...bature (for calculating multiple integrals) formulas have been derived for numerical integration (cf. [[Quadrature formula]]; [[Cubature formula]]).
See also [[Interpolation in numerical mathematics]].
1 KB (164 words) - 08:00, 16 April 2023
...therwise it is called defective. The notion is of particular importance in numerical [[Linear-algebra(2)|linear algebra]].
* D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , '''2''' , Dover, reprint (1988) pp. 741–743
749 bytes (104 words) - 14:06, 19 November 2023
''in numerical analysis''
89 bytes (10 words) - 16:55, 7 February 2011
[[Category:Numerical analysis and scientific computing]]
269 bytes (32 words) - 16:33, 26 October 2014
...heory of approximation of functions and the theory of numerical methods in analysis.
763 bytes (111 words) - 17:25, 7 February 2011
''accuracy analysis''
A systematic study of the precision and errors of numerical and statistical calculation and estimation procedures. See, e.g., [[Error|E
206 bytes (29 words) - 17:24, 7 February 2011
...rule, is assumed to be known. In general, $\tau$ is a parameter that is a numerical characteristic of the approximating set (e.g. its dimension) or of the meth
In numerical analysis, the approximation order of a numerical method having error $O(h^m)$, where $h$ is the step of the method, is the e
2 KB (326 words) - 09:43, 26 April 2014
In the numerical solution of a problem, the error in the result is due to inaccuracies occur
...TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Voevodin, "Numerical methods of algebra" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign=
3 KB (519 words) - 22:31, 1 November 2014
...(cf. [[Fourier integral|Fourier integral]]), which are objects of harmonic analysis.
...[a2]</TD> <TD valign="top"> Y. Katznelson, "An introduction to harmonic analysis" , Dover, reprint (1976) {{MR|0422992}} {{ZBL|0352.43001}} </TD></TR></ta
2 KB (287 words) - 11:59, 27 September 2012
$#C+1 = 34 : ~/encyclopedia/old_files/data/D032/D.0302320 Differentiation, numerical
Finding the derivative of a function by numerical methods. Such differentiation is resorted to when the methods of [[Differen
5 KB (779 words) - 19:35, 5 June 2020
...lign="top">[3]</TD> <TD valign="top"> A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , '''1–2''' , Birkhäuser (1989) (Translate
2 KB (287 words) - 12:05, 13 August 2014
...lign="top">[6]</TD> <TD valign="top"> A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , '''1–2''' , Birkhäuser (1989) (Translate
2 KB (295 words) - 14:06, 14 August 2014
...g of the nodes by their distances from the interpolation point reduces the numerical error in the interpolation.
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
3 KB (368 words) - 08:38, 13 May 2022
...d; this problem is concerned with the testing of hypotheses concerning the
numerical values of several contrasts.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Scheffé, "
Analysis of variance" , Wiley (1959)</TD></TR></table>
1 KB (152 words) - 08:11, 13 February 2024
are known, and the numerical vector $ h = ( h _ {1} \dots h _ {n} ) ^ {T} $
This problem can be solved using numerical methods. In order to solve (5) it is usually necessary to choose some itera
4 KB (528 words) - 08:13, 6 June 2020
...algorithms of [[Statistical modelling|statistical modelling]] used in the numerical research into random processes and phenomena, and algorithms of the [[Monte
...ind (see [[#References|[4]]]). Particularly effective are those stochastic numerical algorithms that allow a number of realizations of the algorithm to be made
5 KB (649 words) - 17:24, 7 February 2011
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a4]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. §10.5</TD></TR></table>
5 KB (691 words) - 16:24, 21 June 2020
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...p">[a4]</TD> <TD valign="top"> P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)</TD></TR><TR><TD valign="top">[a5]</TD>
4 KB (614 words) - 21:36, 1 January 2019
...terature these formulas are known as the Newton–Cotes formulas. A detailed analysis of them can be found in [[#References|[a1]]], [[#References|[a3]]], [[#Refe
...p">[a3]</TD> <TD valign="top"> P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)</TD></TR>
2 KB (240 words) - 21:28, 18 January 2018
$#C+1 = 232 : ~/encyclopedia/old_files/data/I051/I.0501950 Interpolation in numerical mathematics
...rpolation of functionals and operators is also widely used in constructing numerical methods.
26 KB (3,882 words) - 22:13, 5 June 2020
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a2]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR>
2 KB (335 words) - 14:03, 30 April 2023
...o method|Monte-Carlo method]]; [[Stochastic numerical algorithm|Stochastic numerical algorithm]]) and mixed settings.
...atics, statistics, complexity theory, algorithmic analysis, number theory, analysis and measure theory have all been influential. See also [[#References|[a2]]]
6 KB (871 words) - 17:45, 1 July 2020
...">[a3]</TD> <TD valign="top"> C.-E. Froberg, "Introduction to numerical analysis" , Addison-Wesley (1965) pp. 157</TD></TR></table>
3 KB (346 words) - 08:23, 6 June 2020
...roblems (see [[Non-linear equation, numerical methods|Non-linear equation, numerical methods]]). In order to realise these modifications it is important to be a
.../TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Glowinski, "Numerical methods for nonlinear variational problems" , Springer (1984)</TD></TR><TR
6 KB (948 words) - 15:32, 14 February 2020
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a3]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR></table>
4 KB (623 words) - 07:44, 14 January 2024
...valign="top">[3]</TD> <TD valign="top"> G.I. Marchuk, V.I. Lebedev, "Numerical methods in the theory of neutron transport" , Harwood (1986) (Translated
...ve) parabolic problem (cf. [[#References|[a2]]]). Because of the inherent (numerical) stiffness an implicit discretization method, such as BDF, should be advoca
4 KB (649 words) - 05:25, 19 March 2022
Numerical methods for obtaining two-sided estimates (two-sided approximations) are kn
...<TD valign="top"> V.I. Devyatko, "On a two-sided approximation for the numerical integration of ordinary differential equations" ''USSR Comp. Math. Math. P
8 KB (1,164 words) - 08:27, 6 June 2020
...> <TD valign="top"> J. Stoer, R. Bulirsch, "Introduction to numerical analysis" , Springer (1993) pp. 338ff</TD></TR>
936 bytes (133 words) - 22:28, 22 November 2016
The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Bayesian approach|Bayesian appro
...worst-case sense over the class $P$. Alternatively, in Bayesian numerical analysis, one puts an [[A priori distribution|a priori distribution]] $\mu$ on the i
6 KB (908 words) - 18:44, 21 March 2024
...d in [[Variational calculus|variational calculus]]. There are also various numerical approximation methods for finding maximum and minimum points.
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
5 KB (801 words) - 08:00, 6 June 2020
...uations, to produce by means of the well-developed apparatus of continuous
analysis preliminary results on the convergence and optimality of iteration methods,
...ath.org/legacyimages/c/c025/c025600/c02560045.png" /> and using for (12) a
numerical discretization formula at the points <img align="absmiddle" border="0" src=
15 KB (2,038 words) - 17:21, 7 February 2011
...al (cf. [[Chebyshev polynomials|Chebyshev polynomials]]) in the theory of (numerical) interpolation, integration, etc. [[#References|[a1]]].
...valign="top"> L. Fox, I. Parker, "Chebyshev polynomials in numerical analysis" , Oxford Univ. Press (1968)</TD></TR></table>
2 KB (296 words) - 16:59, 22 November 2018
A method to construct an approximating equation for approximate (and numerical) solutions of certain kinds of linear and non-linear integral equations. Th
...="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)</TD></TR></table>
3 KB (401 words) - 17:32, 5 June 2020
...olume of the medium is large. The method has been successfully used in the analysis of the optical fibre oximeter of blood. The use of the so-called Henyey–G
...ow theory]]; [[Transport equations, numerical methods|Transport equations, numerical methods]].
