CC BY
10
УДК 519.644
Optimal Formulas of Numerical Integration with Derivatives in Sobolev Space
Kholmat M. Shadimetov Farhod A. Nuraliev*
Institute of Mathematics Uzbekistan Academy of Sciences Mirzo Ulugbek, 81, Tashkent, 100041 Uzbekistan
Received 05.10.2017, received in revised form 27.03.2018, accepted 09.07.2018
The problem of construction of optimal quadrature formulas in the sense of Sard in the space L^m)(0,1) is considered in the paper . The quadrature sum consists of values of the integrand at internal nodes and values of the first, third and fifth 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 and for any m ^ 6 using Sobolev method. It is based on discrete analogue of the differential operator d2m/dx2m. In particular, for m = 6, 7 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m = 8 new optimal quadrature formulas are obtained.
Keywords: optimal quadrature formula, error functional, extremal function, Sobolev space, optimal coefficients.
DOI: 10.17516/1997-1397-2018-11-6-764-775.
1. Introduction. Statement of the problem
We consider the following general quadrature formula
1 N 3 ( .
/ v(x)dx = Y1 C+ M v(2n-1\0) - ¥>(2n-1)(l) (1.1)
J° f=0 n=1 ^ '
with the error functional
N 3 ( \
l(x) = £[0,1] (x)C W(x - + Anl s(2n-1)(x) - S(2n-1)(x - l) (1.2)
f=0 n=1 ^ '
in the space 4m)(0,1) for m ^ 6. Here C[/], / = 0,N and An, n = 1,2,3 are the coefficients of formula (1.1), h = n, N is a natural number, £[0j1](x) is the characteristic function on the interval [0,1], S(x) is the Dirac delta-function and ^ is an element of the Sobolev space
4m)(0,1) := W : [0,1] ^ R|<p(m_1) is abs. cont. and ^(m) G L2(0,1)} equipped with the norm
{ ri } 1/2
|M4m)(0,1)11 = |Jo (f(m)(x))2dxj (1.3)
* nuraliyevf@mail.ru © Siberian Federal University. All rights reserved
space L2m) (0,1), i.e,. by
1
and/ (p(m)( x))2dx < to.
0
Equality (3) is a semi-norm and ||y>|| =0 if and only if ^(x) = Pm-1(x) where Pm-1(x) is a polynomial of degree m — 1. The difference
1 N 3 ( \ (e,v)= v(x)dx — Y; cimm — ^ aJ ^(2n-1)(0) — ^(2n-1)(i) (1.4)
f=0 n=1 ^ '
is called the error of quadrature formula (1.1). By the Cauchy-Schwarz inequality
\(1,V)\ < |ML2m)(0, i)U-¥\L2rr (0,1)||
error (1.4) of formula (1.1) is estimated by the norm of error functional (1.2) in the conjugate
l\Lm (0,1) = sup \(£,^)\.
||Vi4m)(o,1)||=1
Thus estimation of error (1.4) of quadrature formula (1.1) on functions in the space L2m)(0,1) is reduced to finding the norm of the error functional I in the conjugate space L2m) (0,1).
Obviously the norm of the error functional I depends on coefficients and nodes of quadrature formula (1.1). The problem of finding the minimum of the norm of the error functional I that depends on coefficients and nodes is called Nikol'skii problem. The resulting formula is called optimal quadrature formula in the sense of Nikol'skii. This problem was first considered by S. M. Nikol'skii [11], and by many authors (see e.g. [1-4,12,29] and references therein). Minimization of the norm of the error functional I in terms of coefficients when the nodes are fixed is called the Sard problem. The resulting formula is called the optimal quadrature formula in the sense of Sard. This problem was first investigated by A. Sard [13].
The results of this paper are related to the Sard problem. So here we discuss some of the previous results on optimal quadrature formulas in the sense of Sard which are closely related to our results.
