Rhomberg method integration. Solving differential equations. See the meaning of the Romberg Method in other dictionaries

This method allows one to obtain a higher order of accuracy, ns resorting to more complex formulas such as (3.14), (3.15), which take into account the additional (compared to the minimum) number of approximation nodes. Improving accuracy is achieved by combining a pair of calculations on different grids with different numbers of nodes.

An example of calculating the approximation of the second derivative in (3.17) allows us to assume that in the general case, if the network is some formula g (x, h) for an approximate calculation of the quantity d (x), then the approximation error can be represented as:

Let's carry out one more calculation according to the scheme fhf) for the same point using a uniform grid, but with a step rh. For the new grid, the approximation error will have a form similar to formula (27), i.e.

Having two calculations (3.27), (3.28) for two grids, we can estimate the error R. It can be obtained after subtracting equation (3.27) from (3.28):

where it is assumed that 0 ((r / i) pi1) a 0 (h pi1). Formula (3.29) is usually called Runge's first formula.

Runge's first formula makes it possible to refine the error in the original scheme (3.27), namely, substituting the error from (3.29) into (3.27), we find

Formula (3.30) is called by the second Runge formula, and it allows you to get a numerical result with a higher order of accuracy.

Runge's method is generalized to the case of an arbitrary number of grids q. In this case, by appropriately modifying the calculation scheme, it is possible to increase the accuracy of the approximation to the level 0 (h p + by the Romberg scheme. This scheme is described in more detail in the textbook.

Let's look at an example to illustrate how Runge's method works. Let us choose for testing the numerical differentiation scheme (3.18), i.e., the right finite difference, which, as was established above, has the first order of approximation, i.e. R= 1. Refinement according to Runge should increase the accuracy up to the 2nd order of the grid spacing. As above, the tested function will be y = sin (x).

In fig. 3.4 shows a diagram of the positioning of two grids according to the Runge method, when one of them is twice as detailed as the other.

Fig. 3.4.

Listing 3.4 shows the code for a program that refines the Runge sine derivative after a couple of calculations on the original grid and on a grid with twice the number of nodes. The result of the program is concentrated in the graphs in Fig. 3.5.

ListingPER

% A program illustrating Runge's method for increasing the% accuracy of numerical differentiation by% combining a pair of calculations on two uniform% grids, the second containing twice as many% of nodes% clearing the workspace clear all

% determine the step of the original grid

^ form the original mesh

% determine the number of nodes included in the original% mesh

% determine the step of a more detailed grid

% create a mesh twice as detailed

% we determine the number of nodes included in a more detailed mesh m = I е n g t h (х т);

% calculate the derivative using the right% difference on the original mesh and estimate the% corresponding absolute error

for i = 1: (P-1) dу (i) = (si n (x (i +1)) - si n (x (i))) / h;

erl (i) = abs (cos (x (i)) - dy (j)); end

"/" Calculate the derivative using the right% difference on a more frequent grid and estimate "/" the corresponding absolute error for i = 1: (m-1)

dym (i) = (si n (xm (i + l)) - si n (xm (i))) / hm; er 2 (i) = abs (cos (xm (i)) - dym (i)); end

% we refine the value of the derivative using the% Runge method for i = 1: (P-1)

dyrungefi) = dy (i) - 2 * (dy (i) - dy m (2 * i -1)); e r 3 (i) = abs (cos (x (i)) - dyrunge (i)); end

"/ embed a general graph with all three curves of% errors

pi ot (x ([1: (n-1)]), er1 ([1: (n- 1)]), 1 - o "----

x m ([1: 2: (m-1)]), e g 2 ([1: 2: (m-1)]), "- p 1,... x (), e g 3 ([1 : (n-1)]), 1 -h ');

Comparison of the graphs in Fig. 3.5 confirms the theoretical conclusions. Runge's procedure really dramatically improves the accuracy of the numerical estimate of the derivative in our example.

Fig. 3.5. Errors of two numerical differentiation schemes: for the original scheme and for a scheme with a doubled number of nodes, as well as an error in Runge's refinement procedure

Let's consider one more problem illustrating methods of numerical differentiation. It is required to study the rate and acceleration of population dynamics in Russian Federation... We will take the data from the Russian statistical yearbook. This is a typical example when a function is given in a table and you want to find the first and second derivatives. It is natural to solve this problem on average, since the values of the function are determined with errors. According to the solution procedure, on average, the unknown function is approximated by some polynomial, the coefficients of which are determined by the least squares method. Differentiating the resulting polynomial 1 and 2 times, we find, respectively, the speed and acceleration of demographic dynamics in the Russian Federation.