2 KB (336 words) - 17:10, 7 February 2011
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...p"> G.M. Phillips, P.J. Taylor, "Theory and applications of numerical analysis" , Acad. Press (1973)</TD></TR></table>
4 KB (640 words) - 14:37, 13 January 2024
A typical problem used as a model for investigating and developing numerical methods for some class of problems. For example, in the theory of [[Quadrat
...align="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Ru
2 KB (281 words) - 08:01, 6 June 2020
...cumulated errors in larger calculations. Analysis of accumulated errors in numerical methods permits one to describe methods according to their susceptibility t
...gn="top">[3]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
3 KB (515 words) - 18:22, 18 April 2023
...a particular case of the [[Adjustment method|adjustment method]]. For the numerical solution of (1), or (2), one may, use, e.g., difference methods. In depende
...</TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.G. Evtushenko, "Numerical optimization techniques" , Optim. Software (1985) (Translated from Russia
4 KB (519 words) - 22:10, 5 June 2020
...n="top">[a3]</TD> <TD valign="top"> B. Wendroff, "Theoretical numerical analysis" , Acad. Press (1966) pp. Chapt. 1</TD></TR></table>
3 KB (522 words) - 20:12, 10 January 2024
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
1,010 bytes (152 words) - 08:54, 25 November 2018
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...> <TD valign="top"> Maurice V. Wilkes, "A short introduction to numerical analysis", Cambridge University Press (1966) {{ISBN|0-521-09412-7}} {{ZBL|0149.10902
4 KB (592 words) - 08:09, 26 November 2023
...is [[Degenerate_matrix|singular]] then $\kappa(A)=\infty$. In [[numerical analysis]] the condition number of a matrix $A$ is a way of describing how well or b
974 bytes (157 words) - 19:31, 15 December 2020
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a2]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Addison-Wesley (1956)</TD></TR></table>
4 KB (533 words) - 10:58, 29 May 2020
...)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G.N. Lance, "Numerical methods for high speed computers" , Iliffe (1960)</TD></TR></table>
...e) solution $y^1$ is three. Hence, in general, the difference of these two numerical approximations is only of order three, so that a conservative error estimat
7 KB (1,053 words) - 17:13, 14 February 2020
...use of electronic computers made it necessary to impose new conditions on numerical methods; the main problem at that stage was to develop new, "computer-frie
...arious fields of scientific and everyday activity, and may be described as analysis of mathematical models. The second is the development of methods and algori
13 KB (1,892 words) - 20:11, 31 December 2018
...[a3]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR></table>
3 KB (505 words) - 14:56, 14 February 2020
...opedia/old_files/data/N067/N.0607080 Non\AAhlinear boundary value problem, numerical methods
...roblems for systems of ordinary differential equations, the description of numerical methods usually proceeds without indication of a discretization of the orig
19 KB (2,657 words) - 14:54, 7 June 2020
A method for approximating an [[integral operator]] by constructing numerical methods for the solution of integral equations. The simplest version of a q
...align="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.353
2 KB (335 words) - 16:32, 13 July 2021
...dmund Whittaker, G. Robinson, "The calculus of observations. A treatise on numerical mathematics" (4th ed.) Blackie & Son (1954) p.99. {{ZBL|0058.33603}}</TD></
...<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.S. Householder, "The numerical treatment of a single nonlinear equation" , McGraw-Hill (1970)</TD></TR></
3 KB (496 words) - 09:54, 29 May 2020
...ds of gas dynamics (cf. [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]]), there are 6 different forms of viscosity matrices (see [[#Re
...ference schemes (see [[#References|[10]]]), it is possible, by varying the numerical values of the parameters, to change the values of the terms of the viscosit
7 KB (987 words) - 14:03, 24 December 2020
$#C+1 = 31 : ~/encyclopedia/old_files/data/I052/I.0502080 Interval analysis
A theory intended for the calculation of rounding errors of calculations on numerical computers. Since not all numbers can be represented exactly in a computer w
5 KB (677 words) - 22:13, 5 June 2020
[[Category:Numerical analysis and scientific computing]]
1 KB (217 words) - 18:54, 19 October 2014
...tandard procedure is usually applied to the use of mathematical methods in numerical calculations on a computer (computational mathematics, mathematical statist
...]</TD> <TD valign="top"> V.V. Voevodin, O.B. Arushanyan, , ''Numerical Analysis in FORTRAN'' , Moscow (1979) pp. 73–84</TD></TR><TR><TD valign="top">[2
3 KB (372 words) - 17:12, 7 February 2011
...htarrow \mathbf{C} ^ { n }$, is concerned with the question of whether the numerical discretization inherits the dynamic properties of the differential equation
...solutions is a non-increasing function of $x$. Let $y_1$, $z_1$ denote the numerical solutions after one step of size $h$ with initial values $y _ { 0 }$, $z_0$
9 KB (1,275 words) - 17:43, 1 July 2020
...s for the numerical integration (cf. [[Integration, numerical|Integration, numerical]]) of diffusion problems, introduced by J. Crank and P. Nicolson [[#Referen
and numerical initial condition
14 KB (2,138 words) - 08:45, 28 January 2024
are the numerical parameters of the method. If $ \epsilon ^ {k} = u - u ^ {k} $,
...tor equations, and polynomials deviating least from zero" , ''Mathematical analysis and related problems in mathematics'' , Novosibirsk (1978) pp. 89–108
16 KB (2,300 words) - 19:29, 17 January 2024
...rete mathematics and discrete analysis. In what follows the term "discrete analysis" is understood in the wide meaning of the word, including discrete mathemat
As distinct from discrete analysis, classical mathematics deals mainly with continuous objects. The choice bet
9 KB (1,264 words) - 16:56, 15 April 2012
...p">[a5]</TD> <TD valign="top"> P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)</TD></TR></table>
3 KB (420 words) - 16:58, 14 February 2020
...numerical methods. If the integrand $f$ is given by function values, only numerical methods of the form
.... If adaptive information is superior to non-adaptive information, then an analysis of the trade-off between using adaptive or non-adaptive information on a pa
4 KB (650 words) - 15:07, 14 February 2020
...)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.E. Milne, "Numerical solution of differential equations" , Dover, reprint (1970)</TD></TR></tab
4 KB (538 words) - 17:45, 20 January 2022
...gn="top">[5]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a4]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. Chapt. 8 {{MR|0895822}} {{ZBL|0641.65001}}
4 KB (605 words) - 08:02, 6 June 2020
...e form (2), (3) are already implicit. This complicates significantly their numerical implementation: The values $ k _ {n} $,
The approaches to the construction of numerical methods considered above for equations of type (1) can be extended to ordin
9 KB (1,315 words) - 20:04, 15 January 2024
...ign="top"|{{Ref|Ba}}||valign="top"| N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations", MIR (1977) (Translated from R
...{Ref|KaAk}}||valign="top"| L.V. Kantorovich, G.P. Akilov, "Functional analysis", Pergamon (1982) (Translated from Russian) {{MR|0664597}} {{ZBL|0484.4
3 KB (519 words) - 19:10, 6 April 2012
...d in some way with the existence of an invariant structure (see [[Harmonic analysis, abstract]]), when they are eigen functions of the [[Laplace–Beltrami equ
...ory (see [[#References|[6]]] and [[#References|[7]]]) and in the theory of numerical integration (see [[#References|[8]]]).
4 KB (575 words) - 20:24, 20 December 2016
$#C+1 = 73 : ~/encyclopedia/old_files/data/I051/I.0501430 Integral equations, numerical methods
is a numerical parameter and $ K ( x , s ) $
12 KB (1,747 words) - 13:26, 14 January 2022
...op"> W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, "Numerical recipes" , Cambridge Univ. Press (1986) pp. 105ff</TD></TR>
3 KB (476 words) - 16:30, 29 March 2024
...attice points are also of importance in crystallography, coding, numerical analysis, analytic number theory, Diophantine approximation, computational geometry,
2 KB (239 words) - 19:03, 2 October 2016
...[a2]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR></table>
3 KB (490 words) - 17:39, 14 December 2020
|valign="top"|{{Ref|Ba}}||valign="top"| N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
1 KB (234 words) - 21:21, 15 July 2012
...</TD> <TD valign="top"> N.S. Bakhvalov, "On an estimate of the error at numerical integration of differential equations by Adams' extrapolation method" ''Do
...d-held programmable calculator" G.H. Golub (ed.) , ''Studies in numerical analysis'' , Math. Assoc. Amer. (1984) pp. 199–242</TD></TR></table>
6 KB (888 words) - 21:01, 4 April 2020
...clopedia/old_files/data/E035/E.0305170 Eigen values of integral operators, numerical methods
Numerical methods for computing the complete spectrum of an integral operator or a pa
11 KB (1,669 words) - 19:37, 5 June 2020
$#C+1 = 29 : ~/encyclopedia/old_files/data/W097/W.0907160 Wavelet analysis
In wavelet analysis scaled and displaced copies of the basic wavelet $ g $
6 KB (909 words) - 08:28, 6 June 2020
...rithms. Many problems suffer from the curse of dimension. Examples include numerical integration, optimal recovery (approximation) of functions, global optimiza
...average-case setting (see [[Bayesian numerical analysis|Bayesian numerical analysis]]).
12 KB (1,706 words) - 20:29, 9 December 2023
...ology "best formula" is often encountered in the literature on numerical analysis, but, as was observed in [[#References|[a2]]], p. 75, it should be taken wi
...)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)</TD></TR></table>
4 KB (658 words) - 10:58, 29 May 2020
...ign="top">[1]</TD> <TD valign="top"> S.L. Sobolev, "Some remarks on the numerical solution of integral equations" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''20
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
4 KB (651 words) - 17:23, 14 February 2020
...10/b1100108.png" />) are of no use for an appropriate characterization and
analysis of methods which are able to efficiently integrate a stiff problem. Thus th
...a coefficients entailing B-stability was derived ( "algebraic stability in
numerical analysisalgebraic stability" ). The notion of <img align="absmiddle" border
11 KB (1,517 words) - 17:23, 7 February 2011
...lign="top">[4]</TD> <TD valign="top"> A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , '''1–2''' , Birkhäuser (1989) (Translate
.../TD> <TD valign="top"> U.M. Ascher, R.M.M. Mattheij, R.D. Russell, "Numerical solution for boundary value problems for ordinary differential equations" ,
7 KB (1,036 words) - 19:36, 5 June 2020
...solved, this system has the form of a recurrence relation here. For other numerical methods, see [[#References|[a1]]]. Volterra equations of the first kind are
...<TD valign="top"> A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)</TD></TR></table>
13 KB (1,966 words) - 08:28, 6 June 2020
A very simple finite-difference method for the numerical solution of an ordinary differential equation. Let a differential equation
The numerical algorithm of the Euler method can easily be programmed on a computer.
6 KB (877 words) - 19:38, 5 June 2020
...on fluctuations [[#References|[a1]]]. According to such a linear stability
analysis, all long wavelength concentration fluctuations with wavelengths exceeding
The linear stability
analysis of (a1) yields, writing <img align="absmiddle" border="0" src="https://www.
11 KB (1,535 words) - 17:13, 7 February 2011
...see [[Multi-dimensional statistical analysis|Multi-dimensional statistical analysis]]). If the results of observations $ X _ {1} \dots X _ {n} $
...m. This fact forms the basis of the [[Hotelling test|Hotelling test]]. For numerical calculations one uses tables of the [[Beta-distribution|beta-distribution]]
4 KB (519 words) - 22:11, 5 June 2020
...ferences|[a3]]] and J. Douglas [[#References|[a1]]] as a technique for the numerical solution of elliptic and parabolic differential equations (cf. [[Elliptic p
...valign="top">[a4]</TD> <TD valign="top"> R.S. Varga, "Matrix iterative analysis" , Prentice-Hall (1962)</TD></TR></table>
5 KB (654 words) - 01:55, 21 January 2022
...Therefore for the solution of these problems, alongside analytic methods, numerical simulation on computers is widely used.
...ithms and programs for the computer realization of the discrete models, an analysis of the sensitivity of the model to variations of the parameters, an estimat
15 KB (2,159 words) - 17:08, 7 February 2011
...iven exactly and is small or has special structure. On the other hand, the numerical computation of the Jordan form of a numerically given matrix is an ill-pose
For recent (1998) surveys of algorithms for numerical linear algebra problems, see [[#References|[a2]]], [[#References|[a3]]], [[
10 KB (1,478 words) - 17:02, 1 July 2020
...ncyclopedia/old_files/data/M065/M.0605140 Multi\AAhdimensional statistical analysis,
''multivariate statistical analysis''
27 KB (3,850 words) - 14:44, 7 June 2020
is ensured under the conditions that the numerical realization of the operations $ A _ {k} u ^ {k} $,
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
8 KB (1,124 words) - 11:46, 17 June 2020
...s heuristic principle means that "if one has an important and interesting numerical algorithm, then there is a good chance that its semi-ring analogues are imp
...bi and generalized Bellman equations, cf. [[Idempotent analysis|Idempotent analysis]]) deal with the corresponding finite-dimensional (or finite) "linear appr
7 KB (1,079 words) - 20:39, 16 November 2023
$#C+1 = 49 : ~/encyclopedia/old_files/data/L057/L.0507480 Laplace equation, numerical methods
...elliptic type (see [[#References|[1]]], [[#References|[2]]]) comprise many numerical methods for the Laplace equation. The specific character of the Laplace equ
9 KB (1,360 words) - 13:02, 13 January 2024
...go far beyond approximation considerations, as they appear to have a nice numerical stability (see [[Difference schemes, theory of|Difference schemes, theory o
...nomial spline spaces have been studied in [[#References|[a10]]]. A general analysis of projection methods (of which collocation is a special case) for the solu
12 KB (1,710 words) - 19:39, 18 January 2024
...s to numerical [[Quadrature|quadrature]] and to collocation methods in the numerical solution of functional equations.