There are several methods of construction of optimal quadrature formulas in the sense of Sard such as spline method, ^-function method (see e.g. [1,15]) and the Sobolev method which is based on construction of discrete analogue of a linear differential operator (see e.g. [25,26]). The Sard problem was investigated in various spaces by many authors (see, for example, [1,3,710,14-16,18,20,21,24-28] and references therein).
The main aim of this paper is to construct optimal quadrature formulas (1.1) in the sense of Sard in the space L2m)(0,1).
We use the Sobolev method [25,26] which is based on the discrete analogue of the differential operator d2m/dx2m.
In order to define error functional (1.4) in the space L2m)(0,1) it should satisfy the following conditions (see [24])
(I, xa) = 0, a = 0, 1, 2,...,m — 1. (1.5)
Hence it is clear that for existence of quadrature formulas (1.1) the condition N > m — 4 should be satisfied.
As it was noted above the error of formula (1.1) is estimated by the norm HtH of error functional (1.2). Furthermore the norm of error functional (2) depends on coefficients C[ft], An.
We minimize the norm of error functional (2) in terms of coefficients C\f3\ and An, i.e., we find
i|4m)* = inf £l4m)* . (1.6)
Coefficients C\0\ and An which satisfy (1.6) are called the optimal coefficients and they are denoted by C\fi\ and An. The corresponding quadrature formula is called the optimal quadrature, formula in the sense of Sard. For the sake of convenience the optimal coefficients C\^\ and An will be denoted as C\0\ and An.
Thus to construct optimal quadrature formulas (1.1) in the sense of Sard we need to solve consequently the following problems.
Problem 1. Find the norm of error functional (1.2) of quadrature formula (1.1) in the space
Lm)*(o, 1).
Problem 2. Find coefficients C\fi\ and An which satisfy relation (1.6).
It should be noted that many mathematicians have contributed to the development of the theory of cubature and quadrature formulas among them S.L.Sobolev, I. P. Mysovskikh, V. I.Lebedev, M. D. Ramazanov, G. N. Salikhov, V. I. Polovinkin, M.V. Noskov, V. L. Vaskevich and others.
The paper is organized as follows: in Section 2 we give representation of the norm of error functional (1.2) and we obtain the system of linear equations for the coefficients of the optimal quadrature formulas in the space L^fi, 1). Moreover, the existence and uniqueness of the solution of this system are discussed; in Section 3, using the discrete analogue of the operator d2m/dx2m, explicit formulas for coefficients of optimal quadrature formulas (1.1) are found, and the norm of error functional (1.2) is calculated.
2. The system of equations for optimal coefficients
To solve Problem 1 we get the following expression for the norm of error functional (1.2) [23]
I2 = ("1)r
[É ±C ßc yi«? - 2 h™ i; ,,+
3
. (*2m-2n + q - x) —) dx_ "Jo 2(2m - 2»)!
3 N / / -, n\2m — 2n . / -x j r,\2m — 2n \
-2hAnhcpi«*" ^-»T '+
n=l ß=0 v '
+ ^^ ^^ An Ak + 1
+ 2.^1 (2m _ 2» - 2k + 1)! +
(2m - 2n - 2k + 1)! (2m + 1)!
n=1k=1 V ' V '
(2.1)
Further, in order to solve Problem 2 we apply the Lagrange method, i.e., to find the minimum of ||l||2 under conditions (1.5) we consider the following function
m— 1
||l||2 - 2 • (-1)mE Aa(l(x),xa)
a=0
where Xa are unknown multipliers. The function ^ is the multidimensional function with respect to coefficients C\0\, An and Xa. Setting partial derivatives of ^ with respect to coefficients
C\p\, An to zero and taking into account conditions (1.5), we obtain the following system of linear equations
^ C\ \\hp - hY\2m-1 A . ((hp)2m-2n + (1 - hp)2m-2n) +
C [Y] 2(2m - 1)! An 2(2m - 2n)! +
m — 1
+ \a(hß)a = fm(hß), ß = 0, N, (2.2)
C[ßi((hß) +(1 ~ hß) ) v^_An_) +
[ß] 2(2m - 2k)! n=!2(2m " 2n " 2k + 1)!