Listing 3.5 shows the code for the corresponding program.

Listing 35

% Program for the analysis of demographic dynamics in the Russian Federation We clean the workspace clear all

We enter the data: the year of the population census, Increased to improve the operation of the algorithm by 10 0 0 x = [1. 9 5 9 1.9 7 0 1.9 7 9 1.9 8 9 1.9 9 2 1.9 9 3 * 1.9 94 1.9 9 5 2. 0 0 2];

We enter the data: the number of population in million people. % for the corresponding census year

y = [1 1 7. 5 1 2 9.9 1 3 7.4 1 4 7 1 4 8.3 1 4 8.3 1 4 8 1 4 7.9 1 4 5.2];

Set the degree of the approximating polynomial

We turn to the standard program that calculates the Coefficients of the polynomial approximating the data% on average p = polyfit (x, y, nm);

We determine the values of the approximating polynomial% at the reporting times

phi = polyval (p, x);

We will warn the original data together with the approximating Uspolin

pi ot (1000 * x, y, '*', 1000 * x, phi);

We define the coefficients of the polynomial describing the% rate of demographic dynamics

for i = 1: P m dpl (i) = (nm-i + l) * p (i); end

We determine the values of the approximating polynomial Y of the rate of demographic dynamics in the reporting Time points

dphi 1 = poI yvaI (dpi, x);

% draw a graph of the rate of demographic dynamics

% pl ot (1 0 0 0 * х, dphi 1/1 0 00);

% we determine the values of the approximating polynomial% acceleration of demographic dynamics at the reporting% points in time

for i = 1: (nm-1) dp2 (i) = (nm-i + l) * (nm-i) * p (i); end

% we determine the values of the approximating polynomial% acceleration of demographic dynamics in the reporting time points dphi 2 = poI uvаI (dр2, x);

We will paint a graph of the acceleration of demographic dynamics

% pl ot (1000 * x, dphi 2 / 1eb); grid on

In fig. 3.6 shows a graph of demographic dynamics in the Russian Federation. The tabular values are marked with markers; the line corresponds to a cubic parabola that best approximates our data in the sense of the least squares method. From the graph in Fig. 3.6 it can be seen that the population in the Russian Federation has been steadily declining since the mid-1990s.

Fig. 3.6.

In fig. 3.7, but shows a graph of the rate of demographic dynamics, and in Fig. 3.7, b - acceleration graph.

Fig. 3.7. Demographic dynamics in the Russian Federation: a - speed; b - acceleration

According to the graph in Fig. 3.7, but the rate of demographic dynamics changed sign and became negative in the mid-1990s, which is consistent with the graph of demographic dynamics in Fig. 3.6. The acceleration of demographic dynamics changed sign and became negative in the early 1970s, which is also consistent with the graph of demographic dynamics, where at the beginning of the 1970s. accounted for the peak of the population growth rate.

Differential equations are very common when constructing

models of the dynamics of research objects. They describe, as a rule, changes in the parameters of an object over time. The result solutions to differential equations are functions, and not numbers, as in solving finite equations, as a result of which the methods for solving them are more laborious.

Differential equations also describe the process, heat and mass transfer, change in the concentration of a substance, sugar crystallization processes and many others. When using numerical methods for solving differential equations:

Or y ’= f (x, y) is presented in tabular form, i.e. it turns out
dx is a set of Y i and X i values.

The solution is step-by-step, i.e. one or more starting points (x, y) are used to find the next point in one step, etc. The difference between two adjacent values of the argument h = x i +1 - x i is called step.

The most widespread are the Cauchy problems, in which the initial conditions are given: at x = x 0, y (x 0) = y 0. Having them, it is easy to start the solution process, i.e. find at x 1, y 2 - at x 2, etc.

The main idea of obtaining the simplest computational algorithms in one-step methods is reduced to the expansion of the original solution y (x) in Taylor's row.

The number of members left in the series determines the order and hence the accuracy of the method. Using the resulting expansion, knowing the values of y at the decomposition point y i and the derivative f (x i, y i), find the values of y through step h:
y i +1 = y i + ∆y i.