...). Error estimation by extended interpolation is an important tool for the numerical approximation of linear functionals.
6 KB (756 words) - 20:24, 5 December 2023
...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 275ff</TD></TR><TR><TD valign="top">[a2]</TD
6 KB (778 words) - 14:32, 13 January 2024
In the case of formulas for numerical integration, often the coordinates of the nodes of integration stand out as
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
4 KB (621 words) - 11:59, 12 August 2014
...dimensional quantities) have the same numerical values (see [[Dimensional analysis]]). The converse conclusion is also correct, i.e. if all the corresponding
Dimensional analysis and similarity theory are closely related and are used in experiments with
10 KB (1,521 words) - 19:59, 4 January 2024
.../encyclopedia/old_files/data/L059/L.0509110 Linear boundary value problem, numerical methods
...sufficiently many derivatives. Because of the absence of other assumptions numerical methods differ in their universality.
17 KB (2,540 words) - 19:55, 12 January 2024
<TR><TD valign="top">[a4]</TD> <TD valign="top"> J.M. Ortéga, "Numerical analysis" , Acad. Press (1972) {{MR|0403154}} {{ZBL|0248.65001}} </TD></TR>
..."top">[a1]</TD> <TD valign="top"> C.E. Fröberg, "Introduction to numerical analysis, theory and applications" , Benjamin/Cummings (1985)</TD></TR>
7 KB (1,025 words) - 22:14, 5 June 2020
are numerical parameters.
For numerical computations the matrix of a variational difference scheme should not be to
22 KB (3,256 words) - 17:33, 5 June 2020
...nal inequalities see [[#References|[a6]]], [[#References|[a7]]]; for their numerical study, see [[#References|[a8]]], [[#References|[a9]]]. Eigenvalue problems
...">[a9]</TD> <TD valign="top"> J. Haslinger, I. Hlavaček, J. Nečas, "Numerical methods for unilateral problems" P.G. Giarlet (ed.) J.L. Lions (ed.) , ''
5 KB (737 words) - 20:35, 18 March 2024
...utility theory deals with ordered sets and their monotone mappings into a numerical space (usually one-dimensional). Utility theory arose from researches by ec
...p">[a5]</TD> <TD valign="top"> P.A. Samuelson, "Foundations of economic analysis" , Harvard Univ. Press (1947)</TD></TR>
5 KB (854 words) - 09:14, 7 April 2018
...ut their solutions can be obtained as asymptotic approximations. Moreover, numerical methods are often disregarded if asymptotic approximations can be relativel
...D> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2</TD></TR>
4 KB (660 words) - 19:29, 13 April 2024
...the solution of this equation, is generalized by introducing an auxiliary numerical (or, in general, functional) parameter $ \lambda $
and to the Cauchy problem (2) or (1) one applies a method of numerical integration of ordinary differential equations with step $ h _ {k} $(
12 KB (1,799 words) - 08:05, 6 June 2020
...gn="top">[5]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
7 KB (1,037 words) - 17:08, 13 January 2024
A numerical technique proposed in 1960 by P.D. Lax and B. Wendroff [[#References|[a6]]]
...arch activities aimed at further improvements of methods. A large class of numerical methods for solving (a1) are the so-called conservative methods
16 KB (2,341 words) - 19:00, 26 March 2023
A transform used for the harmonic analysis of functions given on a discrete set of points.
...p">[a1]</TD> <TD valign="top"> C.E. Fröberg, "Introduction to numerical analysis, theory and applications" , Benjamin/Cummings (1985)</TD></TR>
4 KB (602 words) - 14:32, 17 March 2023
''automatic dynamic incremental non-linear analysis system''
...he name ADINA stresses that automatic procedures are used and very general analysis problems can be solved. ADINA has been developed to address many engineerin
8 KB (1,092 words) - 17:08, 7 February 2011
...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 139</TD></TR></table>
3 KB (336 words) - 00:52, 31 December 2021
...he applied [[Spectral analysis of a stationary stochastic process|spectral analysis of a stationary stochastic process]].
...R><TD valign="top">[9]</TD> <TD valign="top"> M.B. Priestley, "Spectral analysis and time series" , '''1–2''' , Acad. Press (1981)</TD></TR><TR><TD valig
6 KB (747 words) - 18:22, 14 January 2021
...erential equations. It developed into a powerful and accurate tool for the numerical solution of complex differential and functional equations. A main idea in i
Theoretical error analysis for the tau method [[#References|[a18]]], [[#References|[a30]]], [[#Referen
20 KB (2,974 words) - 16:52, 1 July 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/C020/C.0200960 Cauchy problem, numerical methods for ordinary differential equations
...ssful solution of the Cauchy problem by numerical methods requires special numerical techniques, despite of the fact that, theoretically speaking, the problem i
16 KB (2,467 words) - 15:35, 4 June 2020
The numerical parameter $ \lambda $
...metric kernel]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]] and [[Ill-posed problems|Ill-posed problems]]. Below Fredholm equ
23 KB (3,417 words) - 02:27, 14 June 2022
...edia/old_files/data/E035/E.0305530 Elliptic partial differential equation, numerical methods
...boundary value problems, and for their approximate solution many different numerical methods have been worked out (see [[#References|[2]]], [[#References|[3]]])
13 KB (1,962 words) - 19:37, 5 June 2020
...lign="top">[3]</TD> <TD valign="top"> A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , '''1–2''' , Birkhäuser (1989) (Translate
.../TD> <TD valign="top"> U.M. Ascher, R.M.M. Mattheij, R.D. Russell, "Numerical solution for boundary value problems for ordinary differential equations" ,
7 KB (1,087 words) - 08:04, 6 June 2020
..."top">[4]</TD> <TD valign="top"> A.S. Householder, G.W. Stewart, "The numerical factorization of a polynomial" ''SIAM Rev.'' , '''13''' (1971) pp. 38–
...[a2]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR></table>
5 KB (723 words) - 07:55, 14 January 2024
a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the
...">[4]</TD> <TD valign="top"> E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , '''1''' , Springer (1979)</TD></TR></table>
5 KB (743 words) - 22:13, 5 June 2020
$#C+1 = 60 : ~/encyclopedia/old_files/data/L059/L.0509050 Linear algebra, numerical methods in
...th the mathematical description and investigation of processes for solving numerical problems in linear algebra.
20 KB (3,033 words) - 22:16, 5 June 2020
...to the accumulation of huge libraries of Fortran subprograms on numerical analysis, statistics, machine graphics, engineering calculations, etc.
...valign="top">[1]</TD> <TD valign="top"> D.D. MacCracken, W.S. Dorn, "Numerical methods with Fortran-IV programming, case studies" , Wiley (1972)</TD></TR
4 KB (631 words) - 19:39, 5 June 2020
''in the numerical solution of algebraic equations''
...numerical methods of linear algebra is the scheme of inverse (or backward) analysis. Applied to the solution of linear algebraic equations
18 KB (2,636 words) - 08:00, 13 May 2022
...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR></table>
3 KB (407 words) - 12:57, 19 December 2020
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Voevodin, "Numerical methods of algebra. Theory and algorithms" , Moscow (1966) (In Russian)</
...>[a7]</TD> <TD valign="top"> A.S. Householder, "Principles of numerical analysis" , McGraw-Hill (1953)</TD></TR></table>
10 KB (1,498 words) - 08:08, 21 March 2022
...op"> W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, "Numerical recipes" , Cambridge Univ. Press (1986) pp. §9.2</TD></TR></table>
5 KB (766 words) - 08:12, 6 June 2020
Methods for obtaining analytical expressions (formulas) or numerical values which approximate to some degree of accuracy the required particular
...employed in theoretical investigations and are used only rarely to obtain numerical solutions of differential equations in practical computations.
32 KB (4,604 words) - 16:55, 7 February 2011
The use of the Euler–MacLaurin sum formula in numerical [[Quadrature|quadrature]] is discussed in [[#References|[a1]]] and [[#Refer
...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="
5 KB (745 words) - 09:18, 6 January 2024
$#C+1 = 187 : ~/encyclopedia/old_files/data/F041/F.0401430 Fredholm equation, numerical methods
...the second kind one uses the language of [[Functional analysis|functional analysis]]. The integral equation (1) can be written as a linear operator equation
20 KB (3,019 words) - 03:31, 14 June 2022
...theory (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]]), plasma physics, including the inverse problems for these most i
...n of the conditions for the appearance of new physical effects, etc. Thus, numerical experiments usually expand the domain of effective utilization of mathemati
13 KB (1,857 words) - 21:26, 8 November 2014
...> <TD valign="top"> J. Stoer, R. Bulirsch, "Introduction to numerical analysis" , Springer (1980) pp. 285–287</TD></TR><TR><TD valign="top">[a2]</TD>
13 KB (1,768 words) - 17:09, 7 February 2011
A second approach to the plan of analysis of stability and convergence dispenses with the need to consider fractional
...for some nonstationary problems" J. Miller (ed.) , ''Topics in numerical analysis'' , Acad. Press (1973) pp. 63–87 (Translated from Russian)</TD></TR><T
12 KB (1,750 words) - 11:54, 26 March 2023
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
5 KB (756 words) - 18:15, 25 June 2020
...nstructed by specialists in the area concerned with the problem. Numerical analysis is concerned with devising methods for approximating the solution to the mo
...n lead to significant errors in approximation. The discipline of numerical analysis involves the design of techniques that take these and other error-producing
29 KB (4,373 words) - 17:21, 2 January 2021
...ust solve a linear boundary value problem, which makes it necessary to use numerical methods and some discretization of the original problem and the linear prob
...chultz, A.H. Sherman, "The application of sparse matrix methods to the numerical solution of nonlinear elliptic partial differential equations" D.L. Colton
9 KB (1,320 words) - 22:17, 5 June 2020
...es|[a7]]] for methods for calculating exact values of the discrepancy. The numerical values differ only slightly from the upper bound in (a8).
...d exactly, since the points $P _ { k }$ are again points of a lattice. The numerical values show that a sequence of type (a9) behaves as good in dimension $k \t
11 KB (1,696 words) - 19:30, 8 February 2024
...that can be solved on computers is high, being determined by the amount of numerical information needed to specify the matrix.
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
9 KB (1,376 words) - 17:46, 4 June 2020
...ssence, the first part of the question is a basic problem of the theory of numerical methods and in most cases it is studied independently of the optimization p
In the practice of numerical integration, active algorithms of the type of integration algorithms with a
12 KB (1,798 words) - 20:07, 12 January 2024
...blem|Stefan problem]], or the phase-transition problem. The most effective numerical method for solving these problems is the calculus of finite differences, wh
...ion between the field of temperatures and other physical phenomena. In the analysis of problems involving the freezing of soil with allowance for the inflow of
4 KB (563 words) - 07:56, 15 July 2014
...sion of the problem increases, the number of operations needed to obtain a numerical solution increases correspondingly, both on account of the larger number of
...dent partial differential equations" , ''The state-of-the-art in numerical analysis. Proc. Conf. Univ. York, 1976'' , Acad. Press (1977) pp. 757–796</TD></
6 KB (763 words) - 19:39, 5 June 2020
One of the notions in [[Non-linear functional analysis|non-linear functional analysis]].