m— 1 1
+ £ a(a - l)...(a - 2k + 2) • Aa = (2m - 2k + 1)!, k = 1 2, 3, (2.3)
a=2k
N
where
WW* = ^, a = 0,1, (2.4)
3=0 +
N 1
[3\(h3)°' - aA1 = a—1, a = 2, 3, (2.5)
3=0 a +
N1
VC[l3\(h/3)a - aA1 - a(a - 1)(a - 2)A2 = -, a = 4, 5, (2.6)
a +1
3=0
N
J2 C[3\(h3)a - aA1 - a(a - 1)(a - 2)A2-
3=0
- a(a - 1)(a - 2)(a - 3)(a - 4)A3 = , a = 6, m - 1, (2.7)
a +1
fmW)= f1 x = (Ml + 2g\ (-h^)2m-1-J , . (2.8)
JmK Jo 2(2m - 1)! (2m)! 2(2m - 1 - j)! • (j + 1)! V ;
System (2.2)-(2.7) is called the discrete system of Wiener-Hopf type for the optimal coefficients [24,26]. Coefficients C[¡\, 3 = 0,N, An and Xa, a = 0, m - 1 are unknowns of system (12)-(17). System (2.2)-(2.7) has unique solution and this solution gives the minimum of ||^||2. Here we omit the proof of the existence and uniqueness of the solution of this system. The proof of the existence and uniqueness of the solution of this system coincides with the proof of the existence
and uniqueness of the solution of discrete Wiener-Hopf type system for the optimal coefficients in
(m) 1 N
the space ¿2 (0,1) for quadrature formulas of the form J f (x)dx =5^ C[¡\f [¡\ (see [24,26]).
0 3=0
It should be noted that the uniqueness of the optimal quadrature formulas in the Sard sense was also discussed [9].
3. The coefficients and the norm of the error functional of the optimal quadrature formulas
In this section we solve system (2.2)-(2.7) and find the explicit formulas for coefficients C[¡\. We use functions of discrete variable and operations on them (see [24,26]).
First we consider some statements that are used in the proofs of main results.
The following theorem is devoted to representation of the Euler-Frobenius polynomial.
Theorem 3.1 (Lemma 3 of [17]). Polynomial
k+1 Aink + 1
Ok (x) = (x - ^ g f-y
i
is the Euler-Frobenius polynomial of degree k, i.e., Qk(x) = Ek(x) where Ai0k = ^ (- 1)i-lC\lk.
1=1
We need the discrete analogue Dm(hp) of the operator d2m/dx2m which satisfies the following equation
hDm(hp) * Gm(hp)= S(hp), (3.1)
\hp\2m-1
where Gm(hp) = ——-—, 5(hfi) is equal to 0 when p = 0, and it is equal to 1 when p = 0,
2(2m - 1)!
i.e., S(hp) is the discrete delta-function.
It should be noted that the operator Dm(hp) was firstly introduced and investigated by
S. L. Sobolev [24].
The discrete analogue Dm(hp) of the differential operator d2m/dx2m which satisfies equation (3.1) was constructed and the following theorem was proved [19].
Theorem 3.2. The discrete analogue of the differential operator d2m/dx2m has the form
m-1 (1 — qk )2m+1qk^'
k=1 QkE2m-1(qk) for ^ ^ 2'
»-m=
1
(1 - qu)2m+1
!+ g Efkr for 131 = '■ (32)
m-1 n „ \2m+1
-22--1 + E q1-^ for 3 = 0.
k=1
where E2m-1(q) is the Euler-Frobenius polynomial of degree 2m - 1, qk are roots of the Euler-Frobenius polynomial E2m-2(q), \qk | < 1, h is a small positive parameter.