If a larger number of terms are retained in the expansion, then it is necessary to calculate f (x i, y i) at several points (in this way, the need to directly calculate the higher derivatives present in the Taylor series expansion is avoided).

The computational algorithms of multistep methods are based on the construction of interpolation or approximating functions, from which the integral is taken.

Numerical methods are used to solve not only individual equations, but also systems of equations (most often of the first order), and most of the methods for solving one equation are easily extended to the solution of systems.

The class of one-step methods includes Euler's methods,
Runge - Kutta and Euler-Cauchy.

Functional equation y ¢ = f (x, y) connecting the independent variable, the required function y (x) and its derivative y (x) is called differential equation of the 1st order.

A solution (particular) solution of an equation on the interval (a, b) is any function at= (x), which, being substituted into this equation together with its derivative ¢ (x) makes it an identity with respect to xÎ (a, b). The equation F. (x, y) = 0 defining this solution as an implicit function is called the integral differential equation... On a plane with a fixed Cartesian rectangular system coordinate equation Ф (x, y) = 0 defines a certain curve, which is called the integral curve of the differential equation.

If in the differential equation y ¢ = f (x, y) function f (x, y) continuous in some region D, plane Ooh and has a limited partial derivative in this area (x, y), then for any point (x 0, y 0) Î D, in some interval x 0 - h £ x £ x 0+ h, there is and, moreover, the only solution y (x) of this equation, satisfying the initial condition

y (x o) - y o.

This statement is known as the Cauchy theorem on the existence and uniqueness of a solution to a differential equation with a given initial condition.

For tasks of this type, allocated to a whole class Cauchy problems, in addition to analytical methods of solution, methods of numerical solution have been developed.

Euler's method

The values of the required function y = y (x) on the segment is found by the formula:

y k + 1 = y k + h × f (x k, y k), (1)

where y k = y (x k), x k + 1 = x k + h, (x n = x), k = 0,1,2, ... n -1 and h =

For a given limiting absolute error e, the initial calculation step h is set using the inequality h 2< .

Euler - Cauchy method

To calculate the values of the function y = y (x) apply the formula:

(2)

where,, ,

For a given limiting error, the initial calculation step h is established using the inequality h 3 < .

Ruge - Kutta method

The values of the required function y = y (x) on the segment are successively found by the formulas:

y k +] = y k + y k, k = 0, l, 2, ... n - l (3)

where y k = (),

p r a v i l o R o mberg a, - method computing a definite integral based on Richardson extrapolation. Let the value I of a certain functional be calculated, while the calculated approximate value T (h) depends on the parameter h, so that the result of the calculation is an approximate equality. Let the information on the behavior of the difference I - T (h) be known. as functions of h, namely: (1) where m - natural number and a depends on the functional to be approximated and the function on which it is calculated, on the method of approximation and (weakly) on h. If, along with Т (h), Т (2h) is calculated, then Richardson's method gives approximation for I (2) This approximation is the better, the weaker a from equality (1) depends on h. In particular, if a does not depend on h, then (2) is exactly equal. R. m. Is applied to the calculation of the integral. The interval is taken for simplicity, it can be any finite. Let (3) Calculations in R.M. are reduced to compiling the following table: where the first column consists of the quadrature sums (3) of the trapezoid formula. The elements of the (l + 2) th column are obtained from the elements of the (l + l) th column by the formula (4) When compiling the table, the main part of the computational work is spent on calculating the elements of the first column. The elements of the following columns are calculated a little more complexly than finite differences. Each element of the table is a quadrature sum that approximates the integral (5) The nodes of the quadrature sum are points, and its coefficients are positive numbers. Quadrature formula (5) is sharp for all polynomials of degree at most 2l + 1.
Under the assumption that the integrand f (x) has a continuous derivative of order 2l + 2 na, the difference has a representation of the form (1), in which m = 2l + 2. It follows that the elements of the (l + 2) -th column calculated by formula (4) are Richardson's improvements to the elements of the (l + l) -th column. In particular, for the error of the quadrature formula of trapeziums, the representation is valid and Richardson's method gives a more accurate approximation to I: it turns out to be the quadrature sum of Simpson's formula, and since the error of this formula is valid for the representation, then again you can use the Richardson method, etc. . m. T 0n is taken as an approximation to I, and it is assumed that there is a continuous derivative f (2n) (x) on. A rough idea of the accuracy of the approximation T 0p can be obtained by comparing T0n and T1, n_1. The method was first described by V. Romberg. Lit.: [l] R o m b e r g W., "Kgl. Norske vid. Selskabs forhandl.", 1955, Bd 28, no. 7, s. 30-36; In a u er F. L., R u t i s h a u s e r H., Stief1 E, "Proq. Symp. Appl. Math.", 1963, v. 15, p. 199-218. I. P. Mysovskikh.