...E$. An operator $A$ is called semi-continuous if for any $u,v,w\in E$ the numerical function $(A(u+tv),w)$ is continuous in $t$. An example of a semi-continuou
4 KB (605 words) - 17:10, 14 February 2020
...atrix|inverse matrix]]. As for the solution of linear systems, methods for numerical inversion can be subdivided into direct and iterative methods; however, ite
...ate inverse matrix. This is an important difference between the problem of numerical inversion of a matrix and the solution of linear systems, where (for exampl
11 KB (1,584 words) - 22:13, 5 June 2020
...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR>
3 KB (445 words) - 18:09, 1 October 2017
.... D. Moore [[#References|[a8]]], [[#References|[a9]]] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting fr
...rasny's method [[#References|[a5]]] used a non-linear filter to remove the numerical noise generated by the physical instability, the convergence of which was p
8 KB (1,185 words) - 17:00, 1 July 2020
is a numerical parameter. Then the contracting-mapping principle implies that if
.../TD> <TD valign="top"> V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Numerical methods" , '''1–2''' , Moscow (1976–1977) (In Russian)</TD></TR><TR><
6 KB (835 words) - 23:24, 7 May 2022
$#C+1 = 61 : ~/encyclopedia/old_files/data/P073/P.0703300 Poisson equation, numerical methods
...mization of the labour of calculation]]) give other characteristics of the numerical methods.
13 KB (1,891 words) - 13:06, 13 January 2024
is called the set of numerical marks of the highest weight $ \Lambda $ .
...ghest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible
9 KB (1,307 words) - 20:04, 27 February 2021
...olution (see [[Transport equations, numerical methods|Transport equations, numerical methods]]).
...S.B. Shikhov, "Questions in the mathematical theory of reactors. Linear analysis" , Moscow (1973) (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign=
6 KB (882 words) - 13:51, 20 April 2014
...ical and practical life insurance mathematics and also forms the basis for numerical procedures, see [[#References|[a8]]].
...mber of them fundamental) to astronomy, mathematical statistics, numerical analysis, and actuarial mathematics. Biographical/bibliographical accounts are given
5 KB (787 words) - 16:55, 1 July 2020
...ng, "The solution of nonlinear finite element equations" ''Internat. J. Numerical Methods Eng.'' , '''14''' (1979) pp. 1613–1626</td></tr><tr><td valign=
13 KB (1,868 words) - 07:21, 13 February 2024
...these fields by the [[Galerkin method|Galerkin method]] through objective analysis (interpolation, extrapolation, smoothing) of empirical data on these fields
...ear mechanics, related to the van der Pol method, in a multiple time scale analysis. An example of this is the quasi-geostrophysic series, filtering fast waves
13 KB (1,880 words) - 08:03, 6 June 2020
c) stochastic analysis naturally yields Brownian functionals defined by stochastic integrals or st
...ial and does not depend on the Borelian version. It causes difficulties in numerical computations and simulation, [[#References|[a4]]].
7 KB (1,053 words) - 22:49, 7 March 2019
...ction theory; [[Analytic number theory|analytic number theory]]; numerical analysis; crystallography (cf. [[Crystallography, mathematical|Crystallography, math
3 KB (421 words) - 22:15, 5 June 2020
In mathematical analysis both convergent and divergent series are used. For the latter various metho
consisting of numerical functions defined on some set $ X $
29 KB (4,393 words) - 19:21, 27 January 2020
...aurin formula|Euler–MacLaurin formula]], the Abel–Plana summation formula (numerical mathematics), and to the basic functional equation for the [[Riemann zeta-f
The first study of sampling and reconstruction in an abstract harmonic analysis setting is due to I. Kluvánek (1965). He replaced the time domain $\mathbf
9 KB (1,428 words) - 07:01, 19 March 2024
[[Category:Numerical analysis and scientific computing]]
3 KB (519 words) - 16:32, 26 October 2014
...cisions. The theoretical side of operations research is concerned with the analysis and solution of mathematical problems of choosing from a given set $X$ of f
...is sometimes of interest to consider a family of problems, depending on a numerical or vectorial value running over some parameter set, and unifying them into
16 KB (2,473 words) - 09:45, 7 August 2014
...he error of the $i$th observation (cf. also [[Numerical analysis|Numerical analysis]]).
...es are more theoretically oriented. An applicable approach for statistical analysis of non-precise data is given in [[#References|[a4]]].
11 KB (1,572 words) - 17:45, 1 July 2020
...ia/old_files/data/H048/H.0408310 Hyperbolic partial differential equation, numerical methods
Methods for solving hyperbolic partial differential equations using numerical algorithms.
24 KB (3,317 words) - 22:11, 5 June 2020
...mic programming]]), computer science, automata and formal language theory, numerical methods, [[parallel programming]], etc. (cf. also [[Idempotent algorithm]]
...valign="top"> A.V. Aho, J.E. Hopcroft, J.D. Ullman, "The design and analysis of computer algorithms" , Addison-Wesley (1976)</TD></TR>
7 KB (1,026 words) - 20:42, 16 November 2023
In this example there is only numerical evidence that $ H _ {a,b} $
a chaotic dynamical system. The mathematical analysis of this example is still in progress (1988).
8 KB (1,192 words) - 16:43, 4 June 2020
...[2]</TD> <TD valign="top"> L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)</TD></TR><T
...align="top">[a1]</TD> <TD valign="top"> J.E. Denis jr., R. Schnable, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-
4 KB (523 words) - 22:14, 5 June 2020
...special challenges to working scientists (cf. also [[Fredholm equation, numerical methods]]). Unlike equations of the second kind, for which the existence o
The analysis of the method of regularization is best carried out in the context of a [[
10 KB (1,476 words) - 17:34, 25 April 2012
...a number of problems, both analytical and numerical; for example, in the (numerical) solution of linear equations and eigenvalue problems. A few well-known fac
...Bu}}||valign="top"| J. Stoer, R. Bulirsch, "Introduction to numerical analysis", Springer (1993) {{MR|1295246}} {{ZBL|0771.65002}}
9 KB (1,294 words) - 22:47, 28 February 2015
...e reproducing formula]].) Such window functions are also used in numerical analysis.
3 KB (466 words) - 07:33, 8 February 2024
A computational algorithm processes the numerical and the symbolic information and usually involves a loss of information and
...stimates may not be sufficiently accurate. A widespread method is interval analysis, in which confidence bounds of the computed magnitudes may be obtained in t
10 KB (1,642 words) - 08:41, 29 August 2014
...error). The latter problem is characteristic, for example, of problems in numerical integration in which the number of points of integration, which measures th
For problem (1), the subject of harmonic analysis, the simplest methods require $ \sim N ^ {2} $
20 KB (2,838 words) - 19:21, 13 January 2024
...ncy: weighting as a possible alternative to trimming in response-time data analysis]. Psikhologicheskie Issledovaniya, 5(19).</ref>. Thus, central observations
Consider a simple numerical example of a dataset consisting of four observations: x<sub>1</sub> = 5, x<
3 KB (531 words) - 16:32, 12 July 2023
In general form the analysis of parametric programming problems has the following aims. 1) To find and c
Numerical methods for solving one-parameter problems in linear parametric programming
7 KB (982 words) - 16:44, 20 January 2024
...ent methods for solving partial differential equations. The complete error analysis of the finite difference and finite volume schemes in [[#References|[a9]]]
...cation of the Bramble–Hilbert lemma in constructive function theory and in numerical methods for differential equations.
11 KB (1,716 words) - 17:52, 3 January 2021
$#C+1 = 113 : ~/encyclopedia/old_files/data/V096/V.0906230 Variational numerical methods
...ier transform, discrete]]). The Galerkin method can also be applied to the numerical solution of evolution equations, leading to a system of ordinary differenti
15 KB (2,260 words) - 08:17, 9 January 2024
...hosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points $ ( x, f ( x)) $,
An extremely important theorem in functional analysis is the so-called [[Closed-graph theorem|closed-graph theorem]]: If $ X $
3 KB (568 words) - 19:42, 5 June 2020
Requirements of engineering have led to the development of numerical methods for solving neutron-transport equations in order to calculate criti
...uation). See [[Transport equations, numerical methods|Transport equations, numerical methods]].
8 KB (1,248 words) - 18:31, 13 January 2024
...144 : ~/encyclopedia/old_files/data/N067/N.0607110 Non\AAhlinear equation, numerical methods
Numerical methods for the solution of a non-linear equation (3) are called iteration
20 KB (2,830 words) - 19:25, 9 January 2024
...]], etc.). In many cases such approximate results match well with accurate numerical predictions.
...pproximations to the solutions of the original problems. In several cases, numerical methods and perturbation methods complement each other. The use of mathemat
11 KB (1,761 words) - 08:32, 23 July 2014
...><TR><TD valign="top">[5]</TD> <TD valign="top"> G.D. Kim, , ''Numerical analysis in Fortan'' , '''3''' , Moscow (1973) pp. 97–113 (In Russian)</TD></TR
7 KB (998 words) - 19:17, 17 January 2024
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
4 KB (587 words) - 02:27, 1 March 2022
Benjamin and Feir used a linearized analysis to show that
...[[#References|[a9]]] to study the long-time evolution of the instability. Numerical experiments indicate that the instability leads to two different types of r
7 KB (969 words) - 10:58, 29 May 2020
...= 135 : ~/encyclopedia/old_files/data/V096/V.0906210 Variational calculus, numerical methods of
The branch of numerical mathematics in which one deals with the determination of extremal values of
23 KB (3,462 words) - 08:27, 6 June 2020
...ernal stability number (see [[Graph, numerical characteristics of a|Graph, numerical characteristics of a]]).