Several properties of function Dm(h3) were proved [19]. Here we consider the following property of function Dm(hfi).
Theorem 3.3. Function Dm(h3) and monomials (hfi)k are related to each other as follows
„„„, n\k i 0 when 0 < k < 2m - 1,
D-mmk H (2m)! when k = 2m, (3.3)
3=-œ
œ | 0 when 2m + 1 ^ k ^ 4m - 1,
E D-(h3)(h3)k = I h2-(mB2m when k = 4m. 3=-œ \ (2m)!
Then taking into account (3.1) and using (2.2), Theorems 3.2, 3.3, we have the following theorem for optimal coefficients C[¡\.
Theorem 3.4. Coefficients C [¡3], ¡3 = 1, 2,... ,N — 1 of the optimal quadrature formulas (1) in the space L^iO, 1), m ^ 6, have the following form
(m— 1 \
1+ {dkql + pk qN, 3 = 1, 2, . . . ,N - 1, (3.4)
where dk,pk are unknowns, qk are given in Theorem 3.2. Theorem 3.4 is proved similarly as Theorem 5.3 [23].
Furthermore we need the following lemmas for the proof of the main results. Lemma 1. The following relation is taken place [23]
' dk qk + Pk qk1)m Ai0„ = ( —1)a+1 p dkqj + PkqN +1(-1)i+1 Ai0a (35)
0
here a and N are natural numbers, dk and pk are constants, Ai0a is given in Theorem 3.1, qk are given in Theorem 3.2.
Lemma 2. The following relations are valid
m- (-i)j-12m—-j B2m-j-ih2m-j-i _ m Bh v-2 (-i,i
j= (j - 1)! i! (2m - j - i)! _ j= j! i! (2m - 1 - j - i)\
2m-2 D uj 2m-2-j , ,
+ y- Bjhj y- _ (-1) 2-^t j ! Z^ i
j ! i! (2m — 1 — j — i)!
j=m+1 i=0 v J '
and
m- (-i)j-1j _ m+i hj Zj-i m- (-i)i
j=1 (j - 1)! p! (2m - 1 - j -p)! _ (j - 1)! ^ i! (2m - 1 - j - i)!"
2m-1 2m-1-j , -
+ y^ hß Zj-i y^ _ (-1)
+ (j- 1)! i
(j - 1)! ^ i! (2m - 1 - j - i)!
j=m+2 i=0
^ _ ^ £ äkqk + pk)+-1) ^ (3.6)
where
dk q^+l + Pk qk (-1)
k=H=0 (1 - qk )
The proof of Lemma 2 is obtained with the use of expansion of the left hand sides of given above relations in powers of h.
Finally we give the main results of the paper.
The following theorem is valid for the coefficients of the optimal quadrature formulas (1.1).
Theorem 3.5. Among quadrature formulas (1.1) with error functional (1.2) in the space i!,m)(0,1), m ^ 6 there exists the unique optimal formula which coefficients
-1 N
( 1 m-1 :,N : \
C[0] _ h (2+ g dk 'M^j, (3.7)
(m-1 \
1+ kC dk (iß + -ß)\ , ß _1,N - 1, (3.8)
( 1 m— 1 q \ C N = H S "k ^J •
(3.9)
;,2nR h2n m —12n—1 NiU+1
^ = ^n J2"k +qk (-+i Aî°2"—i1, « = i, 2,3, (3.10)
(2n)! (2n - 1)!
k = 1 ¿=1
(qk - 1)
where dk satisfy the following system of m — 1 linear equations
m — 1 j
J2J2"k qk + qN^(-5+1 = j, j = 6, m - 1
k=1 ¿=0
(qk - 1)¿+1
N1V+1
j + 1
m — 1 2m — 2j N +./ lV+1
y y "k qk + qk (.+1) ^¿o2m—2j =0, j = 1, 2, 3, Z. Z. k (qk - 1).+1 , j , , ,
k=1 ¿=0 m—1 2j
qk + qk ( + A.02j =0, j = 1, 2. ^ ^ k (qk - 1).+1 ' j '
(3.11)
(3.12)
(3.13)
k=1 ¿=0
Here Ba are Bernoulli numbers, Aijj is the finite difference of of order i, AiQj is given in Theorem 3.1, qk are given in Theorem 3.2.