Watch value Romberg Method in other dictionaries

Method- m. And method w. method, order, grounds; the accepted path for the move, the achievement of something, in the form of general rules. - wild, decent, correct, solid, gradual; ........
Dahl's Explanatory Dictionary

Method- (new official). Reduction, uptr. in new compound words ah in meaning. methodical, for example. method bureau.
Ushakov's Explanatory Dictionary

Method M. And Ustar. Method J.- 1. Method of cognition, approach to the study of natural phenomena and social life. 2. Reception, a system of receptions in some l. areas of activity.
Efremova's Explanatory Dictionary

Method...- 1. The initial part of complex words, which introduces the meaning of the word: methodical (method bureau, method cabinet, method combination, etc.).
Efremova's Explanatory Dictionary

Analytical Forecasting Method- A forecasting method based on obtaining expert estimates by logical analysis of the forecast model.
Political vocabulary

Behavioral Method- (from the English "behavior" - behavior) - in political science involves the study of political phenomena and processes through the analysis of the behavior of individuals and groups in the performance ........
Political vocabulary

Delphi Method, Delphi Method- The collective method peer review, based on the identification of an agreed assessment of the expert group by means of their autonomous survey in several rounds, providing ........
Political vocabulary

Matrix Forecasting Method- A forecasting method based on the use of matrices reflecting the values (weights) of the vertices of the graph model of the forecasting object, followed by transformation of the matrices ........
Political vocabulary

Method- - means of analysis, a way of checking and evaluating knowledge.
Political vocabulary

Destructive Relative Valuation Method- The method of collective idea generation, implemented through two spaced apart during sessions, the first of which fully obeys the rules of collective generation ........
Political vocabulary

Individual Expert Evaluation Method- A forecasting method based on the use of one expert as a source of information.
Political vocabulary

Interview Method- The method of individual expert assessment based on a conversation between an expert and a forecaster according to the "question - answer" scheme.
Political vocabulary

Historical Analogy Method- A forecasting method based on the establishment and use of an analogy of a forecasting object with an object of the same nature, which is ahead of the first in its development.
Political vocabulary

Collective Idea Generation Method- A method of collective expert assessment based on stimulating the creative activity of experts through joint discussion of a specific problem, regulated by ........
Political vocabulary

Peer Review Method- A forecasting method based on identifying a generalized objectified assessment of an expert group by processing individual, independent assessments made ........
Political vocabulary

Method of Mathematical Analogy- A forecasting method based on the establishment of an analogy of mathematical descriptions of the processes of development of objects of different nature with the subsequent use ........
Political vocabulary

Forecasting Scenario Construction Method- An analytical forecasting method based on the establishment of a logical sequence of states of the forecasting object and the forecast background in time at ........
Political vocabulary

Forecasting Method- A method of researching a forecasting object aimed at developing forecasts. Note. Forecasting methods are the basis for forecasting methodologies.
Political vocabulary

Method of Psycho-intellectual Generation of Ideas- The method of individual expert assessment, in which the identification of expert assessment is carried out using programmed control, including an appeal to ........
Political vocabulary

Managed Idea Generation Method- The method of collective idea generation using purposeful intellectual influence (amplifying or suppressing) on the idea generation process.
Political vocabulary

Heuristic Forecasting Method- An analytical forecasting method consisting in the construction and subsequent truncation of an expert judgment search tree using some heuristic.
Political vocabulary

Expert Commission Method- The method of collective expert assessment, consisting in the joint work of experts united in a commission, developing a document on the prospects for the development of the forecasting object.
Political vocabulary

Advanced Forecasting Method- A forecasting method based on the use of the property of scientific and technical information to outstrip the implementation of scientific and technical achievements in public practice.
Political vocabulary

