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Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 75-101. ISSN 2079-6641
MSC 65D32 Research Article
Euler-Maclaurin type optimal formulas for numerical integration
in Sobolev space
A.R. Hayotov1, F.A. Nuraliev1, R.I. Parovik2, Kh.M. Shadimetov1
1 V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, M. Ulugbek str. 81, 100125, Tashkent, Uzbekistan
2 Vitus Bering Kamchatka State University, Pogranichnaya str. 4, Petropavlovsk-Kamchatsky, 683032, Russia
E-mail: hayotov@mail.ru, nuraliyevf@mail.ru, romanparovik@gmail.com, kholmatshadimetov@mai.ru
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N > m — 3 and for any m > 4 using S.L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2m. In particular, for m = 4 and m = 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m = 6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented.
Keywords: optimal quadrature formulas, the error functional, the extremal function S.L. Sobolev space, optimal coefficients
DOI: 10.26117/2079-6641-2020-32-3-75-101
Original article submitted: 01.09.2020 Revision submitted: 10.10.2020
For citation. Hayotov A. R., Nuraliev F. A., Parovik R. I., Shadimetov Kh.M. Euler-Maclaurin type optimal formulas for numerical integration in Sobolev space. Vestnik KRAUNC. Fiz.-mat. nauki. 2020,32: 3,75-101. DOI: 10.26117/2079-6641-2020-32-3-75-101
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Hayotov A.R., et al., 2020
1. Introduction. Statement of the problem
We consider the following general quadrature formula
1 N a
I p(x)ç(x)dx - £ £ Cßj<p(j)(xß) (1.1)
0 ß=o j=о
Funding. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors
with the error functional
N a
i(x) = p(x)em(x) - £ £ (-l)jCpj S (j)(x - xp) (1.2)
P =0 j=0
in a Banach space B. Here Cpj are the coefficients and xp are the nodes of the formula (1.1), N = 1,2,..., a = 0,1,..., p(x) is a weight function, £[0>1](x) is the characteristic function of the interval [0,1], S(x) is the Dirac delta-function, v is an element of the space B.
The difference
^ 1 »7
P ^ N a
(i, v) = i(x)v (x)dx = p(x)v (x)dx - £ £ Cp j v (j)(xp) (1.3)
0 P=0 j=0
is called the error of the quadrature formula (1.1).
By the Cauchy-Schwarz inequality
|(i, v )|<||v |BH-Hi|B^H
the error (1.3) of the formula (1.1) is estimated with the help of the norm of the error functional (1.2) in the conjugate space B*, i.e. by
||i|B*|| = sup |(i, v)|.
II v |B||=1
Thus estimation of the error (1.3) of the quadrature formula (1.1) on functions of the space B is reduced to finding the norm of the error functional i in the conjugate space B*.
Obviously the norm of the error functional i depends on the coefficients and the nodes of the quadrature formula (1.1). The problem of finding the minimum of the norm of the error functional i by coefficients and by nodes is called the S.M. Nikol'skii problem, and the obtained formula is called the optimal quadrature formula in the sense of Nikol'skii. This problem was first considered by S.M. Nikol'skii [21], and continued by many authors, see e.g. [3, 4, 7, 8, 22, 42] and references therein. Minimization of the norm of the error functional i by coefficients when the nodes are fixed is called Sard's problem. And the obtained formula is called the optimal quadrature formula in the sense of Sard. First this problem was investigated by A. Sard [23].
The results of this paper are related to Sard's problem. So here we discuss some of the previous results about optimal quadrature formulas in the sense of Sard which are closely connected to our results.
There are several methods of construction of optimal quadrature formulas in the sense of Sard such as spline method, v- function method (see e.g. [3, 25]) and Sobolev's method which is based on construction of discrete analogue of a linear differential operator (see e.g. [38, 39]). In the different spaces, based on these methods, the Sard's problem was investigated by many authors, see, for example, [1, 2, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and references therein.
In the paper [25], using spline method, optimality of the classical Euler-Maclaurin formula was proved and the error of this quadrature formula is calculated in 4m)(0, n),
where n) is the space of functions which are square integrable with m-th generalized
derivative.
Let WLm (m = 1,2,..., 1 < p < be a class of functions f, having on the [0,1]
(m — 1)- absolute continues derivative and ||f(m) ||p < 1, where || ■ ||p = || ■ ||Lp(o,1). In [42] it is proved, that among quadrature formulas (1.1) when p(x) = 1 the Euler-Maclaurin quadrature formula is optimal in the space WLp. And in [31] optimality of the lattice
cubature formulas of Euler-Maclaurin type is proved in the space L2m).
Using p-function method optimality of the Euler-Maclaurin quadrature formula is proved and the error of this formula is calculated by T. Catinas and Gh. Coman [7] in the space l22)(0, 1). Also using this method in [18] a procedure of construction of quadrature formulas of the form (1.1), which are exact for solutions of linear differential equations and are optimal in the sense of Sard is discussed.
It should be noted, that in applications the formula (1.1) is interesting for small values of a. Optimal quadrature formulas in the sense of Sard for the case a = 0 has already been discussed by many authors, mainly in the space L^ (see [2, 3, 7, 9, 10, 12, 15, 17, 18, 19, 20, 24, 25, 26, 27, 29, 32, 33, 37, 38, 39, 40, 41] and references therein).
The main aim of this paper is to construct optimal quadrature formulas of the form (1.1) in the sense of Sard for the case a = 3 when p(x) = 1 and xp = hp in the space
L2m)(0,1) equipped with the norm
1
1/2
||V(x)||L(m)(0>1) = y (V(m)(x))2dx 1
and /(p(m)(x))2dx < 0
We use the Sobolev method [38, 39] which is based on the discrete analogue of the differential operator d2m/dx2m. We consider the following quadrature formula
N p=0
J <p (x)dx - £ C[j3]p (ÄJ3 )+ a( p'(0) - p'(1)U ^(0) - ^(1)] (1.4)
with the error functional
N
£(x) = e[01](x) — £ C[p]8(x — hp)+ A^8'(x) — 8'(x — 1)^ + ^8w(x) — 8w(x — 1)^
^=0 _ (1.5)
in the space L2m)(0,1) for m > 4. Here C[P], p = 0, N, A and B are the coefficients of the formula (1.4), h = N, N is a natural number.
For the error functional (1.5) to be defined on the space L2m)(0,1) it is necessary to impose the following conditions (see [37])
(£(x), xa ) = 0, a = 0,1,2,..., m — 1. (1.6)
Hence it is clear that for existence of the quadrature formulas of the form (1.4) the condition N m 3 has to be met.
It should be noted that here in after i means the functional (1.5).
As was noted above by the Cauchy-Schwarz inequality, the error of the formula (1.4) is estimated by the norm ||i|L2m)*(0,1)|| of the error functional (1.5). Furthermore the norm of the error functional (1.5) depends on the coefficients C[P], A and B. We minimize the norm of the error functional (1.5) by the coefficients C[P], A and B, i.e., we find
£\&h = inf £\. (1.7)
2 cWA,B 12
The coefficients C[p], A and B which satisfy the equality (1.7) is called the optimal coefficients and denoted by C[p], A and B and the corresponding quadrature formula is called the optimal quadrature formula in the sense of Sard. In the sequel, for the purposes of convenience the optimal coefficients C[p], A and B will be denoted as C[p], A and B.
Thus to construct optimal quadrature formulas in the form (1.4) in the sense of Sard we have to consequently solve the following problems.
Problem 1. Find the norm of the error functional (1.5) of the quadrature formula of the form (1.4) in the space L^*(0,1).
Problem 2. Find coefficients C[p], A and B which satisfy the equality (1.7).
The rest of the paper is organized as follows. In section 2 we give some definitions and known formulas. In section 3 we determine the extremal function which corresponds to the error functional £ and give a representation of the norm of the error functional (1.5). Section 4 is devoted to a minimization of ||£||2 with respect to the coefficients C[p], A and B. We obtain a system of linear equations for the coefficients of the optimal quadrature formula of the form (1.4) in the sense of Sard in the space L2m)(0,1). Explicit formulas for coefficients of the optimal quadrature formula of the form (1.4) are found in subsection 5.1. Moreover we calculate the norm of the error functional (1.5) of the optimal quadrature formula of the form (1.4) in subsection 5.2. Finally, at the end of the paper we present some numerical results.
