...(ed.) ''The Schur complement and its applications'', Numerical Methods and Algorithms '''4''' Springer (2005) {{ISBN|0-387-24271-6}} {{ZBL|1075.15002}}
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A branch of mathematics and computer science concerned with finding efficient algorithms, or computational procedures, for solving geometric problems. Such problems
...metric configurations. It is impossible to list all important problems and algorithms of computational geometry, and some basic concepts and results are consider
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...to find primality proofs for numbers of $ 1000 $ decimal digits. For these algorithms, see the overview in [[#References|[a4]]] and the original references [[#Re
..."> E. Bach, J. Shallit, "Algorithmic number theory" , '''1: Efficient Algorithms''' , MIT (1996)</TD></TR>
3 KB (485 words) - 22:01, 25 October 2014
...D valign="top">[2]</TD> <TD valign="top"> N.M. Nagornyi, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)</TD></TR></table>
2 KB (337 words) - 22:11, 5 June 2020
...ssical real numbers that are computable in some sense or other by means of algorithms. (See also [[Constructive analysis|Constructive analysis]].)
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The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Ba
...for inputs sharing some common properties. In a theoretical comparison of algorithms one either selects a class $P$ of inputs and defines the cost and the error
6 KB (908 words) - 18:44, 21 March 2024
...ociative calculus]]). The detailed definition and general theory of normal algorithms are set forth in [[#References|[1]]] (Chapts. I–V); see also [[#Reference
...riginal algorithm $\mathfrak A$. This convention is known in the theory of algorithms as the normalization principle. The formalization of the concept of an algo
6 KB (927 words) - 15:08, 24 December 2018
...ty, for designing and analyzing parallel combinatorial and graph-theoretic algorithms. The model focuses on the inherent level of computational parallelism witho
...n $O(\log n)$ time using $O(n)$ processors, parallel matrix multiplication algorithms running in $O(\log n)$ time using $O(n^3/\log n)$ processors, and string ma
6 KB (940 words) - 23:19, 25 November 2018
...(i.e. through schemes of general recursion). Let the complexity of normal algorithms and of Turing machines mean, respectively, the length of the encoding and t
The study of the complexity of algorithms solving the finite restrictions of algorithmically unsolvable problems (the
8 KB (1,158 words) - 16:10, 1 April 2020
...tions has been an object of study in the [[Algorithms, theory of|theory of algorithms]] from the very beginning of its creation. In particular, this analogy lies
2 KB (351 words) - 21:15, 29 January 2017
Genetic algorithms (GAs) (also known as evolutionary algorithms [EAs]) assume a discrete search space $H$ and a function $f:H\to \R$. The
[1] Back, T., Evolutionary Algorithms in Theory and Practice, Oxford University Press, 1996
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* Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) {{ISBN|940107691X}} {{ZBL|0709.68004}}
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...ic countries. It is supposed to be a standard language for the exchange of algorithms between socialistic countries. It is based on the language Algol-60 (cf. [[
..."top">[1]</TD> <TD valign="top"> "Description of the language ALGAMS" , ''Algorithms and algorithmic languages'' : 3 , Moscow (1968) pp. 3–56 (In Russian)
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Some authors make a specific distinction between algorithms and procedures. Algorithms are prescriptions of sequences of actions which should always terminate aft
3 KB (398 words) - 16:54, 7 February 2011
...~/encyclopedia/old_files/data/O068/O.0608550 Optimization of computational algorithms
...possessing these properties. For this reason, in the theoretical study of algorithms one introduces a class of problems $ P $
12 KB (1,798 words) - 20:07, 12 January 2024
...e treated as an aspect of the [[idempotent correspondence principle]]. For algorithms, this heuristic principle means that "if one has an important and interest
...algorithms can often be approximated by linear ones. It is well-known that algorithms from linear algebra are convenient for parallel computations (see, e.g. [[#
7 KB (1,079 words) - 20:39, 16 November 2023
...osely connected with the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]), particularly so with the theory of abstract machines, since
...an automaton and that of obtaining corresponding estimates. Fairly simple algorithms have been developed for finite automata making it possible, with the aid of
7 KB (975 words) - 18:49, 5 April 2020
...rithm. Exact definitions of computable functions, effective procedures and algorithms were given in various forms by D. Hilbert, K. Gödel, A. Church, S.C. Kleen
...an algorithm are Turing machines, partial recursive functions, the normal algorithms of Markov, and others.
18 KB (2,635 words) - 19:56, 5 September 2017
..., "Separability with respect to invariants" , ''Studies on the theory of algorithms and mathematical logic'' , '''1''' , Moscow (1973) pp. 205–210 (In Rus
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...The following universal functions play an important role in the theory of algorithms: 1) universal partial recursive functions for the class of all $ n $-
...TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
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...TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
1 KB (148 words) - 17:24, 7 February 2011
...d decoding algorithms that are close in their possibilities to the optimal algorithms whose existence has been proved in the theory of information transmission.