Patent Forecasting Method- An advanced forecasting method based on the assessment (according to the adopted system of criteria) inventions and the study of the dynamics of their patenting.
Political vocabulary

Publication Forecasting Method- An advanced forecasting method based on the assessment of publications about the forecasting object (according to the adopted system of criteria) and the study of the dynamics of their publication.
Political vocabulary

Synoptic Method- A forecasting method based on the analysis by experts of a known set of forecasts of the forecasting object and forecast background with their subsequent synthesis.
Political vocabulary

Systemic Method In Political Science- - consideration of politics as a holistic, complexly organized, self-adjusting mechanism in continuous interaction with the environment.
Political vocabulary

Comparative Method In Political Science- - comparison of the same type of political phenomena and processes and the identification of their common features and specifics in order to find the optimal forms of political organization or by solving problems.
Political vocabulary

Statistical Forecasting Method- Factographic forecasting method based on the construction and analysis of time series of characteristics of the forecasting object.
Political vocabulary

Factographic Forecasting Method- A forecasting method based on the use of sources of factual information.
Political vocabulary

JOURNAL OF COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS, 2009, volume 49,< 2, с. 232-240

UDC 519.644.7

APPLICATION OF THE ROMBERG METHOD TO INCREASE THE ACCURACY OF CALCULATION OF MULTIPLE INTEGRALS

(141980 Dubna, JINR)

e-mail: [email protected]

Received May 24, 2007 Revised March 03, 2008

The well-known Romberg's method, used to improve the accuracy of calculating one-dimensional integrals, is generalized to the case of multiple integrals if the product of compound quadrature formulas is used to calculate them. Under certain conditions, the coefficients of the Romberg formula turn out to be independent of the multiplicity of the integral, which makes it possible to use a simple calculation algorithm developed for one-dimensional integrals. Examples of calculations by the Romberg method for integrals of multiplicity from two to six are given and compared with some other methods. Bibl. 9. Tab. 3.

Key words: numerical integration, multiple integrals, Romberg's method.

1. INTRODUCTION

Romberg's method is well known as a technique for improving the accuracy of calculations of one-dimensional integrals by compiling a certain linear combination of their values obtained on different integration grids. In a multidimensional case, it is difficult to obtain such sets of approximate values if they are calculated by sequential application of some quadrature formula.

Let the function f (x1, ..., xp) be defined on the set 0 = (a1< х1 < Ь1, ..., ар < хр < Ьр}. Будем вычислять интеграл

?) = / //(X1.....Xp) dX1 ... dXp (1)

using the formula

/) = B ... bp B> 1 ... 0] p / (a 1 +] 1Hb ..., ap +] pHp), (2)

> 1 = 0] p = 0

which is the product of compound quadrature formulas with constant integration steps bk:

Sk (Ik) = hk ^ DjkIk (xlf ..., xkak + jkhk), (3)

k = 1, 2, ..., p. Here Mk is the number of partitions of the interval of integration [ak, bk], bk = (bk - ak) / Mk, B> are the coefficients of formula (3),

(x1,, xk) - J - "J f (x1,> xp) dxk +1 ■■■ dxp,

1p (x1, ..., xp) = / (x1, ..., xp). The error of formula (2) R (/) = I (f) - £ (f) can be represented as (CM.)

R = E0 + R1 £ ^ ^ + R1D2 £ 2 EI, 2 +

>1 = 0 ,1=0 ,2 = 0

Y! ... dp2 ... 2 1 _1,

>1 = 0 -1 = 0

where Ek. 1 - error of formula (3).

Suppose that for integration over all variables x1 ..., xp, the same quadrature formula (ZD) is used with an error E = 0 (Ikt). Let us denote by M the total number of partitioning regions of the original region O. If I1 = ... = Ir = I, then M = N and the total error R is estimated as R = 0 (M ~ t / p). With an increase in the multiplicity p, an increase in N leads to an increasingly significant increase in the volume of computations. In this case, with relatively low computational costs, it is possible to obtain only a few approximate values of the integral corresponding to small values of N. Such approximations are usually too rough, but their linear combination according to the Romberg method, which actually means an increase in m (see), may turn out to be satisfactory. for accuracy. Thus, for some p and a certain choice of integration grids, such an approach can be effective in terms of the ratio of computational costs and the accuracy of the results obtained.