(m)
2. Definitions and known formulas
In this section we give some definitions and formulas that we need to prove the main results.
Here the main concept used is that of functions of discrete arguments and operations on them (see. [37, 39]). For the purposes of completeness we give some definitions about functions of discrete argument.
Assume that v and y are real-valued functions of real variable and are defined in real line R.
Definition 2.1. Function v(hp) is called the function of discrete argument, if it is given on some set of integer values of p.
Definition 2.2. The inner product of two discrete functions v(hp) and Y(hp) is called the number
^
[v, Y] = £ v(hp) ■ Y(hp), p=-~
if the series on the right hand side of the last equality converges absolutely.
Definition 2.3. The convolution of two discrete functions v(hp) and y(hp) is called the inner product
v(hp) * y(hp) = [v(hy), Y(hp - hy)] = £ v(hy) ■ Y(hp -hy).
The Euler-Frobenius polynomials Ek(x), k = 1,2,... is defined by the following formula [39]
(1 - x)k+2 / d
k
X
E(x) = ^-^xdxJ (T^, (2.1)
E0(x) = 1.
For the Euler-Frobenius polynomials Ek(x) the following identity holds
Ek(x)= xk£fc( 1) , (2.2)
and also the following theorem is true
Theorem 2.1 (Lemma 3 of [28]). Polynomial Qk(x) which is defined by the formula
k+1 Ai0k+1
Qk(x) = (x - 1)k+1 £ (2.3)
is the Euler-Frobenius polynomial (2.1) of degree k, i.e. Qk(x) = Ek(x), where A'0k =
£l=1 (- 1)i-/c|/k.
The following formula is valid [13]:
n-1 1 k / „ \ '' „n k / „
^ \ A^k ^ V"1 / ^
£0 <<v=r-q Sli-y AW - r-q £lr-q)AV ^ (24)
- qiToV1 - q/ 1 - qiToV1 - q
where A'y^ is the finite difference of order i of y^, q is ratio of a geometric progression. At last we give the following well known formulas from [11]
where Bk+1_y are Bernoulli numbers,
Aaxv = £ CPAa0pxv-p (2.6)
p=o
3. The extremal function and the representation of the error functional norm
To solve Problem 1, i.e., for finding the norm of the error functional (1.5) in the space L2m)(0,1) a concept of the extremal function is used [37]. The function Yi is said to be the extremal function of the error functional (1.5) if the following equality holds
(£, v*)= ^|L2m)* Yd4m) • (3-1)
In the space L2m) the extremal function ^ of a functional i was found by S.L. Sobolev [37, 39]. This extremal function has the form
Vi(x) = (—1)mi(x) * G(x) + Pm—1(x), (3.2)
where
Irl 2m—1
G(x) = ^m—T)! (33)
is a solution of the equation
d2m
^(xH 8(x), (3.4)
Pm—1(x) is a polynomial of degree m — 1, the symbol * is operation of convolution, i.e.
CC cc
f (x) * g(x) = J f (x — y)g(y)dy = J f (y)g(x — y)dy.
CC
( m)
It is well known that for any functional i in L2 the equality
||i|L2m)*(0,1)||2 = (i, y*) = (i(x), (—1)mi(x) * G(x)) =
oo / oo \
I £(x) I (-1)m ^ 4(y)G(x - y)dy I dx
holds [37].
Applying this equality to the error functional (1.5) we obtain the following
Pll2 = (4 %) =
rlh ß_
= (-1)
£ £ rrmnv] - hYl2m-1 2 f r[ ß] 1 |x- hß|2m-1 , £ £rrß]rrY] „ -2 £ rrßßW dx-
,jß=0 Y=0
2 (2m - 1)! /ß=o 0 2 (2m - 1)!
NN
2_
-2_ £ C[/3]P2m-2(h/3) -2B £ C[/3]P2m-4(h/3) + (2 _ 1)! +
= 0 = 0 ( 2 m - 1 ) !
2B+_2 2_B B2 1
+ ^-^ + ^-rVT +
(2m - 3)! (2m - 5)! (2m - 7)! (2m + 1)!
(3.5)
where Pk(h/3 )=(M)ki_ki-M)!.
Thus Problem 1 is solved for quadrature formulas of the form (1.4) in the space
4m)(o, i).
4. The system for optimal coefficients of the quadrature formula (1.4)
Now we investigate Problem 2. For finding the minimum of ||£||2 under the conditions (1.6) the Lagrange method is used. For this we consider the following function
m—1
W = ||£||2 — 2■ (—1)m £ Xa(£(*),xa), a=0
where Aa are unknown multipliers. The function ¥ is the multidimensional function with respect to the coefficients C[!], A, B and Aa. Equating to zero partial derivatives of ¥ by coefficients C[!], A and B, together with the conditions (1.6) we get the following system of linear equations
£CW^ -^)-
m—1
BP2m-4(hß)+ £ "a(hß)« = /m(hß), ß = 0N, (4.1)
a=Ü
N AB m—1 1
£ C'«p2m—2(hß )—(2m—3)7— (2m—5)!+£a ■ = (2m—1)!, (42)
N A B
£ cm p2m—4(hß )—(2m—5)7—(2m—75?+
m—1 1
+ £a(a — 1)(a — 2) ■ "a =(2—3)7, (4 3)
N . 1 _
£ C[ß](hß)- = -—-, - = Ü, 1, (4.4) ß=ü - + 1
N .1 _
£ C[ß](hß)- — -A = —-, - = 2,3, (4.5)
ß=ü -+1
N
£ C[ß](hß)- — -A — -(- — 1)(- — 2)B = -—1—, - = 4,m — 1, (4.6)
ß=ü -+ 1
where pW) = (h!»'*'', /.(*P) = J .y!——dx.
0 v ''
The system (4.1)-(4.6) is called the discrete system of Wiener-Hopf type for the optimal coefficients [37, 39]. In the system (4.1)-(4.6) the coefficients C[!], ! = 0,N, A and B, and also Aa, a = 0, m — 1 are unknowns. The system (4.1)-(4.6) has unique solution and this solution gives the minimum to the ||£||2. Here we omitted the proof of the existence and uniqueness of the solution of the system (4.1) - (4.6). The proof of the existence and uniqueness of the solution of this system is as the proof of the existence and uniqueness of the solution of the discrete Wiener-Hopf type system of the optimal coefficients in the space L2m)(0,1) for quadrature formulas of the form (1.1) for the case a = 0 (see [37, 39]). It should be noted, that in [18] the uniqueness of optimal quadrature formulas in the Sard sense of the form (1.1) is discussed.
5. The coefficients and the norm of the error functional of optimal quadrature formulas
In the present section we study the solution of the system (4.1)-(4.6). To solve this system we use the approach which was suggested by S.L. Sobolev in [38].
5.1. The coefficients of optimal quadrature formulas of the form (1.4)
Suppose that C[/] = 0 for / < 0 and // > N. Using Definition 2.3 we rewrite the equation (4.1) in the convolution form:
C[//] * 2'(2m_ 1); - AP2m-2(h// ) - BP2m-4(h// )+£ A« (h/ )a = /m (h/ ), /= 0, N, (5.1) where
f (h/) (*/)2" + V (_hP)2"_'_j (52)
/m(h/) = W + j== 2(2m _1 _ j)! ■ ( j + 1)! ■ (5-2)
We consider the following problem.
Problem A. Find the discrete function C[/] and unknown coefficients A, B, Xa, which satisfy the system (4.1)-(4.6).
Further, instead of C[/] we introduce the functions
v(h/)=C[/]* IS, (53)
m_1
U(h/) = v(h/) _ AP2m_2(h/) _ BP2m_4(h/)+ £ Aa(h/)a. (5 . 4)
a=0
In this statement it is necessary to express C[/] by the function u(h/). For this we need such operator Dm(h/), which satisfies the equation
hDm(hß) * G(hß) = 5(hß), (5.5)
where G(hp) = ^2^1)! is the discrete argument function corresponding to G(x) defined by (3.3), 8(hp) is equal to 0 when p = 0 and is equal to 1 when p = 0, i.e. 8(hp) is the discrete delta-function. The equation (5.5) is the discrete analogue of equation (3.4). So the discrete function Dm(hp) is called the discrete analogue of the differential operator d2m/dx2m [37].