For square of the norm of error functional (1.2) of optimal quadrature formulas (1.1) we give the following result without proof. The proof is given in Theorem 5.5 [23].
Theorem 3.6. For square of the norm of error functional (1.2) of the optimal quadrature formula (1.1) in the space L2n^(Q, 1), m ^ 6 the following relation holds
ê\L"]*(0,1)
(-1)
m+1
B2mh2m 2h2m+1
12
(2m)! (2m)! k= ¿=0
^e qk+qN+:(-y+1 A.02"
(qk - 1)i+1
where dk are determined from system (3.11)-(3.13), B2m are Bernoulli numbers, A®Q2m is given in Theorem 3.1, qk are given in Theorem 3.2.
In the proof of Theorem 3.5 we use the following relations [6]
n— 1
Eq7 7k =
k
7=0
1 - q 1 - q
(l-J A.0k - ^ £( ^
a.7 k
(3.14)
where A® yk is the finite difference of Yk of order i, q is the ratio of a geometric progression. Let us consider the following well known relation [5]
ß — 1 k+1 E^k = E
7=0 j=1
k! B,
k+1—j
j! (k + 1 - j)!
ttt ßj,
(3.15)
where Bk+\-j are Bernoulli numbers
V
AaxV = J2 CpAaQpxV-p. (3.16)
p=0
Proof of Theorem 3.5. Let us consider the first sum of equation (2.2). For this sum we have
2
1
7= n
N
2m—1
S V C\Y1\hß - hY\ S = 1)!
C \0l(hß)2m—1 A N C\ n(hß - hY )2m—1
= C [0] (m-îy. + 7= C [Y] (2m - 1)! - 7=0CY] 2(2m - 1)! .
Let us introduce
S V C\ -,1 (hß - hY)2m—1 S f^ C\ . ,(hß - hY)2m—1 S1 =7^C[Y] (2m - 1)! ' S2 = 7=0C[Y] 2(2m - 1)! .
Using (3.4) and (3.14), (3.15), for S1 we have
S1
ß m—1
EM 1 + E
=1 k=1
h2m
(2m - 1)!
h2m (2m - 1) !
1—ß qk
qk -1
"k qk + Pkqk
1
N—7})
(hß - h-)
2m 1
ß—1
E-2m—1
=0
2m fn 1\ID m—1 \ - (2m - 1)!B2m—j ßj + y^
^ j ! • (2m - j)! ß ^
j=1 k=1
q— ^ Aß^l + N—ß
- ^tt E q _ 1). K pkqk ß
¿=0
ß 2m-1 qß
(qk - 1).
(2m - 1)!
m-1 ß-1 ß-1
E ("k qßE q—7-2m—1 + Pk qN—ßJ2 q7-2m—1 k=1 V 7=0
{2m —
(k- g (k-) }+ Pk 4T-qk, I? ( qk-r )' A'°2m—1 -
2m 1
1 - qk
m-1
A¿ß,
2m 1
Taking into account that qk is the root of the Euler-Frobenius polynomial E2m-2(q) and using Theorem 3.1, relation (3.16), the expression for Si is reduced to the following form
S1 =
(hß)2m+h • (hß)2m:1 B1+h
(2m)!
(2m - 1)! '
2m—2
2m V^ B2m—j ßj +
j=1 j!(2m - j)! +
2m 1
+h2m j
ß
2m— 1— j m— 1 j
EE
j!(2m - 1 - j)!
j=0 k=1 ¿=0
-"k qk + Pk qk +¿(-1) (qk -
^¿0^.