In this article, Romberg's method is applied to the case when the integration over all variables is performed using the trapezoidal formula.

2. CALCULATION METHOD Consider the compound formula of trapeziums:

1k) = Yk 2 ^) k1k (X1 "" Xk-1, ak +> kJk),

G 1, if] k = 1, 2, ..., Ik -1> k = 11/2, if] k = 0, Ik,

k = 1, 2, ..., p. In the one-dimensional case, under the assumption that the (21 + 2) th derivative of the integrand f (x) is finite, the error of formula (5) can be represented in the form (see)

Rtr (/) = - 2 72m2m (/ (2m -;) (b) - / (2m -;) (a)) + r (d),

where a, b are the limits of integration,

r (d) = -y 21 + 2 / (21 + 2) (^) d2 "+ 2 (b - a), a b.

Numbers Y; satisfy the equality

Below are the meanings of some of them:

Y 2 = 1 1 "Y 4 = TOP" Y 6 = aloil "Y 8 = 1opplpp" Y10 =

12 14 720 16 30240 18 1209600 110 47900160"

Therefore, if the functions 1k (x1, ..., xk), k = 1, 2, ..., p, have bounded partial derivatives (1k (x;, xk)) 2m of order 21 + 2, the error of formula (5) can be written as

E "-1 (4) = 2 Stk (1k) (Yk) 2t + Gk (Yk),

& mk (1k) = -Y 2 [(1k (X;,.> Xk - 1 »bk)) 2m-1 - (1k (X ;, Xk-1, ak)) 2m-1],

Gk (Yk) = -Y 21 + 2 (1k (x1; Xk-; Lk)) 21 + 2 (Yk) 21 + 2 (Lk - ak).

ak< < Ьк. Подставляя это выражение при к = 1, 2, ..., р в формулу (4), получаем

R = 2 ° t; (I;) (J;) 2 t + r; (th;) +

Y1 2 2 St2 (12) (Y2) 2t + G2 (Y2)

Th; .Air-1 2- 2 ^ p-1 2 ° tr (1p) (Yp) 2 t + Gr (YR)

>1 = 0 -1 = 0

Let us introduce the notation

Wk (1k) = Y1 Yk-1 2- 2 - BbStk (1k).

>1 = 0 -1 = 0

W1 (4) = ^ t1 (11), k = 1 2,., P.

Then the expression for R can be written in the form

R = 2 W; (I;) (J;) 2t + g; (J;) +

2 W2 (12) (Y2) 2t + Y; 2 ^ Г2 (Я2) + ...

2 Ttr (1 r) (dp) t + d1-dp-; 2- 2 burnR (th?) =

> 1 = 0) p-1 = 0

22 Wk (1k) (Jk) 2t + R *,

R * = G; (Y;) + Y; 2 ^ Г2 (J2) + ... + J; - Yr _; 2- 2 ^; Gr (Dp). > 1 = 0> 1 = 0 -1 = 0 Suppose now that integral (1) is calculated I + 1 times by formula (2) for

Ir = Ir, X Ir, ... D ^,

where X is some positive constant.

Let us introduce the index r to denote the number of the corresponding calculation of the integral for N1, ..., Mp. All quantities depending on Mk will also have the index r. Then the partition numbers N1 and the steps of the integration grid H "k can be expressed in terms of the powers of X as follows:

Mk = bk = X "A °, k = 1, 2, ..., p, r = 0, 1, I.

We have the equalities

1 (/) = d (/) + /) = d (/) + XX r (1k) (bk) 2m + A *,

r = 0, 1, ..., I. We form a linear combination of these relations with some coefficients σ, requiring that the condition

1 (/) = X cd; (/) + X X X sWk I (4) (bk) 2m + X sL *.

0 k = 1t = 1 z = 0 I = 0

For the coefficients Wkg (1k), the following inequalities are fulfilled:

\ Wkg (4) |< л1 ... Лк_ 1 X- X В>k-11 Rev (1k) |<

< Ртк(1к)П(Ь«-ап) ^ -1- X- X Вк-l,

N1 Nk-1> 1 = 0> k-1 = 0

Pmk (4) = max | ^ mk (4) | , k = 1, 2, ..., p, m = 1, 2, -, /.