It should be noted that the operator Dm(hp) was firstly introduced and investigated by S.L. Sobolev [37].
In [30] the discrete analogue Dm(hp) of the differential operator d2m/dx2m, which satisfies equation (5.5), is constructed and the following theorem is proved.
Theorem 5.1. The discrete analogue of the differential operator d2m/dx2m has the form
m_1 (1 _ )2m+1qlpI
qkE2m_1(qfe) f IP I_ '
m_1 (1 _ q )2m+1
1 + EiE_4m+- lor IPI = 1' (5.«)
12m— 1
Dm(hß )=(2m - 1)!
h2m
E2m-1(#/0
m— 1
_22m-1 + E (1 _ )2m+1 for p = 0'
where E2m-1(q) is the Euler-Frobenius polynomial of degree 2m _ 1, are the roots of the Euler-Frobenius polynomial £2m_2(q), |qk I < 1, h is a small positive parameter.
Furthermore several properties of the discrete argument function Dm(hp) were proved in [30]. Here we give the following property of the discrete argument function Dm(hp) which we need in our computations.
Theorem 5.2. The discrete argument function Dm(hp) and the monomials (hp)k are related to each other as follows
ß
,, 0w, „a I Ü when Ü < 2 < 2m — 1, D'»(hß)(hß)k H (2m) ! when Ü < 2m< , (5 7)
^ | 0 when 2m + 1 < k < 4m — 1,
pE Dm(hp)(hp)k = | qMiBm when k = 4m,
p=—^ ^ (2m)!
where B2m is the Bernoulli number.
Then, taking into account (5.5) and Theorems 5.1, 5.2, for the optimal coefficients C[p] we have
C[p]= hDm(hp) * u(hp). (5.8)
So, if we find the function u(hp), then the optimal coefficients C[P] will be found from equality (5.8).
To calculate the convolution (5.8) it is required to find the representation of the function u(hp) for all integer values of P. From the equality (5.1) we get, that u(hp) = fm(hp) when hp e [0,1], where fm(hp) is defined by equality (5.2). Now we need to find the representation of the function u(hp) when p < 0 and p > N. Since C[p] = 0 when hp / [0,1], then
C[p]= hDm(hp) *u(hp) = 0, hp e [0,1].
hp 2m— 1
Now we calculate the convolution v(hp) = C[p] * 2(2m—1)! when hp e [0,1]. Suppose p < 0, then taking into account (4.4)-(4.6), we have
"<hp) = c[p ] * ^=e
EC-1SEC«(hY)a — E |g—-1—(SECM(hY).
(hp)2m—1 , (hp)2m—2 — (hp)2m—3 /1+\ , (hp)2m—4 /1+3A
2(2m- 1)! 4(2m-2)! 4(2m-3)! V3 J 2(2m-4)!3! V4
m—1 (hp )2m-1-a (_ 1)a / i
E ^-1 MM —+ aA + a (a - 1)(a - 2)B
a=4 2(2m- 1 - a)!a! \a + 1 v yv '
2m-1 fhR\2m-1-a/_ 1\a N
aE (£-. - 0).a ECM(hY)a•
2m-1 , ,p\2m-1-at i\a N
Hence, denoting by Rm- 1(hp) = E ^_1_M' E C[y](hy)a for the case p < 0 we
a=m ( ^ y=0
get
v(hp) = (hp)2m-1 , (hp)2m-2 (hp)2m-3 / 1 , -A , (hp)2m-4 / 1 + 3A
V(hp) 2(2m - 1)! + 4(2m - 2)! 4(2m - 3)^3 + y + 2(2m - 4)!3^4 + 3
m—1 (hß )2m-1-a (_ 1)a / i \
—a£4 2(2m-1-a)!a! (o+T + aA + a(a- 1)(a-2)B) -1(hß^ (5 9)
Now suppose ß > N then for v(hß) we get
v(hß) _ (hß)2m-' (hß)2m-2 . (hß)2m-3 / T + 2A\ _ (hß)2m-4 fT + 3A\
V(hß) 2(2m- 1)! 4(2m-2)! + 4(2m-3)^3 + J 2(2m-4)!3! V4 + 3 )
m—1 (hß)2m-1-a(_ 1 )a / 1 \
+ £ (2(ßm-1- 0).«! (o+T + aA + a(a— 1)(a— 2)B) + R»-1(hß). (510)
Denoting
m-1
Rm--1(hß) _ £ Aa(hß)a -Rm-1(hß), (5.11)
a _0
m-1
Rm+-1(hß) _ £ Aa(hß)a + Rm-1(hß), (5.12)
a=0
and taking into account (5.9), (5.10), (5.4) we have the following problem Problem B. Find the solution of the equation
hDm(hß) * u(hß)_ 0, hß / [0,1] (5.13)
having the form:
u(hß)_ <
(hß )2
2m-1 | (hß)2m-2 (hß)2m-3 /1 , 2 a 2(2m-1)! + 4(2m-2)! 4(2m-3)! V3 + 2AJ
^2(Sm-!m-!43! (4 + 3A) - AP2m-2(hß) - BP2m-4(hß)
m1
- £
2m— 1—a
m_
2(2m-1-a )!a
a_4
fm(hß ),
(hß )2m-1
2(2m-1)! 4( 2m-2)! ' 4(2m
¿O-Ot (0+1 + aA + a (a - 1)(a - 2)B) + R^ (hß), ß < 0
< ß < N,
0
(hß)2m-2 (hß)2m-3 /1 , 2 a 4(2m-2)! + 4(2m-3)! U + 2AJ
2m 2
(hß)
2m 3
2m- 4
2(2m-4-!3! (1 + 3A) - AP2m-2(hß) - BP2m-4(hß)
m- 1
+ £ 2(2m-1 —a )!a a_4
(hß ^T-0-a)a (0+1 + aA + a (a — 1)(a — 2)B) + Ri^m ß
where R^^P) and Rm+\(hP) are unknown polynomials of degree m _ 1 are unknown coefficients.
If we find RH (hp) and R+^hP), then from (5.11), (5.12) we obtain
> N,
(5.14) and A, B
m-1
£ Aa (hß)
a_0
a _ 2 (Rm
(+)1(hß)+ Rm—)1(hß)) ,
Rm-1(hß) _ 1 (Rm+-}1(hß)—Rm-—1(hß))
Unknowns Rm_)1(hP), Rj+^hP), A and B can be found from equation (5.13), using the discrete argument function Dm(hp). Then we can obtain the explicit form of the function
u(hp) and respectively we can find the optimal coefficients C[p] (p = 0,1,...,N), A and B. Thus Problem B and respectively Problem A can be solved.
But here we will not find rV-4(hp), RwMhp). Instead, using Dm(hp) and the form (5.14) of the discrete argument function u(hp), taking into account (5.8), we find the expressions for the optimal coefficients C[p] when p = 1,2,...,N - 1. We introduce the following notations
= (2m- 1)!(1 -qk)2m+1 E J (hy)2m—1 , (hy)2m—2 , (hy)2m—3 /1 + 2A k h2mqkE2m- 1(qk) yE1 ^ 2(2m- 1)! + 4(2m- 2)! + 4(2m- 3)^3 +
+ ^1 (1 + 3A) + E-V^^ ( a+r + aA + a (a - 1)(a - 2)B 12(2m-4)! \4 J a~42(2m- 1 - a)!a!\a + 1 v /v '
-AP2m-2(-hy) - BP2m-4(-hy)+ MY) - /m(-hy) j , (5.15)
_ (2m- 1)!(1 - qk)2m+1 ~ J (h(y + N))2m—1 (h(y + N))2m—2 Pk = h2mqkE2m- 1(qk) 2(2m - 1)! 4(2m - 2)!
