...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=
3 KB (453 words) - 18:48, 5 April 2020
...ersion of the first one. The methods of solution here consist of arbitrary algorithms and non-deterministic formally-describable procedures.
==Equivalent transformations of algorithms.==
15 KB (2,362 words) - 18:45, 27 April 2014
...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
18 KB (2,562 words) - 18:06, 1 April 2020
...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
5 KB (759 words) - 09:49, 27 March 2023
.... 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
5 KB (718 words) - 04:11, 6 June 2020
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.
12 KB (1,899 words) - 19:37, 5 June 2020
...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.
4 KB (544 words) - 08:07, 6 June 2020
* Rajeev Motwani, Prabhakar Raghavan, "Randomized Algorithms", Cambridge University Press (1995) {{ISBN|978-0-521-47465-8}} {{ZBL|0849.6
1 KB (173 words) - 07:29, 14 November 2023
...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>
4 KB (587 words) - 20:53, 8 December 2015
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}}
1 KB (186 words) - 16:57, 25 November 2023
...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