...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