, (h(y+N))2m—3 (1 + N _ (h(y+N))2m—4 (1 + 3A + 4(2m - 3)! V3 + ) 12(2m - 4)! V4 +3
+ m-1 (h(r+iV ))2m-1-a!(-!l)a (1 + aA + a (a _ ^ - 1)B
a=4 2(2m- 1 - a)!a! \a + 1
-AP^-2(h(y+N)) - BP2m-4(h(y + N))+ rV+— 1 (h(r + N)) - /m(h(y + N))j, (5.16)
where k = 1,2,...,m- 1, E2m-1(q) is the Euler-Frobenius polynomial of degree 2m- 1, qk are given in Theorem 5.1. Note that because of |qk| < 1 the series in the (5.15) and (5.16) are convergent. The following holds
Theorem 5.3. The coefficients C[p], p = 1,2,...,N -1 of optimal quadrature formulas of the form (1.4) in the space l2v)(0, 1) for m > 4 have the following form
(m—1 . . \
1 +E(d*qp + PkqN-J j , p = 1,2,..., N - 1, (5.17)
where dk, pk are defined by (5.15), (5.16), qk are given in Theorem 5.1.
Proof. Suppose p = 1,2,...,N - 1. Then from (5.8), using Definition 2.3, equalities (5.6), (5.14), we have
C[p]= hDm(hp) * u(hp)= h £ Dm(hp - hy)w(hy) =
Y=—^
-1 N ^
h| E Dm (hp - hy)w(hy) + EDm(hp - hy)u(hy)+ E Dm(hp - hy)w(hy) ) .
vr=—^ y=0 y=N+1
Now, adding and subtracting the expressions h E Dm(hp _hy)/m(hy) and h E Dm(hp _
y=-^ y=n+1
hy)/m(hy) to and from the last expression and taking into account Definition 2.3 we get
™ Jn/^w^u ß (2m — 1)! (1 — gfc)2m+1 f y C[ß]_ hs. Dm(hß) * fm(hß)+ £ gß h2m q, ~ , (q, ) £
k_1 ^ h2m qkE2m-1(qk) Y_1
(hy)
2m 1
2 (2m — 1)!
(hy)
2m 2
+
(hy)
2m 3
4(2m — 2)! 4(2m — 3)^3
1 (hy)2m-4
- + 2A +■ v "
1 + 3A
m1
1
£ (hy)2"-1-0 + £4 2(2m — 1 — a)!a! V a + 1
12 (2m — 4)! \4
+ aA + a (a — 1)(a — 2)B
—AP2m-2( —hy) — BP2m-4 ( — hy) + £"-1 ( —'W — fm( —hy)
+ qN-ß (2m — 1)! (1 — qk)2m+1 ^ qy + £ qk h2m qkE2m-1 (qk) y£1^
2m 1
(h(y+N)) 2 (2m — 1)!
(h(y+N))2m-2 + (h(y + n))2m-3 /1 + 2 \ (h(y + N))2m-V1 + 3A 4(2m — 2)! + 4(2m — 3)! V3 + ) 12(2m — 4)! V4 +3
"t1 (h(y + n ))2m-1-a (—1)a + ¿4 2 (2m — 1 — a )!a!
1
a +1
+ aA + a (a — 1)(a — 2)B
—AP2m-2(h(y+N)) — BP2m-4(h(y + N)) + (h(y + N)) — fm(h(y + N))
Hence taking into account the notations (5.15), (5.16) we obtain
m1
C[ß] _ h Dm(hß) * fm(hß) + £ (dkqß + PkqN ) .
k_1
(5.18)
Now using Theorems 5.1, 5.2 and equality (5.2) we get
Dm(hß) * f (hß)_ Dm (hß) *
(hß)
2m 2m 1
+ £
(—hß )2m-1-j
(2m)! j_0 2(2m — 1 — j)!(j + 1)!
(hp )2m = ^m (hp) * ^ = 1
Putting (5.19) to equation (5.18) we get (5.17). Theorem 5.3 is proved.
Furthermore we need the following lemmas in the proof of the main results. Lemma 5.1. The following identity is taken place
(5.19)
a 4qk + PkqN+'(—1)'+1 ia _ ( 1 )a+1 ° dkqk + PkqN+1(—1)i+1 A0a (5 20) _ ^ A0 _(—1) _ A0 , (5.20)
here a and N are natural numbers, dk and are defined by (5.15), (5.16), A'0a is given in Theorem 2.1, qk are given in Theorem 5.1.
+
Proof. For the purposes of convenience the left and the right hand sides of (5.20) we denote by L1 and (—1)a+1L2 respectively, i.e.
= £ dkqk + PkqN+i(—1)'+1 A,0a and L2 = £ dkqk + PkqN+1+—1)i+1 A,0a.
1 i=0 (9k -1)'+1 2 ,=0 (1 - 9k )'+1
First consider L1. Using the equality (2.3) and the identity (2.2) for L1 consequently we get
j a dkqk + PkqN+''(-1),+1 A,0 a 4qk E (q ) + PkqN+a(-1)a+1 E ( 1 L1 = £—(q—F1—A 0 = 79—Ea—1(qk)+ (qk - 1)a+1 Ea—^ qk
dkqk F , , , PkqN+a(-1)a+1 Ea—1 (qk) dkqk + PkqN+1(-1)a+1 F , ,
- E a—1 (qk-Trad--=-77.--E a—1(qk).
(qk — 1)a—1 ™ky ' (qk — 1)a+1 qfea—1 (qk — 1)a+1
(5.21)
Similarly for L2 using (2.3) and (2.2) we have
j f^qj + PkqN+1(—1)-—1 A-(a ^ E / 1 \ + PkqN+1 E (q )
L2 = =—(1—qkF11—A Ü 1 Ea —1l M + (qk—TF+1 Ea—1(qk)
dkqk E , , , PkqN—1 E f , dkqk(— 1)a—1 + PkqN—1 E f , = (1 — qk)a+1Ea—1 (qk) + (qk — 1 )a+1Ea—1 (qk) =-(qk — 1)a+1-Ea—1 (qk)
= (—1)a+1 dkqk + PkqNI1a(+^1)a+1 Ea—1(qk ). (5.22)
From (5.21) and (5.22) it is clear, that L1 = (—1)a+1L2. Lemma 5.1 is proved. We denote
Z-=£ (1 —S+' A'°p (523)
Lemma 5.2. The following identities are valid
j=Ü (j + 1)! -=1 (2m — j — -)! = j=2 j! -=1 -! (2m + 1 — j — -)! +
+ j=t+1 j -=1 (2m + 1 — j — -)
and
m_1 (—1)j+12m_W hp+1Zp _ m+1 hj'Zj—1 « (—1)-
j=( (j + 1)! ¿1 P! (2m — 1 — j — p)! = (j — 1)! -=1-! (2m + 1 — j — -)! +
2m
£ (—1)-£ fi - U ! £
j=m+2 (j — ')' i=i i! (2m + 1 — j — 0!"
where Bj are Bernoulli numbers and Zj is defined by (5.23).
The proof of Lemma 5.2 is obtained by expansion in powers of h of the left sides of the given identities.
For the coefficients of the optimal quadrature formulas of the form (1.4) the following theorem holds.
Theorem 5.4. Among all quadrature formulas of the form (1.4) with the error functional (1.5) in the space L2m)(0,1) for m > 4 there exists unique optimal formula which coefficients are determined by the following formulas
C[0]= h(1 + m_1, (5.24)
V2 k=1 1 _ /
m_1
]= h( 1 + (q/ + qN)) , / = ^^, (5.25)
k=1
C[N]= hi 1 + £ dkf_^ , (5.26)
V2 k=1 1 _ qk J a /2 ( 1 qk+qN+1^\
A=h [n _k=1dk (T_kFj, (5.27)
B = h4 (B4 1 d f qk + (_1)i+1qN+'Ai0s\ (5 28)
=h ^ _ 3! k=1 dki=0 (qk_ 1)i+1 A0J , (5.28)
where dk satisfy the following system of m _ 1 linear equations
m 1 j i+1 N+i
E<E +(_1) * A '0j = ^, j = 4^7, (5.29)
k=1 k= (qk _ 1)i+1 j+1 , ( )
m_1 j q + (_ 1 )i'+1qN+'
£ dk£ qk +( ^ A '0j = 0, j = 2,2m_ 4,2m_ 2. (5.30)
k=1 i=0 (qk 1)
Here Ba are Bernoulli numbers, A'yj is the finite difference of order i of yj, A'0j is giyen in Theorem 2.1, qk are giyen in Theorem 5.1.
