...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