ^ { q ^ { d } } - x$ is the product of
12 KB (1,739 words) - 13:11, 26 March 2023
...le computation one has to use specialized [[Software|software]]. To encode algorithms on a computer, special language means are required. These are also consider
2 KB (270 words) - 17:01, 7 February 2011
.... As shown in [[#References|[a3]]], this suffices to guarantee linear-time algorithms for many otherwise hard graph problems when instances are confined to Halin
...Borie, R.G. Parker, C.A. Tovey, "Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed
5 KB (722 words) - 20:45, 16 March 2023
* Donald E. Knuth, "Introduction to combinatorial algorithms and Boolean functions", ser. The Art of Computer Programming 4.0 (2008) pp.
2 KB (261 words) - 18:11, 14 October 2023
...for the modern precise statement of this problem as given by the theory of algorithms (cf. [[Church thesis|Church thesis]]). Other examples of such group calculi
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)</TD></TR></table>
9 KB (1,334 words) - 19:42, 5 June 2020
...nts take the values 0 or 1 only. Conditions a) and b) enable one to devise algorithms for the solution of the transportation problem that are essentially simpler
The best known of these algorithms are the method of potentials and the so-called Hungarian method. The method
5 KB (746 words) - 14:56, 7 June 2020
...rogram, i.e. a collection of machine commands describing the computational algorithms, organizing the realization of the computational process in the given compu
...onal steps, method of]]). An important task in the theory of computational algorithms is their optimization.
10 KB (1,642 words) - 08:41, 29 August 2014
Already in the Middle Ages a number of algorithms for constructing magic squares of odd order $ n $
Many algorithms for constructing magic squares are known (resulting in squares with non-lin
5 KB (762 words) - 19:27, 12 January 2024
..."A space-efficient on-line method of computing quantile estimators" ''J. Algorithms'' , '''2''' (1981) pp. 164–177</TD></TR></table>
2 KB (281 words) - 17:22, 7 February 2011
...n="top"> J.P. Duval, "Factorizing words over an ordered alphabet" ''J. Algorithms'' , '''4''' (1983) pp. 363–381</TD></TR>
6 KB (946 words) - 13:41, 20 March 2023
...se of development of the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]), there emerged a number of modifications of the original defi
===Representing Algorithms by Turing Machines===
14 KB (2,355 words) - 12:40, 28 December 2013
...is proved by providing exponentially large lower bounds that hold for all algorithms. Many problems suffer from the curse of dimension. Examples include numeric
...o [[Optimization of computational algorithms|Optimization of computational algorithms]]. Whether a problem suffers from the curse of dimension depends on the exa
12 KB (1,706 words) - 20:29, 9 December 2023
...nce, or they may vary during the calculation. For this reason, one designs algorithms that enable one to use grids that are more closely spaced in such zones.
In order to set up reliable numerical algorithms, the problem of constructing a grid is often formulated as a task of minimi
5 KB (759 words) - 10:59, 16 April 2014
...>[a3]</TD> <TD valign="top"> J.M. Robertson, W.A. Webb, "Cake-cutting algorithms: be fair if you can" , A.K. Peters (1998)</TD></TR><TR><TD valign="top">[a
2 KB (317 words) - 16:50, 30 December 2018
...ly analyzed as well as implemented. Especially, primal-dual interior-point algorithms (i.e., methods generating primal and dual solutions in each iteration) prov
...ign="top"> Yu. Nesterov, A.S. Nemirovskii, "Interior point polynomial algorithms in convex programming" , ''Studies in Applied Mathematics'' , '''13''' , SI
5 KB (710 words) - 22:13, 5 June 2020
.../TD> <TD valign="top"> M. Minoux, "Mathematical programming: theory and algorithms" , Wiley (1986)</TD></TR></table>
3 KB (372 words) - 08:07, 22 November 2014
...includes, in particular, the self-scaling variable metric algorithms (SSVM algorithms), which share most properties of the Broyden family and automatically compe
...op"> S.S. Oren, D.G. Luenberger, "Self-scaling variable metric (SSVM) algorithms I" ''Management Science'' , '''20''' (1974) pp. 845–862</td></tr><tr><
12 KB (1,897 words) - 19:37, 9 February 2024
...[[Network|network]] or [[Graph|graph]] that satisfy a given property. Such algorithms are also used, e.g., to optimize a function on a graph or network or on a s
There is an obvious analogous search algorithm for non-oriented graphs. These algorithms run in $ O ( \# V + \# E ) $
11 KB (1,442 words) - 08:12, 6 June 2020
In practice, time-consuming scanning is combined with algorithms for finding a local extremum: by scanning and a priori reduction of $ f (
1) Algorithms of the heavy-sphere type (cf. [[Heavy sphere, method of the|Heavy sphere, m
9 KB (1,335 words) - 08:01, 6 June 2020
...and hence the Berlekamp–Massey algorithm, has connections to several other algorithms, most notably the extended [[Euclidean algorithm|Euclidean algorithm]] [[#R
...gorithms in numerical analysis such as Lanczos recursion and Levinson–Shur algorithms for Toeplitz matrices, as well as problems of minimal realizations in syste
8 KB (1,155 words) - 18:48, 26 January 2024
...TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987) {{MR|0904271}} {{ZBL|0634.5200
2 KB (301 words) - 08:28, 6 June 2020
Hierarchical clustering algorithms can be characterized as ''greedy''
The algorithms, which have the potential for ''exactly'' replicating
13 KB (1,912 words) - 18:57, 7 March 2024
...estimate of the complexity of the amount of work and the specification of algorithms (see also [[Algorithmic information theory|Algorithmic information theory]]
...hnik, "On the unsolvability of the reducibility problem in the theory of algorithms" ''Dokl. Akad. Nauk SSSR'' , '''108''' : 2 (1956) pp. 194–197 (In Ru
6 KB (921 words) - 17:32, 5 June 2020
...recent survey of the simplex algorithm, the Karmarkar algorithm (interior algorithms) and ellipsoid methods in relation to each other, cf. [[#References|[a8]]].
...p">[a6]</TD> <TD valign="top"> A.R.G. Heesterman, "Matrices and simplex algorithms" , Reidel (1983)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">
8 KB (1,246 words) - 06:35, 7 May 2022
A method of defining functions studied in the theory of algorithms and other branches of mathematical logic. This method has been used for a l
...nition of recursion consists not only in its significance in the theory of algorithms, but also in that it permits one to look from an "algorithmic" (in the ge
13 KB (2,044 words) - 08:10, 6 June 2020
factorization. Such decompositions play an important role in numerical algorithms, [[#References|[a2]]], [[#References|[a3]]] (for instance, in computing eig
2 KB (350 words) - 08:26, 6 June 2020