Proof. We consider the first sum of equation (4.1). For this sum we have
S = fc[y] ^ _ hY l2m_1 = S Y=0C[Y] 2(2m _ 1)!
= C[0](h/ )2m_1 + f rh ^ _ hy)2m_1 Erh ^ _ hy)2m_1
= + Y=1C[Y] (2m _ 1)! Y=0C[Y] 2 (2m _ 1)! .
The last two sums of the expression S we denote by
S = f C[r](h/ _ hY)2m_1 S2 = yCh^ _ hY)2m_1
S1 = Y=1C[Y] (2m _ 1)! , S2 = Y=0C[Y] 2 (2m _ 1)!
and we calculate them separately.
By using (5.17) and formulas (2.4), (2.5) for S1 we have
ß
m1
S1 = £h 1 +£ (^kqk + Pkqfe
Y=Ü
N
Y(hß — hY)
2m 1
h
2m
(2m— 1)!
h2m (2m— 1)! 1
qk
(2m— 1)! ,ß—
k=1
ß — 1 m — 1 / ß — 1 ß — 1
£ y2«—1 + £ dkqß £ q— YY2m—1 + PkqN—ß £ qkky2"—1
Y=Ü k=1 \ Y=Ü Y=Ü
2m 1 A-Ü2m 1
2p (2m— 1)!B2m—j ßj + m-1 j=1 j! ■ (2m—j)! ß + ¿1
iß] qk dkqk '
qk
1
1 -=ü
(qk— 1)-
1 — ß 2m — 1 A-ß 2m — 1
qk-
1
1 -=Ü
(qk 1)
v , N—ß
- K Pkqk
1 2m 1
1 — qk
£ Y—
-=ü v1
qk
qk
A-Ü2m—1
qß 2m — 1 qk
1 — qk
£ 1—
-=ü v1
qk
qk
- o 2m— 1
A-ß
Taking into account that qk are roots of the Euler-Frobenius polynomial E2m—2(q) and using Theorem 2.1, we have
2m— 1 A-02m—1 1
= (qk—^ =(qk— 1)2m — 1 E2m—2(qk )= Ü,
and additionally using the identity (2.2), we get
2m 1 -
£ ( A-Ü2m—1
-=0
1 — qk
( ± _ 1)2m—
vqk 7
1 E2m—2(1/qk ) = Ü.
Keeping in mind the last two equalities, for S1 we obtain
S =
h
2m
(2m— 1)!
2m (2m— 1)!B2m j£1 j!(2m — j)!
j ß j + "j;1 — ^ + Pkg^N+-( —1)-A--ß 2m—1
k=1
(qk— 1)+
2m 1 (2m 1 ) !
Hence, using the equality Ap2m—1 = E j!((z/V—a — j)!p2m—1—jA,0j which is based on the formula (2.6), the expression for S1 we reduce to the following form
(hß)2m , (hß)2m— 1 B1 + h2m 2£— 2 *2m—j^ ß j +
S (2m)! +h (2m— 1)f
Ö j!(2m — j)!
1
2m— 1 ß 2m— 1—j m— 1 j' _d,q, + PqN+-' ( — 1)
+h2m £ .J 1 „ £ £ dkqk + 1) A-(j. (5.31)
j=Ü J!(2m— ^ J)! kt1-t( (qk —1)-+1 ( )
Now we consider S2. By using equations (4.4)-(4.6) we rewrite the expression S2 in powers of hp
= f C[r1(hp_ hY)2m_1
S2 = Y=0C[Y1 2(2m_ 1)!
= -1 (hp)2m_1-j(_1)j £j + 2E-1 (hp)2m_1-j(_1)j N C[r1(hr)j
j=0 2(2m_ 1_ j)!j! ¿0^^ + j=m 2(2m_ 1_ j)!j! ^^
= (hp)2m_1 (hp)2m_2 + (hp)2m_3 /1 + \_ (hp)2m_4 /1 + \ 2(2m_ 1)! 4(2m_ 2)! + 4(2m_ 3)^3 + J 2(2m_ 4)!3^4 + J
+1 (gj ( 7+T + jA + j( j- 1)( j_ 2)B
2m_1 (hp)2m_ 1_j(- 1)j N
+ £(&_ 1 _ j j e cmw. («2)
Substituting (5.2) and S into equation (4.1) and using (5.31), (5.32) we have
(hp)2m + C[0l(hp)2m _1 + h (hp)2m_1 B + 2m_2 B2m _ jh2m_ j(hp)j + (2m)! + C[01(2m- 1)! + h (2m_ 1)!B1 + j=1 j!(2m_j)! +
+ 2E_1 hj+1(hp)2m _1-j m-1 7 _dfeqfe + pfeqN+''(_ 1)'' A0j_
+ j=0 j!(2m_ 1_j)! k=1 '=0 (qk_ 1)'+1
(hß)2m-1 . (hß)2m-2 (hß)2m-3 /1 + 2 \ + (hß)2m-4 /1 + \_ 2 (2m- 1)! + 4(2m-2)! 4(2m-3)^3 + y + 2(2m-4)13^4 + 3J
Cj ( 7+T + jA + 7C j- 1)( j-2)B)-
- I HÊ3-177 £ C[Y](h Y) 7 - A IS
A ^ (hß )2m-2-7 (-1)7 _ (hß )2m-4 B ^ (hß )2m-4-7 (-1)7 + mv1 . (,_)t A 7^0 2■ 7! ■ (2m-2-7)! -B2(2^)! -B 2■ 7! ■ (2m-4-7)! + ¿0(hß)
_ (hß)2m + ^ (-hß)2m-1-7
(2m)! 77=0 2■ (2m- 1 -7)! ■ (7 + 1)!'
Hence equating coefficients of (h/)j for j = 0, m_ 1 we have
1 B A 1 N 21
^ = 2(2m)! + 2(2m_4)! + 2(2m_ 2)! + 2(2m_ 1 _ j)! yÇ0C[r](_hr) m ,
= (_1)j + B(_1)j + A(_1)j
2 (2m - 7)! 7! 2 (2m - 4 - 7)! 7! 2(2m - 2 - 7)! 7!
h2m-7 mf ^7 -dkqk + Pkgf+''(-1)''Ai02m-1-7 (2m- 1 - 7)!7! k=1 ¿0 (qk - 1)i+1
i-ECMMY)--j — j = Y-m"Y, (5.33)
2(2m — 1 — j)!j! Y=( j!(2m — j)!
and equating coefficients of (h!)j for j = m, 2m — 5 and for j = 2m — 4,2m — 1 we respectively get equations (5.34) and (5.35)-(5.38)
?j+4 d'q' H(p'qNi;;+71)J+' a'0j+4 = BM, j=(5.34)
'=1 i=0 (q'— 1) j +5
^12 — d'q' + p'qN+i(—1)'' ■ 2
I I (q 1v+i ) A'02 = 0, (5.35)
'=1 i'=0 (q' 1)
C[0]=h (2+1 , (5.36)
A =h2 ( E^^T) , (5.37)
B = hw 1 + 1 "ff dkqk + — 1)i+1 Aina (5 38)
B"h ^720 + 31 k?1 = to—7+ A0J . (538)
Substituting the expressions (5.36) into (4.4) when i = 0, also taking into account (5.17), we find C[N] which have the following form
C[N] = h [ 1 + "j-1 dkqf—pkqk | . (5.39)
Now, putting the values (5.33) of Aj into (4.2) and (4.3), we respectively get equation (5.40) and the system of equations (5.41)-(5.45) for unknowns dk and pk, i.e. we get the following system of equations
"f12"— 2 dkqk + PkqN+1( - 1)i+1 Ai02m—2 = "V-12"— 2 dkqN+i + M(- 1)i+1 —2 (5 40) k=1 i=0 (1- qk)i+1 A 0 k=1 = (1- qk)+ A 0 , (5.40)
110 dk^^Ai0j-1 = j j = 5" (5.41)
m—1 f dkqk + PkqN+-(—1)-+1 A-(3 = V f dkqN+- + Pkqk(—1)-+1 A(3 (_ 4?)
kL1 -h (qk—1)-+1 A Ü = kL1 -h (1—qk)-+1 A Ü, (5 2)
■■•12 dkqN+- + pkqk(—1)-+1 . 2 / N
£ £ (1 —^+1 ) A-Ü2 = Ü, (5.43)
k=1 -=Ü
m-1 dkqk + PkqN+1 = m—1 dkqN+1 + Pkqk ( 44)
kh (qk—1)2 kh (1—qk)2 , )
m—1^ dkqk + PkqN+1(—1)-+1 A-'(2m—4 = V2m—4 dkqN+- + Pkqk(—1)-+1 A-(2m—4 (5 45)
kL1 h (1—qk)-+1 A Ü kL1 -hü (1—qk)-+1 A Ü . (545)
Thus, for 2m — 2 unknowns dk and (k = 1,m — 1) we have got 2m — 2 equations (5.34)-(5.35) and (5.40)-(5.45).