X ... X B ... = 1,

"1" k- 1> 1 = 0 L-1 = 0

k = 2, 3, ..., p, r = 0, 1, ..., I. Consequently,

\ Wk g (4) |< ртк(1к)П(Ьп-ап) для к = 2, 3, ..., р, т = 1, 2, ..., I,

Wkg (11) |< Рт1(4) длЯ I = 0, 1, ..., I.

Then, taking into account (6), we have

I (/) - X (/) - X with l

< X Рт1(11)(Ъ0)

XX rtk (4) P (bn-an) ()

We now require that the conditions

2CX "" = 0, m = 1, 2, ..., I.

These conditions, together with condition (7), form a system of I + 1 linearly independent equations, from which I + 1 unknown coefficients c are determined. Substituting these coefficients into inequality (9), we obtain

I (/) = 2 С & (/) + RRomb, (10)

^ from b = 2 C; = 0

R * = r; (d1) + 1 2 Γ2 (+2) + - + 1 -air-; 2- 2 - ^ Gr (dp),

> 1 = 0> 1 = 0) p-1 = 0

Tk (dk) = -Y21 + 2 (1k (X ;, ..., Xk-1, ^ k)) 21 + 2 (dk) 21 + 2 (bk - ak).

Theorem. Let the function f (x1, ..., xp) have bounded partial derivatives f21 + 2 (x1, ..., xp),

k = 1, 2, ..., p. Then the error in formula (10) satisfies the inequality

| ^ from |<| У 21 + 2 Рг П(Ь" - а")2 таХ /Х2" + 2 (Х1" Хр)|(Й0)21 +2, (11)

A A O I xk I

Pr = 2 C X-2 "1 +!).

Evidence. According to the properties of definite integrals and the properties of integration with respect to a parameter, we have

(1k (x1, -, Xk)) 2r + 2 = -TGG2 I -I / (x;, -, xp) Lxk + 1- LXr =

1 -1 / X2 + 2 (X1 "-" Xp + 1 - Lxr,

max | (4 (X;, -, Xk)) 21 + 2 = max

1 -1 / h +2 (X1, -, XP) ^ to +1 -

<тах|/г2г+2(X;,-, Xp)| П (Ьп - ап), к = 1,-, р - 1.

For k = p we have b ^, ..., X,) = f (x1, ..., xp) and

max | (1p (X;, -, Xp)) ^ = maX / x2r + 2 (X;, -, Xp) | ...

X, -, xp \ lp I О I

Then the quantity m * = max \ mk \ satisfies the inequality

t * (bk)< |у21 + 2таХ/х21+2(Х1, Хр)|П(Ьп - ап)(Ък)2" + 2, к = 1, 2,., р.

I + 2G "" L | .Y T2 "+ 2

Taking into account (8), we obtain

\I*\< т*(Л1) + (Ь - а1)т*(Л2) + ... + (Ь1-а1)...(Ьр-1-ар-1)т*(лр) =

IY 21 +2 P (bn - an) X maX / X2 "+ 2 (X1, Xp) | (bk)

A A O I Xk I

^ mn = XI cfn<

<|У21 + 2 П(Ьп - ап)XтаХ+2(Хl, ХрIX 1с""1(Ък)

Using relation (6), from this inequality we obtain estimate (11).

3. SOFTWARE IMPLEMENTATION

It should be noted that the coefficients σ of formula (10) do not depend on the multiplicity of the calculated integrals, since the integration grids (6) over all variables x1, ..., Xp are formed using the same value of the constant X. This makes it possible to apply in the multidimensional case a well-known algorithm developed for calculating one-dimensional integrals. For X = 2, this algorithm is defined by the recursive formula:

^ = ^-1) + ($-1) - ^-1))/(22(>-1)-1).

Here £ (1), r = 0, 1, ..., I, are the values of integral (1) obtained using the composite formula

trapezoids (5) with the corresponding integration steps b "k, k = 1, 2, ..., p. The quantities B (2), ...,

B ■ "+ 1) are the values refined using formula (10). The procedure for obtaining refined estimates using formula (12) can be represented in the form of the following diagram (see):

S (1) s (1) s (1) s (1) B0 B1 B2

On this basis, a computer program was written, according to which the calculation of the integral is performed until one of the conditions is fulfilled

tt | 1 - 1 -1)<еаь8,

The value £ abs or erel is user-definable. The result is the value S- "+

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