...align="top">[9]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Ru
10 KB (1,477 words) - 11:05, 26 March 2023
be its point spectrum, i.e. the set of numerical functions $ \lambda = \lambda ( A ) $
...locally compact Abelian group (see [[Harmonic analysis, abstract|Harmonic analysis, abstract]]), which consists of finding conditions under which the subspace
9 KB (1,382 words) - 08:22, 6 June 2020
...The theory of Banach spaces is a thoroughly studied branch of functional analysis, with numerous applications in various branches of mathematics — directl
The Banach spaces encountered in analysis are mostly sets of functions or sequences of numbers which are subject to
25 KB (4,253 words) - 23:54, 13 April 2019
...lently, for every Borel set $B$). In the usual terminology of mathematical analysis this means that $P_X$ is absolutely continuous with respect to [[Lebesgue m
...a random variable $X$ are sufficiently characterized by a small number of numerical characteristics. For example, the [[median (in statistics)]] and [[Quantile
6 KB (904 words) - 20:43, 1 January 2019
...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
...[a2]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="
10 KB (1,519 words) - 02:51, 21 March 2022
...= 132 : ~/encyclopedia/old_files/data/R110/R.1100080 Renormalization group analysis
...ale, from which observed phenomena can be explained. Renormalization group analysis allows one to determine effective theories at each length scale, from micro
10 KB (1,447 words) - 08:11, 6 June 2020
..., problems on equivalent transformations of automata, etc. The problems of analysis concern the description of the behaviour of a given automaton or the determ
...TD valign="top">[2]</TD> <TD valign="top"> V.M. Glushkov, "Synthesis of numerical automata" , Moscow (1962) (In Russian)</TD></TR><TR><TD valign="top">[3]<
7 KB (975 words) - 18:49, 5 April 2020
...6160.htm','Variational calculus','../v/v096190.htm','Variational calculus, numerical methods of','../v/v096210.htm','Venn diagram','../v/v096550.htm','Zeta-func
as numerical functions and to apply to them the methods of analysis. In general, the value of a field quantity at a point depends on the choice
7 KB (1,075 words) - 16:43, 4 June 2020
As in most numerical methods, an approximate solution, $u _ { N}$, is sought which is the sum of
...pectral methods are, in general, more complicated to code and require more analysis to be done prior to coding than simpler methods.
9 KB (1,381 words) - 16:55, 1 July 2020
Observatory) T.N. Thiele was a leading personality in astronomy, numerical
analysis, actuarial science, and mathematical statistics. He invented the
11 KB (1,576 words) - 19:39, 7 October 2023
...s the extremal (maximal or minimal) value of the objective function — some numerical characteristic of the process. A multi-stage process is one the structure o
Conventional methods of mathematical analysis are altogether inapplicable to problems of dynamic programming or else invo
10 KB (1,500 words) - 16:50, 20 January 2024
...ectrum of $T$, and provide a powerful tool of [[Spectral analysis|spectral analysis]].
In order to carry out a more detailed spectral analysis of the Sturm–Liouville operator $T$, one has to consider the spectral mea
23 KB (3,372 words) - 17:44, 1 July 2020
Very often the analysis of such schemes, after expressing it in the canonical form
...undary value problem, numerical methods|Non-linear boundary value problem, numerical methods]]).
15 KB (2,220 words) - 08:52, 10 April 2023
...y stable cycle. This assertion is mainly based on the results of numerical analysis (significantly less was obtained by purely mathematical methods, [[#Referen
4 KB (552 words) - 22:11, 5 June 2020
An application of the Budan–Fourier theorem in numerical analysis may be found in {{Cite|BoSc}}, where it is used in the interpolation by spl
4 KB (552 words) - 13:01, 24 April 2012
..." . It is especially suited for the description of algorithms of numerical analysis. The language is machine-independent and does not provide standard ways of
4 KB (632 words) - 17:21, 7 February 2011
...y leads to non-smooth functions (see [[Maximin, numerical methods|Maximin, numerical methods]]). Apparently, all of the commonly-used methods for optimization h
.../TR><TR><TD valign="top">[18]</TD> <TD valign="top"> Yu.G. Evtushenko, "Numerical methods for finding global extrema (case of a non-uniform mesh)" ''USSR Co
13 KB (1,911 words) - 08:00, 6 June 2020
...importance of Butcher series stems from their use to derive and to analyze numerical methods for differential equations. For example, consider the $ s $-
...D valign="top">[a1]</TD> <TD valign="top"> J.C. Butcher, "The numerical analysis of ordinary differential equations" , Wiley (1987) {{ZBL|0616.65072}}</TD>
8 KB (1,232 words) - 08:28, 12 November 2023
A system of ordinary differential equations in the numerical solution of which by explicit methods of Runge–Kutta or Adams type, the i
...is not always essentially decreased after asymptotic transformation. Thus, numerical methods are used to solve systems of a general form. The main property of t
30 KB (4,292 words) - 05:40, 24 February 2022
...ability are topological invariants. Among the numerical invariants (taking numerical values on every topological space), the most important are the dimensional
...often related to the consideration of additional structures on the space (numerical, algebraic, order), naturally compatible with its topology. Thus, one disti
10 KB (1,522 words) - 19:35, 25 March 2023
...<TD valign="top">[11]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} <
11 KB (1,584 words) - 11:51, 8 April 2023
...describes phenomenological models of hysteresis and is convenient for the analysis of closed-loop systems with hysteresis non-linearities is presently (1996)
...nity (including the [[Hopf bifurcation|Hopf bifurcation]]), asymptotic and numerical methods (including an averaging principle), etc. have also been investigate
4 KB (584 words) - 22:11, 5 June 2020
...y, based on considering the dimensions of these quantities. In dimensional analysis one considers the problem of establishing various systems of units of measu
Dimension formulas make it possible to determine the numerical scale factors for the computation of corresponding characteristics when cha
16 KB (2,568 words) - 20:00, 4 January 2024
...ial and matrix computations is exploited in order to devise very effective numerical solution algorithms. Famous special cases are the Hilbert matrix and Hilber
...ormal orthogonal polynomials in the context of the algorithms of numerical analysis, including Lanczos' tridiagonalization process, rational interpolation sche
14 KB (2,126 words) - 16:45, 1 July 2020
...of operations is to be achieved in minimal time for given resources, this analysis reduces to finding the critical paths and determining the least time in whi
5 KB (745 words) - 08:02, 6 June 2020
One of the methods of mathematical analysis which in many cases makes it possible to reduce the study of differential o
is a constant function) is a numerical operator; insofar as $ [ \alpha ] [ \beta ] = [ \alpha \beta ] $,
12 KB (1,635 words) - 14:54, 7 June 2020
...thematical models and mathematical methods, which permit the isolation and analysis of quantitative correlations occurring in statistical data, and their subst
...ted with the preparation and foundation of optimal economic solutions, the analysis of necessary conditions for their effective realization, the modelling of p
14 KB (2,040 words) - 19:36, 5 June 2020
...a1]</TD> <TD valign="top"> K.E. Atkinson, "An introduction to numerical analysis" , Wiley (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
5 KB (774 words) - 17:33, 5 June 2020
models for the analysis of economic processes" ["'3''', p.300].
economics to experimental and numerical verification and thus to
15 KB (2,119 words) - 18:50, 24 March 2023
...ce methods for solving certain boundary value problems of Stefan type" , ''Numerical Methods in Gas Dynamics'' , Moscow (1965) pp. 139–183 (In Russian) {{MR|0
...ces|[a17]]] (general theory), [[#References|[a18]]]–[[#References|[a19]]] (numerical methods).
12 KB (1,620 words) - 08:23, 6 June 2020
As yet (1989) there exists no rigorous mathematical analysis of the solvability of the boundary value problems of hydro- and aeromechani
...of problems of practical importance, using well-developed approximate and numerical methods.
23 KB (3,271 words) - 08:02, 6 June 2020
...ient for the proof of the fundamental results of constructive mathematical analysis, is not, in contrast to the Heyting one, a subsystem of classical arithmeti
In the case of applied systems (arithmetic, analysis) it is required that the formulas under consideration be closed. The majori
10 KB (1,524 words) - 20:25, 4 June 2020
...top">[a2]</td> <td valign="top"> D. Gottlieb, S.A. Orszag, "Numerical analysis of spectral methods: Theory and applications" , SIAM (1977)</td></tr><tr><
6 KB (866 words) - 18:56, 24 January 2024
a) The volume of a three-dimensional body is a numerical characteristic of the body; in the simplest case, when the body can be deco
...valign="top"> V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , '''1–2''' , MIR (1982) (Translated from Russian)</TD></TR></table>
8 KB (1,344 words) - 08:28, 6 June 2020
...itz method]] and the [[Trefftz method|Trefftz method]]; it is also used in numerical methods: in Krylov's method for obtaining the coefficients of the secular e
...]</TD> <TD valign="top"> P. Henrici, "Applied and computational complex analysis" , '''1''' , Wiley (1974) pp. 553; 562</TD></TR><TR><TD valign="top">[a7]
9 KB (1,382 words) - 08:27, 6 June 2020
...or, more generally, [[Spline|splines]], etc. One can say that mathematical analysis begins at the moment when the concept of convergence has been introduced on
...ential equations there are various convergent difference methods for their numerical solution, which are suitable for use in modern computers.
22 KB (3,726 words) - 10:31, 2 September 2017
[[Numerical analysis|numerical analysis]] and in applied mathematics,
11 KB (1,808 words) - 21:52, 5 March 2012
...or [[#References|[a9]]] (cf. also [[Dynamical system|Dynamical system]]). Numerical methods are described in [[#References|[a3]]], [[#References|[a6]]] and [[#
It can perform a bifurcation analysis of (a1). It can compute branches of stable and unstable periodic orbits and
13 KB (1,928 words) - 17:00, 1 July 2020
$#C+1 = 84 : ~/encyclopedia/old_files/data/G043/G.0403340 Gas dynamics, numerical methods of
...y computational algorithms. Below the fundamental aspects of the theory of numerical methods for solving problems in gas dynamics will be considered by writing
28 KB (3,843 words) - 19:41, 5 June 2020
...cience. In this article, three important approaches to the description and analysis of discrete event systems will be given. It depends on the field of applica
...added because it is the neutral element with respect to maximization. The numerical evaluation of the recurrence relation is completely standard, except that t
14 KB (2,165 words) - 19:35, 5 June 2020
...application to analytic functions of a complex variable and to problems of numerical mathematics.
...gn="top">[4]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
27 KB (3,799 words) - 17:14, 12 November 2023
...s with a large number of channels is conducted both by means of asymptotic analysis of explicit formulas, which become effective for large $ n $,
these relations are of little use in the search for the numerical values of $ p _ {m} $.
10 KB (1,560 words) - 08:02, 14 January 2024
Analysis, Mathematical Physics, Rational Mechanics and Probability
from 1929. Eleven papers on Mathematical Analysis, all published
12 KB (1,866 words) - 14:20, 10 March 2024
...rences|[a8]]], [[#References|[a9]]], [[#References|[a10]]]. For a thorough analysis with explicit error bounds see [[#References|[a8]]], for major modification
...egar (ed.) M. Shub (ed.) S. Smale (ed.) , ''The Mathematics of Numerical Analysis'' , ''Lect. Appl. Math.'' , '''32''' , Amer. Math. Soc. (1996) pp. 895–
8 KB (1,086 words) - 17:43, 1 July 2020
...face [[#References|[a7]]] to optimization problems and applications in the numerical solutions of partial differential equations in high dimensions.