Now, subtracting (5.34) from (5.41) and (5.35) from (5.43) and combining with (5.42), (5.44) we get the following m — 1 equations
m_1 j q + qN+''(_ 1)''+1 _
I (dk _Pk)E qkA'0j = 0, j = (5.46)
k=1 i=0 (qk 1)
Taking into account uniqueness of the optimal coefficients, we conclude, that the homogeneous system of linear equations (5.46) has trivial solution. This means, that
dk = ' k = 1,2'...' m — 1. (5.47)
Then, using (5.50), from (5.34), (5.35), (5.40) and (5.45) we get (5.29), (5.30), and from (5.17), (5.36)-(5.39) we obtain (5.24)-(5.28). Theorem 5.4 is proved.
5.2. The norm of the error functional of optimal quadrature formulas of the form (1.4)
The following holds
Theorem 5.5. For square of the norm of the error functional (1.5) of the optimal quadrature formula (1.4) on the space L2m)(0'1) for m > 4 the following equality is valid
2 r ». 1.2m o;„2m+1 m—1 2m _i_ „N+'7_ 1 V+1
, ir\2m
2
£ |l2" (0,1) _(— 1)m+1
B2mh2m 2h2m+1 "f1 2™ qk + qN+'(—1)'+1 Ai02 (2m)! (2m)! ¿1 (qk — 1)i+1 0
where dk are determined from the system (5.29)-(5.30), B2m are Bernoulli numbers, A'02m is given in Theorem 2.1, qk are given in Theorem 5.1.
Proof. Computing definite integrals in the expression (3.5) of ||£||2 we get
Ml2 _(-1)"
lern EcM|hß - hy|2m—1
_ß=0 Uy=0 2(2m —1)!
—fm(hß ) — AP2m-2(hß ) — BPm-4(hß ) j — fm(hß ) — APm-2(hß ) — BP2m-4(hß )
2A 2B + A2 2AB B2 1
+ ^-7t7 + ^-^-+ ^-zttt +
(2m — 1)! (2m — 3)! (2m — 5)! (2m — 7)! (2m + 1)!j ' where /m(hp) is defined by formula (5.2). As is obvious from here according to (4.1) the
m—1
expression into curly brackets is equal to the polynomial — E (hp)a. Then ||£||2 has
the form
_ (—1)
a_0
' N / m—1
£e[ß ] — £ Aj (hß)j—fm(hß)
ß _0 V j_0
NN —A £ e[ß]P2m-2(hß) — B £ e[ß]P2m-4(hß) ß _0 ß _0
2A 2B + A2 2AB B2 1
+ ^-+ ^-^--+
(2m — 1)! (2m — 3)! (2m — 5)! (2m — 7)! (2m + 1)!
Hence using (4.2) and (4.3) we get
m-1 N N
M2 = (-1)" -E Aj E C[p](hp)j - EC[p]/m(hp) . j=0 p=0 p =0 m- 1 m- 1
+AE jAj + BE j( j - 1)( j - 2)Aj
j=2 j=4
A B 1
+ ^-+ ^-^7 +
(2m - 1)! (2m - 3)! (2m + 1)! From here, after some simplifications, using (5.33), (5.2) and (4.4)-(4.6), we have
2
= (—1)m
m—1 (_ 1) j+1 N
h ( j+¿¿—1—j)! Y=Ü C[Y](hY)2m"1"j
V h2m—j m-1 d 2m^-^k + qN+-(—1)-+1 A-Ü2m"l"j
jèû (j + 1)!(2m — 1 — j)! kE1 dk -hü (qk — 1)-+1 ü
V (—1)j B(—1) j
j=ü (j + 1) ! (2m — j) ! j=( (j + 1) ! (2m — 4 — j) ! m—3 A(— 1)j t m^1 B2m—j
(5.48)
E (j + 1)! (2m - 2 - j)! + £ (j + 1)! (2m - j)!
1 N A B 1
-(Z")! pEC[p](hp) " + (2m- 1)! + (2m- 3)! + (2m + 1)!
When a > m — 1, using (5.17) and formulas (2.4)-(2.6), we get
£ «-a+i + E jl^ha+1—j
+h a+1£ 4qk + PkqN+1(-1)i+1 Ai0 a +h k=1 fe <1 - qk)i+1 A 0
_ « a!hj+1 j dkqN+'' + Pkqk(-1)'+1 A0j (5 49)
j=1 j!(a - j)! k=1 fe (1 - qk)i+1 A 0 . <5.49)
Using Lemmas 5.1, 5.2 and taking into account (2.1), (2.3) and (5.49), after some simplifications, from (5.48) we have
I2 = (—1)m
r , „ B2mh2m , h2m+1Z2m
i,2m+1 m—1 2m q + qN+1 (_ 1 ) -+1 h_ V j V qk + qk ( 1) A-n2m
— (2m)! kE dk-hü (1—qk)-+1 Aü
^B^'™ (—1)- « hj'Zj'"l ^ (—1)-
£ j! £-!('9m_i_ 1 -_ ^ ! + £
jh2 j! -hü -!(2m + 1 — - — j)! jhi (j — 1)!-hü-!(2m + 1 — - — j)!
m—4 B(—1)j + £7 ( )
m-2 m -2
A(—1)j
j_0 j!(2m — 3 — j)! j_0 j!(2m — 1 — j)!
(5.50)
where
2m
K1 _ £
hjZ
j-1
2m+1-j
2m
I
hjZ
j-1
m+1 (j — 1)! i_1 i!(2m + 1 —i — j)! j_t+1 (j — 1)!(2" + 1 — j)!
and
2m
h jZ
2m+1-j
(—1)'
£ h Zj-1 £ (—1) _ hjZj-1 (1 — 1)2m+1-j _ 0
j_T+1 (j — 1)! ¿0 i!(2m + 1 —' — j)! j_t+1 (j — 1)! (2m + 1 — j)! ,
2m-1 j ■ 2m-j
K2 _ — £ j £
(—1)'
j_m+1
j!
'=0
i!(2m + 1 — i — j)!
2m-1
+ E
j_m+1
hjBj
j!(2m + 1 — j)!
j_m+1 j! \ i_0
(—1)'
1
i!(2m + 1 — i — j)! (2m + 1 — j)!
= _ 2m—1 hj (—1)j — 1 \ = 0
j=m+1 j! \(2m + 1 — j)!J
Since K1 = K2 = 0 then from (5.50) we get the following
_ (-1)m
B2mh2m + h2"^m-1 qk + qN+'(—1)i+1 Ai02m
(2m)! + (2m)! ¿1 ^¿0 (qk — 1)i+1 0
h2m+1 2" qk + qN+1(— 1)i+l 2m'
(2m)! £ dki_0
(1 - qk)
i+1
Ai02
Hance using Lemma 5.1 we have the statement of the theorem.
Theorem 5.5 is proved.
In particular, from Theorems 5.4 and 5.5 for the cases m = 4' 5 we get the following corollary which confirm optimality of the classical Euler-Maclaurin quadrature formula in the space L2m)(0' 1) when m = 4'5.
Corollary In the space 4m)(0' 1), m = 4' 5 among all quadrature formulas of the form (1.4) with the error functional (1.5) there exists unique optimal formula whose coefficients are determined by the following formulas
c[p] = / 2' p = 0'N'
h, ß _ 1, N — 1,
h2 h4
A = —' B =--.