(3.17)
Now we consider S2. Using equations (2.4)-(2.7), we rewrite the expression for S2 in powers of hp
S=£ C M^Y )1r1 =
Y=o 2(2m -1)!
2 2k+1
:EE
(hß)
2m—1—j ( _ 1)j
2(2m - 1 - j
(hß)2m—1—j (-1)j 2(2m - 1 -
k=0 j=2k m— 1
+
j=6
2m—1 (hß)2m—1—j (-1)j + ^ 2(2m - 1 - j)!j!
j=m 7=0
tn( J--j )j! \j + 1
'(-1)j ( - j)j! V
+ E j(j - 1)... (j - 2n + 2)aJ +
(3.18)
+ E j (j - 1)... (j - 2« + 2)An +
n=1
m—1—j (-1)j N _1 A-! ! E C \-]( hY )j
Let us substitute (2.8) and S into equation (2.2). Then using (3.17), (3.18) and equating coefficients of the same powers of hp we have
m-1 m-1 j
£ Aj (hß )j _ £ j
j=0 j=0 j!
(-1)j A (-1)3 +
2(2m - j)! ^=1 " 2(2m - 2n - j)!
- 1 2 m- 1 - j
+ h2m-j ^12m-1-j dk qk + pk +i(-1)i+1 A,02m-1-j + (3 19)
+ (2m - 1 - j)! kC ^ -^-A0 + (3.19)
N
+
(_1)2m-1-j
V / \ ^ /T-,1 / L.,)2m-1-j
2(2m - 1 -,,.
v 7=0
^ E C Mh )2
1
_ V- B2m-j h2m-j (hß)j
j= j!(2m - j)! :
£1 £ dk:k 1)i+1 Ai0j _ j, j (3.20)
k=1 i=0 m- 1 2
(ik - 1)i+1
dkik + pk'N+i(- 1)i+1
(ik - 1)i+1
dkik + pk'N+i(- 1)i+1
J2J2 qk q, _k1)i+i Ai02 =0, (3.21)
k = 1 i=0 qk m—1 4
VV Ai04 = 0, (3.22)
k= ¿0 ( qk- 1)i+1
C[0] = hi2+ , (3.23) 2 k=1 1 - qk
= -T^-^ Y y1 dkqk + ^+'+- 1)i(1 Ai02"—1, n =1, 2, 3. (3.24)
n ( 2n)! ( 2n - 1)! k=1 ^ ( qk - 1)i+1 ' >> y >
Taking into account (3.23) and (3.4), when a = 0 we obtain from (2.4)
C [N ] = h (1 + V dk * - Pk qk ). (3.25)
V2 fo 1 - qk )
Substituting Xj from (3.19) into (2.3), we obtain the following equations for unknowns dk and pk
m—1 2^2 dk qk + Pk qN +1 (-1)i+1 Ai02m —2 = m—1 2^2 dk qN+i + Pk qk (-1)i+1 Ai02m—2 (3 26)
k= ¿0 (1 - qk)i+1 = ¿1 ¿0 (1 - qk)i+1 A0 , (3.26)
V1 V dkqk+ + Pkqk+- 1)i+1 Ai0j—1 = Bj j = 77m (3.27)
i=0 (1 - qk )i+1 j
mdk qk + Pk qN+i(-1)i+1 Ai03 v1^ dk qN+i + Pkqk(-1)i+1 (328)
-(qk - 1)i+1-A0 = -(1 - qk)i+1-A0 , (328)
k=1 i=0 yHk ' k=1 i=0 v Hk'
(1 - ik )i+1
dk ik + pk iN+i(- 1)i+1
(ik - 1)i+1
dk ik + pk iN+i(- 1)i+1
(ik - 1)i+1
dk iN+i + pk ik (- 1)i+1
m^^ dkik + pk:N+i(-1)i+1 Ai05 _ m-1 ^ dk:N+i + pkik(-1)i+1 a,05
2.2. (ik - 1)i+1 A0 (1 - ik)i+1 A0 ' (3.29)
k=1 i=0 yHk ' k=1 i=0 v Hk>
-1 2
yydk qk -TP^-I) Ai02 _0, (3.30)
^^ (1 - ik )i+1
k=1 i=0
-1^ "kqN + + Pkqk(-1)¿+1 .¿04 (1 - qk^
1 A r, _L n „N +1 m—1 A „N+1
yy"k qk +Pkqk(-1) A¿04 = 0, (3.31)
¿1 ¿0 (1 - qk )¿+1
v-^ "k qk + Pk qk + = v-^ "kqk + + Pk qk (3 32)