...basis function approximation" W.A. Light (ed.) , ''Advances in Numerical Analysis II. Wavelets, Subdivision, and Radial Functions'' , Oxford Univ. Press (19
8 KB (1,246 words) - 15:13, 18 February 2013
...e that as suggested by the presence of chaotic solutions and by a Painlevé analysis [[#References|[a7]]] (cf. also [[Painlevé test|Painlevé test]]), the Kura
...iciently large $L$ [[#References|[a13]]] (see Fig.a1), and there is strong numerical evidence that the "simple" solutions destabilize to an (apparently unique
21 KB (3,050 words) - 17:43, 1 July 2020
a ''probability distribution'' which has a numerical feature of
# Because a sample is only part of the population, the numerical
22 KB (3,172 words) - 20:20, 12 March 2016
...models analytic solutions can be obtained only for the simplest cases, and numerical methods are widely employed in practical work in aerodynamics. These method
...ers are employed, there is no point in using these methods, since accurate numerical methods for solving these problems belong to the simplest problems of compu
25 KB (3,871 words) - 20:01, 4 January 2024
...icult, since solving this integral equation (2) is an ill-posed problem in numerical mathematics.
See the entry [[Discriminant analysis|Discriminant analysis]] for the applications of the empirical Bayesian approach to hypotheses tes
8 KB (1,315 words) - 10:33, 29 May 2020
...These methods are collectively known as correlation analysis. Correlation analysis of statistical data consists of the following basic practical steps: 1) the
...es increases or decreases on the average as the other increases). Prior to numerical processing, the results are usually grouped and presented in the form of a
12 KB (1,686 words) - 04:11, 15 June 2022
...TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H.J. Stetter, "Analysis of discretization methods for ordinary differential equations" , Springer
11 KB (1,674 words) - 14:30, 7 April 2020
...R.S. Varga, "Functional analysis and approximation theory in numerical analysis" , ''Reg. Conf. Ser. Appl. Math.'' , '''3''' , SIAM (1971)</TD></TR><TR><T
..."Approximation by functions of fewer variables" R.E. Langer (ed.) , ''On numerical approximation'' , Univ. of Wisconsin Press (1959) pp. 275–327</TD></TR>
21 KB (3,060 words) - 20:29, 10 January 2021
...align="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.353
Recently these ideas are meeting new interest among numerical analysts. They are essentially used to solve boundary value problems on com
6 KB (852 words) - 08:12, 6 June 2020
...equation has many applications in stability and control theory; efficient numerical algorithms for solving it are available.
..."top">[a2]</td> <td valign="top"> R.E. Bellman, "Introduction to matrix-analysis" , McGraw-Hill (1960)</td></tr><tr><td valign="top">[a3]</td> <td valign="
5 KB (876 words) - 16:52, 1 July 2020
...ogram focusing on exact mathematical formula manipulation. It differs from numerical packages in that it manipulates symbols rather than numbers; thus, it does
...to a certain extent GAP (see below), and SENAC (a Software Environment for Numerical and Algebraic Computation).
9 KB (1,292 words) - 20:51, 18 September 2016
...[a7]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 209</TD></TR><TR><TD valign="top">[a8]</TD>
5 KB (752 words) - 08:29, 6 June 2020
...Markov process]]), whereas for those frequently encountered processes with numerical values the phase space reduces simply to $\mathbf R$ with the standard stru
...TD valign="top"> Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds and physics" , North-Holland (1977) (Translated from French) {{M
6 KB (906 words) - 21:49, 30 April 2014
$#C+1 = 209 : ~/encyclopedia/old_files/data/S084/S.0804600 Sequential analysis
The basic principles of sequential analysis consist of the following. Let $ \xi _ {1} , \xi _ {2} \dots $
18 KB (2,425 words) - 14:55, 7 June 2020
...e are special methods for solving specific functional equations, including numerical methods, for example, the [[Grid method|grid method]], etc. The method of v
An analytic application to the Cauchy problem (2) of methods for the numerical integration of ordinary differential equations leads to a whole class of di
23 KB (3,277 words) - 19:16, 16 January 2024
which need not be numerical) have the solution $ x _ {0} $
the complete analysis is effected by the method of the Newton diagram. As regards the study of th
13 KB (1,815 words) - 06:29, 30 May 2020
...s of a limit, a derivative, a differential, and an integral. Infinitesimal analysis was born; in the first place in the form of the [[Differential calculus|dif
...nd for transferring questions of geometry into the language of algebra and analysis and for solving them neatly by algebraic and analytic methods. On the other
22 KB (3,256 words) - 21:54, 30 March 2012
For applications in a variety of fields of non-linear analysis and physics, see [[#References|[a1]]], [[#References|[a4]]], [[#References|
...J.F. Colombeau, "Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics" , Springer (1992)</td></tr><tr><td va
7 KB (996 words) - 20:13, 4 February 2024
Inequalities in which there occur quantities that can take distinct numerical values may be true for some such values and false for others. For example,
...rov inequality]]; [[Imbedding theorems|Imbedding theorems]]. In functional analysis, in the definition of the [[Norm|norm]] in a function space, one requires t
7 KB (1,040 words) - 22:12, 5 June 2020
[[Category:Analysis]]
Also called ''total variation''. A numerical characteristic of functions of one or more real variables which is connect
16 KB (2,511 words) - 16:07, 12 October 2012
...the alternatives under the respective criteria is expressed in grades on a numerical scale;
...them seem to be innumerate in the sense that they are insensitive to what numerical values mean within a particular context. Many decisions are made in groups
18 KB (2,526 words) - 15:30, 1 July 2020
...gn="top">[8]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
14 KB (1,916 words) - 19:59, 12 January 2024
Three sources of errors are behind the lack of accuracy in numerical computations: data errors associated with the physical model, truncation er
taken as an indication of the accuracy, cannot be calculated, and numerical analysts have devised bounds for the latter in terms of the variation $ \
30 KB (4,205 words) - 18:47, 5 April 2020
...lable for wings of other shapes for stationary and non-stationary motions. Numerical methods for computing the flow around wings resulting from non-linear flow
...ution of three-dimensional problems is much more difficult and is based on numerical methods.
23 KB (3,690 words) - 20:00, 4 January 2024
numerical variables, the equality sign, the function symbols $ + , \cdot, {} ^ \
...ant number-theoretic theorem which can be demonstrated without recourse to analysis is known which could not be demonstrated by formal arithmetic. To write dow
12 KB (1,840 words) - 19:36, 9 February 2024
...top">[a2]</td> <td valign="top"> D. Gottlieb, S.A. Orszag, "Numerical analysis of spectral methods: Theory and applications" , SIAM (1977)</td></tr><tr><
7 KB (1,028 words) - 17:44, 1 July 2020
...e method for the construction of geometry and its foundations. This deeper analysis of the foundations of geometry was enhanced by the discovery in 1826 of the
...ole system, cleared the consistency of the system by the construction of a numerical model and established the independence of the groups of axioms, as well as
25 KB (3,631 words) - 19:39, 5 June 2020
$#C+1 = 125 : ~/encyclopedia/old_files/data/S086/S.0806330 Spectral analysis
...of functions of one or several operators find wide application in spectral analysis. These may be very simple (the spectrum of a polynomial in an operator cons
16 KB (2,424 words) - 08:22, 6 June 2020
...ogramming|dynamic programming]], the explanation of the role of functional analysis in the theory of optimal systems, the discovery of connections between solu
...f a control), research into sufficient conditions, and the construction of numerical algorithms. The relations between solutions of problems of the mathematical
21 KB (3,161 words) - 08:04, 6 June 2020
...proposed by F.Yates in1934, and it is known as ''Yates' correction''. For numerical improvements due to the continuity corrections above, we refer to Kendall a
Finally, extensive numerical computations in Bhattacharya and Chan (1996) show that the chisquare approx
11 KB (1,719 words) - 20:18, 12 March 2016
multivariate $T^2$, principal components analysis, and canomical
did his fundamental research establishing the field of sequential analysis.
13 KB (1,905 words) - 13:07, 18 March 2023
...un, see [[#References|[a8]]] and [[#References|[a10]]]; these also contain numerical analyses concerning the production, costs and work productivity.
...n="top">[a6]</td> <td valign="top"> W.W. Haynes, "Managerial economics: Analysis and cases" , The Dorsey Press (1963)</td></tr><tr><td valign="top">[a7]</t
6 KB (958 words) - 15:30, 1 July 2020
...ns, can be solved and investigated with the aid of differential operators. Numerical tools may operate on the solution sets of equations rather than on the equa
...TR><TD valign="top">[a2]</TD> <TD valign="top"> A.D. Talantsev, "On the analysis and synthesis of certain electrical circuits by means of special logical op
8 KB (1,166 words) - 06:28, 30 May 2020
...ustomer satisfaction while keeping the total workload unchanged. Extensive analysis (and optimization in operation) of queues with priority has been done by be
A classic monograph on the analysis of priority queues is [[#References|[a2]]].
21 KB (3,138 words) - 20:48, 11 February 2024
...'s theorem). (All the complete locally convex spaces arising in functional analysis, with their usual topologies, belong to both of these classes.) In fact, th
...ially) in connection with the theory of random processes, in the theory of numerical cylindrical measures on locally convex spaces, have been obtained.
41 KB (6,085 words) - 08:26, 6 June 2020
...#References|[a4]]]. Comparatively little is known for unbounded intervals. Numerical results reported in [[#References|[a5]]] show that complex zeros arise for
An important fact for the analysis of the Stieltjes polynomials $E _ { n + 1}$ is their close connection with
7 KB (1,048 words) - 20:47, 11 February 2024
The branch of numerical mathematics studying methods for finding approximate solutions to different
...R><TD valign="top">[6]</TD> <TD valign="top"> G.I. Marchuk, "Methods of numerical mathematics" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD
26 KB (3,757 words) - 17:33, 5 June 2020
...rted the interest in various questions of approximation theory and Fourier analysis in ${\bf R} ^ { n }$ connected with the study of hyperbolic cross partial F
...="top"> L.V. Zhizhiashvili, "Some problems of multidimensional harmonic analysis" , Tbilisi State Univ. (1996) (In Russian) (Edition: Second)</td></tr></
17 KB (2,613 words) - 19:24, 12 February 2024
analysis.
analysis of variance.
8 KB (1,250 words) - 14:47, 18 March 2023
The simplest ordinary differential equation is already encountered in analysis: The problem of finding the primitive function of a given continuous functi
...[Eigen value|Eigen value]]) and also with the [[Spectral analysis|spectral analysis]] of ordinary differential operators.