12' 720
Furthermore for square of the norm of the error functional the following is valid
£|L2m)*(0,1) _(— 1)m+1
h2mB2m (2m)!
, m _ 4,5.
2
6. Numerical results
It should be noted that constructed optimal quadrature formulas of the form (1.4) with the error functional (1.5), the coefficients which are determined by formulas (5.24)-(5.28) are exact for monomials xa, a = 0,...,m — 1. This statement is also checked numerically.
Clearly, the optimal coefficients (5.24)-(5.28) depend only on the roots qk (where |qk| < 1) of the Euler-Frobenius polynomial E2m—2(q), which is defined by formula (2.1). Therefore to obtain numerical values of the coefficients C[p] (p = 0,1,...,N), A, B it is sufficient to calculate the roots of the Euler-Frobenius polynomial E2m—2(q), whose absolute values are less than 1.
It should be noted that for m = 2,3,...,7 the Euler-Frobenius polynomials E2m—2(q) and their roots are given in [23]. Below we consider some particular cases.
We consider the case m =6. In this case we obtain the optimal quadrature formulas of the form (1.4) in the space l26)(0,1). Here we need the roots qk of the Euler-Frobenius polynomial E10(q), in which |qk| < 1. From (2.1) we get
E10<q) = 1+2036q+152637q2+2203488q3+9738114q4
+ 15724248q5 + 9738114q6 + 2203488q7 + 152637q8 + 2036q9 + q10.
and the roots qk of this polynomial, whose absolute values less than 1 are
q1 = -0.00051055753444650205713591952840749392417989252,
qz = -0.01666962736623465609658583608981508371547272055,
q3 = -0.08975959979371330994414267655614154254756196601, (6.1)
q4 = -0.27218034929478588568629528025828776815123525956,
q5 = -0.66126606890073470691013126292248166961816286716.
Now we give table of values of the coefficients of the optimal quadrature formula of the form (1.4) for the case m = 6 and N = 10. For m = 6 and N = 10 solving the system (5.29)-(5.30) and using (6.1) from (5.24)-(5.28) we get the following optimal quadrature formula of the form (1.4) in the space l24)(0, 1)
1 10
/ 9(x)dx = EC[p(0.1p)+ A(p'(0) - p'(1)) + B(pw(0) - pw(1)). (6.2)
0 p=0
The coefficients of the optimal formula (6.2) are presented in Table 1.
Using Theorem 5.5 we get the following estimation of the formula (6.2)
|(£,9)| < |L26)(0,1)|-0.24989492918 x 10—10. (6.3)
As it was noted in introduction the error of the optimal quadrature formula (1.4) is estimated by the norm ||£|| which is given in Theorem 5.5., i.e.
|(£,9)l<|9l4m)(0,1)|| PlL/M,1)||.
Numerical results of ||£|| for the cases m = 4,5,6,7 and N = 10, 50, 100 are given in Table 2.
Table 1
The coefficients of the optimal quadrature formula (6.3).
C[0] = 0.0497668123352824127234689329023
C[1] = 0.1004149880826437216520334950106
C[2] = 0.0996945183368890756257025927425
C[3] = 0.1002114315494163091062049537068
C[4] = 0.0998395174169098402529374943719
C[5] = 0.1001454645577172812793050625314
C[6] = 0.0998395174169098402529374943719
C[7] = 0.1002114315494163091062049537068
C[8] = 0.0996945183368890756257025927425
C[9] = 0.1004149880826437216520334950106
C[10] = 0.0497668123352824127234689329023
A = 0.0008207935968426995516167321994353
B = -0.00000012495765008009292800730370565
Table 2
Numerical results of ||£|| for the cases m = 4,5,6,7
and N = 10, 50, 100.
N = 10 N = 50 N = 100
m = 4 0.909241x 10-7 0.145478 x 10-9 0.909241 x 10-11
m = 5 0.144487 x 10-8 0.462361x 10-12 0.144487 x 10-13
m=6 0.249894 x 10-10 0.149789 x 10-14 0.231969 x 10-16
m=7 0.610131 x 10-12 0.548027 x 10-17 0.398205 x 10-19
These numerical results show that the error of the optimal quadrature formula decreases as m and N increase.
Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.
References
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(2)
the class W£2)(M;ON) with equally spaced nodes", Optimization methods and operations research, applied mathematics, 191 (1976), 174-177 (in Russian).
Mevers L. F., Sard A., "Best approximate integration formulas", J. Math. Phvsics, 29 (1950), 118-123.
Nikol'skii S.M., "To question about estimation of approximation by quadrature formulas", Uspekhi Matem. Nauk, 5:2(36) (1950), 165-177 (in Russian).
Nikol'skii S.M., Quadrature Formulas, Nauka, Moscow, 1988 (in Russian).
Sard A., "Best approximate integration formulas; best approximation formulas", Amer. J. Math., 71 (1949), 80-91.
Sard A., Linear approximation, AMS, 1963.
Schoenberg I.J., "On monosplines of least deviation and best quadrature formulae", J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 144-170.
Schoenberg I. J., "On monosplines of least square deviation and best quadrature formulae II", SI AM J. Numer. Anal., 3:2 (1966), 321-328.
Schoenberg I. J. , Silliman S. D., "On semicardinal quadrature formulae", Math. Comp., 126 (1974), 483-497.
Shadimetov Kh.M., Optimal formulas of approximate integration for differentiable functions, Candidate dissertation, Novosibirsk, 1983, 140 pp.
Shadimetov Kh.M., "Optimal quadrature formulas in Lf(Q) and L^R1)", Dokl. Akad. Nauk UzSSR, 1983, №3, 5-8 (in Russian).
Shadimetov Kh.M., "The discrete analogue of the differential operator d2m/dx2m and its construction", Questions of Computations and Applied Mathematics, 1985, 22-35.
[31] Shadimetov Kh. M., "Optimal Lattice Quadrature and Cubature Formulas", Dokladv Mathematics, 63:1 (2001), 92-94.
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[33]
in 4m)(0,1) space", J. Comput. Appl. Math., 235 (2011), 1114-1128.
[34] Shadimetov Kh.M., Havotov A.R., "Optimal quadrature formulas in the sense of Sard in W2(m,m-1) space", Calcolo, 51 (2014), 211-243.
[35] Shadimetov Kh. M., Havotov A. R., Azamov S.S., "Optimal quadrature formula in ^(P) space", Applied Numerical Mathematics, 62 (2012), 1893-1909.
[36] Shadimetov Kh.M., Havotov A.R., Nuraliev F. A., "On an optimal quadrature formula in
Sobolev space Llm)(Q, 1)", Journal of Computational and Applied Mathematics, 243 (2013), 2
[37] Sobolev S.L., Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974 (in Russian).
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[39] Sobolev S.L., Vaskevich V.L., The Theory of Cubature Formulas, Kluwer Academic Publishers Group, Dordrecht, 1997.
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[41] Zhamalov Z.Zh., Shadimetov Kh.M., "About optimal quadrature formulas", Dokl. Akademii Nauk UzSSR, 7 (1980), 3-5 (in Russian).
[42]
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References (GOST)
[1] Akhmedov D.M., Hayotov A.R., Shadimetov Kh.M. Optimal quadrature formulas with derivatives for Cauchy type singular integrals // Applied Mathematics and Computation. vol. 317. 2018. pp. 150-159.
[2] Babuska I. Optimal quadrature formulas // Dokladi Akad. Nauk SSSR. vol. 149. 1963. pp. 227-229 (in Russian).
[3] Blaga P., Coman Gh. Some problems on optimal quadrature // Stud. Univ. Babes-Bolyai Math. 2007. vol. 52. no. 4. pp. 21-44.
[4] Bojanov B. Optimal quadrature formulas // Uspekhi Mat. Nauk. 2005. vol. 60. no. 6(366). pp. 33-52 (in Russian).
[5] Boltaev N.D., Hayotov A. R., MilovanoviC G.V., Shadimetov Kh.M. Optimal quadrature formulas for numerical evaluation of Fourier coefficients in W2(m,m-1) // Journal of Applied Analysis and Computation. 2017. vol. 7. no. 4. pp. 1233-1266.