¿1 (qk -1)2 = k= (1 - qk)2 ' ( )
m — 12m—4 " i + P qN +1( 1)i+1 m —12m—4 " qN+¿ + P q ( 1)¿+1
"kqk + Pkqk (-1) A¿{)2m—4 = ^ V^ "kqk + Pkqk(-1V A¿02m—4 (3 33)
¿0 (1 - qk)¿+1 ¿1 ¿0 (1 - qk)¿+1 ' 1 " J
m—1 2m—6 j j N + 1/ -, n7 + 1 m— 12m—6 , N+i , / 1 N¿ + 1
y^ "k qk + Pk qk (-1) Ai02m—6 = V^ y^ "k qk + Pkqk(-1r Ai02m — 6 (3 34)
¿0 (1 - qk )¿+1 ¿1 ¿0 (1 - qk)¿+1 ■ 1 ■ J
Thus, from (3.20)-(3.21), (3.22) and (3.26)-(3.34) we obtain
m— 1 2m—2j m + qN+¿(_1)¿+1
E("k - Pk)£ qk 1(-+11) A¿02m—2j =0, j = 1, 2,3, (3.35)
k=i i=o (qk -1)
m-1 2j m + qN+i (_1)<+1
E(dk - Pk)£ qk 1+7 A-Q2j = Q, j = 1,2, (3.36)
k=i i=o (qk 1)
m-1 j q, + qN+i (_1)i+1 _
E(dk - Pk)£ qk + 1+1 AiQj = Q, j = 6~m—1. (3.37)
k=1 i=0 (qk 1)
Taking into account uniqueness of the optimal coefficients, we conclude that homogeneous system of linear equations (3.35)-(3.37) has trivial solution. This means that
dk = Pk, k = 1, 2,... ,m — 1. (3.38)
Then using (3.38), from (3.35)-(3.37) we obtain (3.11)-(3.13), and from (3.4), (3.23)-(3.24), (3.25) we obtain (3.7)-(3.10).
Theorem 3.5 is proved. □
Remark 1. It should be noted that in the cases m = 6 and m = 7 we obtain from Theorems 3.5 and 3.5 the classical Euler-Maclaurin quadrature formula.
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Оптимальные формулы численного интегрирования с производными в пространстве Соболева
Холмат М. Шадиметов Фарход А. Нуралиев
Институт математики имени В. И. Романовского Академии наук Республики Узбекистан Мирзо Улугбека, 81, Ташкент, 100041
Узбекистан
В настоящей статье 'рассмотрена проблема построения оптимальных квадратурных формул в смысле Сарда в пространстве Ь^ (0,1). Здесь квадратурная сумма состоит из значений подынтегральной функции в узловых точках и значений первой, третьей и пятой производных подынтегральной функции в концевых точках интервала инегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального N и для любого m ^ 6 с использованием метода Соболева, основанном на дискретном аналоге дифференциального оператора d2m/dx2m. В частности, для m = 6, 7 оптимальность классической квадратурной формулы Эйлера-Маклорена доказана. Начиная с m = 8 получены новые оптимальные квадратурные формулы.
Ключевые слова: оптимальная квадратурная формула, функционал погрешности, экстремальная функция, оптимальные коэффициенты.