...bed on an intuitive level and have no clear mathematical description. Such algorithms always include learning processes with respect to a list of objects that ar
7 KB (993 words) - 17:28, 7 February 2011
In the theory of algorithms more special kinds of $tt$-reducibility are also considered, for instance b
...gn="top"> A.N. Degtev, "Reducibilities of tabular type in the theory of algorithms" ''Russian Math. Surveys'' , '''34''' : 3 (1979) pp. 155–192 ''Uspek
3 KB (414 words) - 21:57, 16 January 2016
...(ed.) ''The Schur complement and its applications'', Numerical Methods and Algorithms '''4''' Springer (2005) {{ISBN|0-387-24271-6}}{{ZBL|1075.15002}}
759 bytes (114 words) - 12:25, 12 November 2023
..., W.H. Payne, "Generalized feedback shift register pseudorandom number algorithms" ''J. ACM'' , '''20''' (1973) pp. 456–468</td></tr><tr><td valign="top
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The use of adaptive algorithms is now (1998) very widespread across such varied applications as system ide
The study of adaptive algorithms is generally made through the theory of [[Stochastic approximation|stochast
8 KB (1,242 words) - 15:31, 1 July 2020
...n the computational complexity of a problem; the analysis of all permitted algorithms together with the structure of a problem can provide lower bounds (cf. also
Consider a computational model for uniform algorithms over a set $\Sigma$. Two prominent examples are
7 KB (1,161 words) - 16:46, 1 July 2020
.../TD> <TD valign="top"> M. Minoux, "Mathematical programming: theory and algorithms" , Wiley (1986)</TD></TR>
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...hms with regular closure does not coincide with the set of actually stable algorithms, construction of a closed algorithm frequently helps in investigating the s
...>[6]</TD> <TD valign="top"> A.F. Shapkin, "Closure of two computational algorithms, based on the idea of orthogonalization" ''Zh. Vychisl. Mat. i Mat. Fiz.''
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...<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
1 KB (172 words) - 16:25, 1 May 2014
...ly-increasing function from the natural numbers onto $A$. In the theory of algorithms the important characteristics of a direct counting of a set are recursivene
1 KB (167 words) - 11:07, 17 March 2023
...the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the [[Simplex method|simplex method]]. Some of
...<TD valign="top"> M. Grötschel, L. Lovász, A. Schrijver, "Geometric algorithms and combinatorial optimization" , Springer (1987)</TD></TR><TR><TD valign=
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...ersion of the first one. The methods of solution here consist of arbitrary algorithms and non-deterministic formally-describable procedures.
==Equivalent transformations of algorithms.==
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...discrete optimization problems are naturally formulated in terms of local algorithms.
The starting theorems for local algorithms are the uniqueness theorem and the theorem on the existence of a best algor
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...many geometric and analytical properties. There are elegant and efficient algorithms for evaluation, differentiation, subdivision of the curves, and conversion
...s that allows exact representation of conic sections. These properties and algorithms make the curves popular in areas where shape is important. In addition to C
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.... Combinatorial Gray codes are useful because they often lead to efficient algorithms for generating (i.e., listing) the objects. Some of the more important comb
...TD valign="top"> C.D. Savage, "Gray code sequences of partitions" ''J. Algorithms'' , '''10''' (1989) pp. 577–595</TD></TR><TR><TD valign="top">[a6]</TD>
7 KB (1,089 words) - 19:42, 5 June 2020
...hastic Lyapunov functions have been used to prove convergence of recursive algorithms driven by stochastic processes. Convergence problems of this type arise in
...Schuppen, "Convergence results for continuous-time stochastic filtering algorithms" ''J. Math. Anal. Appl.'' , '''96''' (1983) pp. 209–225</TD></TR></tab
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A basic concept in the branch of the theory of algorithms called enumeration theory, which investigates general properties of classes
...ation of this idea it was noticed that many known results in the theory of algorithms turn out to be consequences of general regularities in enumeration theory.
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...of abstract computers by other classes (e.g. in order to simulate parallel algorithms by sequential ones).
...ty of their structure and also by the type of information processed by the algorithms. The main classes of abstract computers are listed below.
9 KB (1,295 words) - 17:46, 4 June 2020
$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of
...[Turing machine|Turing machine]]). Subsequent development of the theory of algorithms is due to the studies of Kleene, Post [[#References|[6]]], [[#References|[7
19 KB (2,846 words) - 16:10, 1 April 2020
...programming language has a universal character, allowing one to formulate algorithms for solving various problems that are to be solved on different computers.
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* Rajeev Motwani, Prabhakar Raghavan, "Randomized Algorithms", Cambridge University Press (1995) {{ISBN|978-0-521-47465-8}} {{ZBL|0849.6
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...ctionals. In discrete programming the algorithms employed are analogues of algorithms of local optimization, descent, random search, etc. For a review of approxi
...to find the extremum or a good approximation to it in the class of simple algorithms; this collection is often effectively described. This was first noted in so
10 KB (1,391 words) - 12:30, 17 February 2021
...such as the memory management aspects and computational complexity of such algorithms (cf. also [[Complexity theory]] and [[Algorithm, computational complexity o
<TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987)</TD></TR>
5 KB (738 words) - 19:10, 17 December 2015
<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
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian) {{ZBL|0663.03023}}</TD></TR>
3 KB (544 words) - 07:12, 10 November 2023
...used method for abbreviating proofs, and is often useful in improving the algorithms for the establishment of derivability. One of the most important results in
1 KB (207 words) - 19:21, 17 October 2014
...s present no difficulties). Also, rational numbers, algebraic polynomials, algorithms, and calculi of various well-defined types (cf. [[Algorithm|Algorithm]]; [[
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Reidel (1988) (Translated from Russian) {{ZBL|0663.03023}}</TD></TR>
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There are efficient algorithms to solve this problem. Let $ X $
...valign="top">[a9]</TD> <TD valign="top"> J. Kennington, R. Helgason, "Algorithms for network programming" , Wiley (1980)</TD></TR></table>
7 KB (1,074 words) - 20:15, 15 March 2023
...mathematical logic — the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]). The growing stream of information and the related problems o
...re of discrete analysis, related to problems in finite structures, is that algorithms by which these problems may be solved exist in most cases, while in classic
9 KB (1,264 words) - 16:56, 15 April 2012
* Eric Bach, Jeffrey O. Shallit, ''Algorithmic Number Theory: Efficient algorithms, Volume 1'', MIT Press (1996) {{ISBN|0262024055}}
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