33 KB (4,933 words) - 01:50, 23 January 2022
...arises. The Ising model also became a very useful testing ground for many numerical methods: e.g. systematic expansions of $F$ at low $T$ (in the variable $u =
16 KB (2,407 words) - 19:47, 17 February 2024
...]) and mathematical [[Cybernetics|cybernetics]] and their study aims at an analysis and formalization of the concept of an algorithm and at modelling real devi
In many cases an abstract machine is constructed to compute numerical (word) functions and predicates. The computation of a function $ f( x _ {
18 KB (2,656 words) - 04:11, 6 June 2020
...tivity (cf. [[Graph, connectivity of a|Graph, connectivity of a]]). In the analysis of the reliability of electronic circuits or communications networks there
...mbedding|Graph imbedding]]; [[Graph, numerical characteristics of a|Graph, numerical characteristics of a]]); some of them were suggested by other branches of m
13 KB (1,976 words) - 20:08, 15 March 2023
...mmodated in the method of utilization and interpretation of the results of analysis.
...compact convex set. By expanding the space of products the problem of the analysis of the efficient methods here may be reduced to the case where $ Z $
37 KB (5,562 words) - 11:11, 7 January 2024
...acity zero in accordance with various problems of mathematical physics and analysis (see [[#References|[6]]]).
is defined axiomatically as a numerical set function $ K \rightarrow C (K) $
17 KB (2,487 words) - 06:29, 30 May 2020
...of the first kind; differentiation of functions known only approximately; numerical summation of Fourier series when their coefficients are known approximate
==Numerical methods for solving ill-posed problems.==
33 KB (5,246 words) - 00:23, 25 April 2012
...outside of the coding context. These include fast algorithms in numerical analysis such as Lanczos recursion and Levinson–Shur algorithms for Toeplitz matri
8 KB (1,155 words) - 18:48, 26 January 2024
if its numerical range $ \{ {( Tx, x) } : {\| x \| = 1, x \in D ( T) } \} $
...st lower bound of its spectrum is equal to the greatest lower bound of the numerical range of $ T $.
20 KB (2,924 words) - 19:38, 5 June 2020
...onal cases. The latter include the classification of algebraic surfaces by numerical invariants, criteria of rationality and ruledness of algebraic surfaces, an
b) Numerical invariants of algebraic surfaces. The concept of the geometric genus $ p
26 KB (3,736 words) - 13:08, 8 February 2020
...modern mathematics such as topology, algebra, function theory, functional analysis, and differential equations.
The selection of uniformly-continuous functions from the class of all numerical functions of one or several variables became one of the starting points of
15 KB (2,197 words) - 17:15, 7 February 2011
[[Mayr, Georg von|von Mayr]], is said to be exhaustive numerical description of an
science", where causes and effects are inferred from numerical
10 KB (1,517 words) - 07:18, 8 March 2024
''Diophantine analysis''
...er date, while a systematic study of Diophantine equations — not only with numerical, but also with coefficients which are functions — only began in the 1950s
24 KB (3,602 words) - 11:48, 26 March 2023
...not by individual properties of the control sequences but only by certain numerical parameters. Examples of the second type are theorems on the so-called singl
...f customers. Next to a single-service facility, networks of queues require analysis. In a network a customer or a message has to be transported via the nodes c
12 KB (1,824 words) - 08:09, 6 June 2020
The name of problems in the mechanics of deformable solids dealing with the analysis of stresses and displacements in cylinders with tunnel cracks (with crack-l
...ass of functions in which the solution of the integral equation is sought. Numerical solutions of linear problems have also been found by the methods of edge co
15 KB (2,200 words) - 14:56, 7 June 2020
Among the problems of dynamics undergoing intensive mathematical analysis is the problem of free and forced vibrations of a shell. The structure of t
Methods of numerical mathematics have been extensively used in shell theory. For the case of sep
8 KB (1,200 words) - 17:04, 14 February 2020
Keywords: ''Analysis of variance; borrowing strength; clusters;
correlation structure; empirical Bayes analysis;
23 KB (3,576 words) - 22:10, 12 March 2016
...eddies symmetrically staggered, as in Fig.a1. However, a linear stability analysis with respect to small disturbances shows that the first configuration is al
...shape, of the body. This justifies the continuous actual research for the numerical solution to the problem, one of the challenging problems of present computa
19 KB (3,002 words) - 17:46, 1 July 2020
...utation of the input data is equally likely. For biased distributions, the analysis becomes significantly more complicated, and for arbitrary distributions the
...been conjectured that this boundary behaves like a phase transition with a numerical value around $4.2$, like it is known for properties of random graphs [[#Ref
21 KB (3,198 words) - 18:47, 11 December 2020
...als, quadrature formulas, moment problems, and other problems of classical analysis (see [[#References|[7]]]–). The origin of the study of the rows of the Pa
...th century onwards, Padé approximation has become an independent object of analysis and constitutes an important chapter in the theory of rational approximatio
15 KB (2,227 words) - 13:32, 11 November 2023
recommendation, he was appointed Professor of Analysis and Mechanics
statistical analysis " always to collect a large number of elements,
13 KB (1,947 words) - 18:43, 4 March 2024
...the study of regularity and other qualitative properties of solutions; the analysis of many asymptotic problems, including large deviations and homogenization
9 KB (1,306 words) - 08:28, 6 June 2020
...a complex variable is intimately connected with other branches of complex analysis, and with mathematics in general. Methods and results on conformal mapping,
...">[a6]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5</TD></TR><TR><
18 KB (2,637 words) - 07:20, 26 March 2023
|valign="top"|{{Ref|1}}||valign="top"| Anderberg, M.R. (1973), ''Cluster Analysis for Applications''. Academic Press, New York.
...|18}}||valign="top"| Murtagh, F. and Heck, A. (1987), ''Multivariate Data Analysis'', Kluwer Academic, Dordrecht.
13 KB (1,912 words) - 18:57, 7 March 2024
and public policy, enables students to develop skills at data analysis,
Unfortunately, aggregation makes analysis at finer levels difficult and often
11 KB (1,492 words) - 20:20, 12 March 2016
Such probabilities may be numerical (the smaller class) or non-numerical
All his life Keynes strongly advocated the collection and analysis
17 KB (2,523 words) - 19:38, 21 March 2023
...l numbers. Most often, the ranges of the functions are ordinary continuous numerical intervals, the set of integers, or various finite subsets of it. The $ x
are involved in an electrical network, they can be defined by numerical codes $ 0 $
35 KB (5,424 words) - 11:17, 26 March 2023
...$\int f ( \theta , \phi ) d \phi$, an important advantage for statistical analysis.
...is somewhat more difficult when the E-step is difficult to compute, since numerical integration can be very expensive computationally. The Monte-Carlo EM algor
26 KB (3,861 words) - 17:03, 1 July 2020
of the same type. If the functional equation contains another numerical (or, in general, functional) parameter $ \lambda $,
...tional-differential, as well as more complicated equations in mathematical analysis are all equations of the type (1), as are also systems of algebraic equatio
19 KB (2,803 words) - 19:40, 5 June 2020
...ome primitive cultures, the necessity to communicate information about the numerical size of this or that collection has led to distinguishing certain standard
...te counting methods take shape, special devices are used for counting, and numerical notations emerge. Numbers are separated from the objects being counted, and
23 KB (3,482 words) - 08:03, 6 June 2020
...quire an exact meaning. The following fundamental concepts of mathematical analysis are connected with that of a limit: [[continuity]], [[convergence, types of
such functions are called numerical) and $ E \subset X $,
32 KB (5,224 words) - 19:36, 25 March 2023
...t. From the point of view of probability theory, these characteristics are numerical or functional properties of a certain probability distribution, while the s
...The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin (1969)</TD></TR></table>
11 KB (1,705 words) - 08:12, 6 June 2020
A numerical characteristic assigned to planar figures of a certain class (for example,
...t square is 1. The term "area" is used also in a more general sense as a numerical characteristic of two-dimensional surfaces in three-dimensional space, for
21 KB (3,223 words) - 18:48, 5 April 2020
...us of variations in the large a less complete information in terms of some numerical invariant — the Lyusternik–Shnirel'man category $ \mathop{\rm cat}
...valign="top">[7]</TD> <TD valign="top"> J. Eells, "A setting for global analysis" ''Bull. Amer. Math. Soc.'' , '''72''' (1966) pp. 751–807 {{MR|020374
14 KB (2,052 words) - 08:27, 6 June 2020
...l equations with deviated arguments includes all equations of mathematical
analysis. Nevertheless, ordinary differential equations with deviated arguments are
...ated above, if simple approximation or iteration methods are employed. The
numerical methods of solution are the same, in principle, as for <img align="absmiddl
27 KB (3,819 words) - 16:54, 7 February 2011
...the whole, belongs to the corresponding domains of algebra and functional analysis.
An important role in functional analysis is played by a criterion for compactness of a set $A$ of continuous functio
33 KB (5,289 words) - 19:37, 25 March 2023
it covers all of the physics and mathematical analysis of the period. Poisson
display numerical regularity, but gave rise to many memorable
11 KB (1,710 words) - 19:08, 6 March 2024
which are used in the numerical solution of (B), (7'}prm) in the second approximation. The subsequent appro
...corresponding boundary value problem has been analyzed and algorithms for numerical solution have been given [[#References|[14]]]. Here the functions $ P, Q
28 KB (4,176 words) - 05:30, 19 March 2022
...Ericsson, A. Ruhe, "The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems" ''Math
The standard reference to numerical linear algebra is the text book [[#References|[a7]]], which provides an exc
19 KB (2,830 words) - 19:42, 27 February 2021
...stability of inversion procedures, as well as the development of efficient numerical algorithms. In tomography a fundamental role is played by the [[Radon trans
...n function. Algebraic reconstruction techniques use iterative methods from numerical linear algebra (cf. [[Iteration methods|Iteration methods]]) applied to the
16 KB (2,342 words) - 17:25, 13 June 2020
...listic. This idea has recently been applied in cluster analysis, where the numerical dissimilarity coefficient has been replaced by an element of $ \Delta ^
12 KB (1,757 words) - 08:07, 6 June 2020
A numerical method based on simulation by random variables and the construction of stat
...D> <TD valign="top"> N.M. Korobov, "Number-theoretic methods in applied analysis" , Moscow (1963) (In Russian)</TD></TR><TR><TD valign="top">[22]</TD> <TD
24 KB (3,433 words) - 13:09, 13 January 2024
...y and special functions, linear algebra, the calculation of integrals, the numerical solution of ordinary and partial differential equations, the method of leas
...ing stage and the form of the output of the information, the assignment of numerical information, describing the parameters of the tracking camera (the constant
19 KB (2,814 words) - 17:16, 7 February 2011
with Gabriel Cramer (1704-1752), professor of analysis and geometry
domain of geometry as well as that of analysis" Buffon presented
13 KB (2,017 words) - 09:45, 15 August 2023
$#C+1 = 166 : ~/encyclopedia/old_files/data/L110/L.1100140 Lie symmetry analysis
...can serve as benchmarks in the design, accuracy testing, and comparison of numerical algorithms.