[6] Boltaev N.D., Hayotov A. R., Shadimetov Kh.M. Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space LÎ,m) // Numerical Algorithms. 2017. vol. 74. pp. 307-336.
[7] Catina§ T., Coman Gh. Optimal quadrature formulas based on the 0-function method // Stud. Univ. Babe§-Bolyai Math. 2006. vol. 51. no. 1. pp. 49-64.
[8] Chakhkiev M. A. Linear differential operators with real spectrum, and optimal quadrature formulas // Izv. Akad. Nauk SSSR Ser. Mat. 1984. vol. 48. no. 5. pp. 1078-1108 (in Russian).
[9] Coman Gh. Quadrature formulas of Sard type // Studia Univ. Babes-Bolyai Ser. Math.-Mech. 1972. vol. 17. no. 2. pp. 73-77.
[10] Coman Gh. Monosplines and optimal quadrature formulae in Lp // Rend. Mat. 1972. vol. 6. no. 5. pp. 567-577.
[11] Gelfond A.O. Calculus of Finite Differences. Moscow: Nauka, 1967 (in Russian).
[12] Ghizzetti A., Ossicini A. Quadrature Formulae. Berlin: Akademie Verlag, 1970.
[13] Hamming R. W. Numerical methods for Scientists and Engineers. USA: McGraw Bill Book Company, Inc., 1962.
[14] Hayotov A. R., Jeon S., Lee C.-O. On an optimal quadrature formula for approximation of Fourier integrals in the space L^ // Journal of Computational and Applied Mathematics. 2020. vol. 372. 112713.
[15] Hayotov A. R., Milovanovic G.V., Shadimetov Kh.M. On an optimal quadrature formula in the sense of Sard // Numerical Algorithms. 2011. vol. 57. no. 4. pp. 487-510.
[16] Hayotov A. R., Nuraliev F. A., Shadimetov Kh.M. Optimal Quadrature Formulas with Derivative in the Space z2m)(0,1) // American Journal of Numerical Analysis. 2014. vol. 2. no. 4. pp. 115-127.
[17] Kohler P. On the weights of Sard's quadrature formulas // Calcolo. 1988. vol. 25. 1988. pp. 169-186.
[18] Lanzara F. On optimal quadrature formulae // J. Ineq. Appl. 2000. vol. 5. pp. 201-225.
[19] Maljukov A.A., Orlov I.I. Construction of coefficients of the best quadrature formula for the class WZ(22)(M; ON) with equally spaced nodes // Optimization methods and operations research, applied mathematics. 1976. vol. 191. pp. 174-177 (in Russian).
[20] Meyers L. F., Sard A. Best approximate integration formulas // J. Math. Physics. 1950. vol. 29. pp. 118-123.
[21] Nikol'skii S.M. To question about estimation of approximation by quadrature formulas // Uspekhi Matem. Nauk. 1950. vol. 5. no. 2(36). pp. 165-177 (in Russian).
[22] Nikol'skii S.M. Quadrature Formulas. Moscow: Nauka, 1988 (in Russian).
[23] Sard A. Best approximate integration formulas; best approximation formulas // Amer. J. Math. 1949. vol. 71. pp. 80-91.
[24] Sard A. Linear approximation. AMS, 1963.
[25] Schoenberg I.J. On monosplines of least deviation and best quadrature formulae // J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1965. vol. 2. pp. 144-170.
[26] Schoenberg I. J. On monosplines of least square deviation and best quadrature formulae II // SIAM J. Numer. Anal. 1966. vol. 3. no. 2. pp. 321-328.
[27] Schoenberg I. J. , Silliman S. D. On semicardinal quadrature formulae // Math. Comp. 1974. vol. 126. pp. 483-497.
[28] Shadimetov Kh. M. Optimal formulas of approximate integration for differentiable functions. Candidate dissertation. Novosibirsk, 1983. 140 c.
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[30] Shadimetov Kh.M. The discrete analogue of the differential operator d2m/dx2m and its construction // Questions of Computations and Applied Mathematics. 1985. pp. 22-35.
[31] Shadimetov Kh. M. Optimal Lattice Quadrature and Cubature Formulas // Doklady Mathematics. 2001. vol. 63. no. 1. pp. 92-94.
[32] Shadimetov Kh. M. Construction of weight optimal quadrature formulas in the space L2m)(0,N) // Siberian J. Comput. Math. 2002. vol. 5. no. 3. pp. 275-293 (in Russian).
[33] Shadimetov Kh.M., Hayotov A. R. Optimal quadrature formulas with positive coefficients in L2m)(0,1) space // J. Comput. Appl. Math. 2011. vol. 235. pp. 1114-1128.
[34] Shadimetov Kh.M., Hayotov A. R. Optimal quadrature formulas in the sense of Sard in W2(m,m-1) space // Calcolo. 2014. vol. 51. pp. 211-243.
[35] Shadimetov Kh.M., Hayotov A. R., Azamov S.S. Optimal quadrature formula in K2(P2) space // Applied Numerical Mathematics. 2012. vol. 62. pp. 1893-1909.
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[39] Sobolev S. L., Vaskevich V.L. The Theory of Cubature Formulas. Dordrecht: Kluwer Academic Publishers Group, 1997.
[40] Zagirova F. Ya. On construction of optimal quadrature formulas with equal spaced nodes. Preprint No. 25. Novosibirsk: Institute of Mathematics SD of AS of USSR, 1982. 28 c (in Russian).
[41] Zhamalov Z. Zh., Shadimetov Kh. M. About optimal quadrature formulas // Dokl. Akademii Nauk UzSSR. 1980. vol. 7. pp. 3-5 (in Russian).
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Вестник КРАУНЦ. Физ.-Мат. Науки. 2020. Т. 32. №. 3. С. 75-101. ISSN 2079-6641
УДК 519.644 Научная статья
Оптимальные формулы типа Эйлера-Маклорена для численного интегрирования в пространстве Соболева
А. Р. Хаётов1, Ф.А. Нуралиев1, Р. И. Паровик2, Х.М. Шадиметов1
1 Институт Математики имени В. И. Романовского Академии наук Узбекистана, г. Ташкент, ул. Мирзо Улугбека 85, 100170, Республика Узбекистан
2 Камчатский государственный университет имени Витуса Беринга, г. Петропавловск-Камчатский, ул. Пограничная 4, 683032, Россия
E-mail: hayotov@mail.ru, nuraliyevf@mail.ru, romanparovik@gmail.com, kholmatshadimetov@mai.ru
В настоящей статье рассматривается задача построения оптимальных квадратурных формул в смысле Сарда в пространстве L2m)(0,1). Здесь квадратурная сумма состоить из значений подынтегральной функции в узлах и значений первой и третьей производных подынтегральной функции на концах интервала интегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального числа N > m — 3 и для любого m > 4, используя метод С.Л. Соболева который основывается на дискретный аналог дифференциального оператора d2m/dx2m. В частности, при m = 4 и m = 5 получен оптимальность классической формулы Эйлера-Маклорена. Начиная с m = 6 получены новые оптимальные квадратурные формулы. В конце работы приведаны некоторые численные результаты.
Ключевые слова: оптимальные квадратурные формулы, функционал погрешности, экстремальная функция, пространство Соболева, оптимальные коэффициенты.
DOI: 10.26117/2079-6641-2020-32-3-75-101
Поступила в редакцию: 01.09.2020 В окончательном варианте: 10.10.2020
Для цитирования. Hayotov A. R., Nuraliev F. A., Parovik R. I., Shadimetov Kh.M. Euler-Maclaurin type optimal formulas for numerical integration in Sobolev space // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 75-101. DOI: 10.26117/2079-6641-2020-32-3-75101
Конкурирующие интересы. Авторы заявляют, что конфликтов интересов в отношении авторства и публикации нет.
Авторский вклад и ответсвенность. Все авторы участвовали в написании статьи и полностью несут ответственность за предоставление окончательной версии статьи в печать. Окончательная версия рукописи была одобрена всеми авторами.
Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Хаётов А. Р. и др., 2020
Финансирование. Исследование выполнялось без финансирования