23 KB (3,178 words) - 22:16, 5 June 2020
...{Ref|He}}||valign="top"| P. Henrici, "Applied and computational complex analysis", '''2''', Wiley (1977) {{MR|0453984}}
...o, "The theory of convergent continued fractions and its applications in numerical mathematics", Moscow (1983) (In Russian)
11 KB (1,649 words) - 11:41, 4 September 2013
...y, a lot of work has been recently (as of 2000) done in order to develop "analysis on fractals" , using the Sierpiński gasket (and sometimes the Sierpiński
A number of other topics from classical harmonic analysis, probability theory, partial differential equations, mathematical physics,
26 KB (3,837 words) - 19:21, 17 March 2024
...In the narrow case a stochastic process can be regarded either simply as a numerical function $ X ( t) $
|valign="top"|{{Ref|Du}}|| R. Durrett, "Brownian motion and martingales in analysis" , Wadsworth (1984) {{MR|0750829}} {{ZBL|0554.60075}}
22 KB (3,177 words) - 20:25, 16 January 2024
develop the means of doing so; a remarkable facility with numerical
summaries without technical analysis; a very sharp intellect,
9 KB (1,384 words) - 13:08, 18 March 2023
is called the linear deformation. The numerical value of the principal scale $ M $
...problems in]]), after N.A. Tissot who in 1859 first published his classic analysis of the distortion occurring under a map projection.
12 KB (1,890 words) - 09:14, 26 March 2023
...a branch of spatial statistics that deals with statistical methods for the analysis of spatially referenced data with the following properties. Firstly, values
...ymbol{K}$ but unknown parameter vector $\boldsymbol{\beta}$. From Bayesian analysis we know that it is natural to assume a prior of the form $\boldsymbol{\beta
25 KB (3,705 words) - 19:33, 5 October 2023
...letcher, "An overview of unconstrained optimization" ''Dundee Numerical Analysis Report'' , '''NA/149''' (1993)</td></tr><tr><td valign="top">[a9]</td> <td
12 KB (1,897 words) - 19:37, 9 February 2024
by giving theoretical proofs and many numerical examples,
to develop, perhaps in vain, the basis of a real analysis of
13 KB (1,926 words) - 18:33, 17 April 2024
convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets $ \{ x _ \alpha \} _
...ign="top"> B.Z. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in partially ordered spaces" , Moscow (1950) (In Russian)</TD></TR><TR><T
14 KB (2,252 words) - 08:09, 21 January 2024
...antial deviations of the perturbed solution from the true solution. In the numerical implementation of some finite methods rounding errors arise, and the accumu
...<TD valign="top"> K.H. Borgwardt, "The simplex method: a probabilistic analysis" , Springer (1987)</TD></TR>
12 KB (1,882 words) - 22:25, 1 September 2016
...s: A set of charts covering the manifold. To use manifolds in mathematical analysis it is necessary that the coordinate transitions from one chart to another a
Since, in mathematical analysis, manifolds are important as carriers of differentiable mappings, they are s
30 KB (4,462 words) - 07:59, 6 June 2020
...e canonical equations. Canonical equations are extensively employed in the analysis and synthesis of automata.
...TD valign="top">[2]</TD> <TD valign="top"> V.M. Glushkov, "Synthesis of numerical automata" , Moscow (1962) (In Russian)</TD></TR><TR><TD valign="top">[3]<
19 KB (2,906 words) - 07:33, 26 March 2023
The analysis of completions satisfying additional metric or spectral constraints is a re
...norm of the inverse and the condition number of a matrix by completion" ''Numerical Linear Alg. Appl.'' , '''2''' : 2 (1995) pp. 155–171</TD></TR><TR><TD
14 KB (2,045 words) - 15:19, 28 February 2022
symmetrical contracts between players. However, an analysis of the
Fermat and Pascal he gave the numerical results without the appropriate
16 KB (2,633 words) - 12:03, 1 November 2023
Euler (1768) proposed a method for the approximate (numerical) solution of problems in variational calculus, which received the name of E
...e concept of a generalized solution. Such a reduction is very important in numerical mathematics as well, since direct methods of variational calculus can be em
28 KB (4,182 words) - 07:56, 16 April 2023
...ctions of lower order. The procedure for such an expansion follows from an analysis of the physical singularities of the system and of the small parameter whic
these integrals are calculated using numerical methods). The mathematical problems which arise when a virial expansion is
14 KB (2,025 words) - 08:23, 6 June 2020
...al success, by different optimality principles. Thus, the construction and analysis of optimality principles are an essential part of the theory of games. Ever
...ntain algorithms for finding solutions. Therefore, particular analytic and numerical methods for the location of solutions are important. Moreover, the problem
25 KB (4,000 words) - 19:41, 5 June 2020
...moment problem subsumes the multi-variable quadrature problem of numerical analysis (cf. [[#References|[a6]]], [[#References|[a13]]], [[#References|[a23]]], [[
...</tr><tr><td valign="top">[a22]</td> <td valign="top"> M. Putinar, "Linear analysis of quadrature domains" ''Ark. Mat.'' , '''33''' (1995) pp. 357–376 {{MR|1
23 KB (3,450 words) - 17:45, 1 July 2020
e.g. piecewise-continuous functions) whose numerical values satisfy (2). Any function $ u(t) \in U $
...automatic control. In particular, one may mention the methods of frequency analysis; methods based on the first approximation to [[Lyapunov stability theory|Ly
43 KB (6,618 words) - 07:32, 26 March 2023
...of the existence, uniqueness and stability, and also by creating efficient numerical methods for their solution. Existence theorems have been obtained for local
...Lavrent'ev, "Some improperly posed problems of mathematical physics and analysis" , Amer. Math. Soc. (1986) (Translated from Russian)</TD></TR><TR><TD val
19 KB (2,714 words) - 13:11, 13 January 2024
...ion is useful when some functions are not differentiable. Using non-smooth analysis, one can replace derivatives by other objects such as subgradients (see, e.
...meters, controls, and states, and for formulations of duality theories and numerical methods. The multipliers describe sensitivity of the optimal value function
16 KB (2,514 words) - 17:28, 23 October 2017
...of functions, one of the most rapidly developing branches of mathematical analysis.
...es of specific approximation methods and on advanced results of functional analysis, turned out to be also very fertile in ideas, because, by proceeding in thi
29 KB (4,328 words) - 18:47, 5 April 2020
Hence numerical maximization is needed to obtain the maximum likelihood estimates.
t \geq 0 $. From the above it becomes clear that the analysis of the
16 KB (2,671 words) - 20:19, 12 March 2016
...elonging to continuous mathematics, although the search for an approximate numerical solution already presupposes the approximation of continuous functions by d
An analysis of the lower bound for $ L ( n) $
27 KB (4,142 words) - 14:55, 7 June 2020
Numerical values of <img align="absmiddle" border="0" src="https://www.encyclopediaof
...encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600193.png" />-adic
analysis stimulated the development of the theory of Diophantine approximations in t
54 KB (7,359 words) - 18:32, 31 March 2017
...es were greatly extended in the 1950s. They stimulated the requirements of numerical mathematics, penetrating ever deeper into problems of an optimization chara
...ral duality theorems, reflecting fundamental relations in geometric convex analysis (see [[#References|[7]]], [[#References|[8]]]). If, e.g. $ X $
27 KB (3,803 words) - 07:21, 26 March 2023
is a summable numerical sequence, $ \{ x _ {i} ^ \prime \} $
...fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press (1968) (Translated from Russian)</TD></TR><TR><T
24 KB (3,574 words) - 18:23, 21 January 2021
The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... usi
...the arithmetization program signified efforts to develop a foundation for analysis, i.e. the calculus, in terms of the natural numbers
91 KB (14,186 words) - 16:58, 23 November 2023
....encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100215.png" /> and
numerical calculations show that it should be <img align="absmiddle" border="0" src="
.... Pelczynski, "Structural theory of Banach spaces and its interplay with
analysis and probability" , ''Proc. Int. Congress Mathematicians 1983'' , North-Holl
88 KB (12,202 words) - 17:19, 23 October 2017
...op"> N.I. Akhiezer, "The classical moment problem and related questions in analysis" , Hafner (1965) (Translated from Russian) {{MR|0184042}} {{ZBL|0135.33803}
...ent problem in the setting of the theory of continued fractions and linear analysis has intensified; cf. [[#References|[a16]]]–[[#References|[a20]]].
33 KB (4,743 words) - 01:58, 17 June 2022
...in the theory of categories (topoi, Abelian categories), and in functional analysis (representation theory). Conversely, the ideas and methods of these discipl
...sults which one now calls the theory of elliptic curves arose as a part of analysis (rather than geometry) — the theory of integrals of rational functions on
29 KB (4,414 words) - 17:20, 17 December 2019
method is no less yours than mine." "Analysis'' (as it has
probability, illustrated by a numerical example, when its
25 KB (4,000 words) - 07:43, 14 August 2023
...TD valign="top">[1]</TD> <TD valign="top"> L. Euler, "Einleitung in die Analysis des Unendlichen" , Springer (1983) (Translated from Latin)</TD></TR><TR><
Some more facts. Although numerical tables do not suggest it, the function $ \pi ( x) - \mathop{\rm li} ( x)
36 KB (5,530 words) - 08:29, 14 January 2024
...ion or bounded deviation from a [[Conformal mapping|conformal mapping]]. A numerical characterization of the distortion under a mapping $ f : D \rightarrow D
...TD valign="top"> Yu.G. Reshet'nyak, "Stability theorems in geometry and analysis" , Novosibirsk (1982) (In Russian)</TD></TR><TR><TD valign="top">[39]</TD
28 KB (3,937 words) - 14:54, 7 June 2020
...is the following problem on the complexity of realization. By some means a numerical measure (the complexity of a formula) is introduced on the set of all eleme
...lay an important role in mathematical logic, model theory and mathematical analysis. For countable-valued logics it has been established that the set of pre-co
34 KB (5,105 words) - 19:20, 16 January 2024
The problem of determining the numerical values of the probabilities $ p _ {k} $
...of a random variable completely, one uses a, not too large, collection of numerical characteristics. The ones most often used are the [[Mathematical expectatio
42 KB (6,282 words) - 08:07, 6 June 2020
Quadratic forms with any algebraic numerical coefficients.
...l calculus in the large|Variational calculus in the large]] for the global analysis problems that emerged later. For the much related topic of optimal control,
29 KB (4,109 words) - 19:54, 18 March 2018
By numerical computations combined with analytic theory it has been shown that the first
|valign="top"|{{Ref|Eu}}||valign="top"| L. Euler, "Einleitung in die Analysis des Unendlichen" , Springer (1983) (Translated from Latin) {{MR|0715928}} {
45 KB (7,251 words) - 02:20, 29 June 2022
By about 1820, mathematicians had developed deductively a large part of analysis using the real numbers and their properties as a starting point.
...Cauchy, Weierstrass, Dedekind, Cantor, and others succeeded in “reducing” analysis to the arithmetic of natural numbers $\mathbb{N}$.
189 KB (29,059 words) - 14:31, 19 March 2023