Journal of Computations & Modelling, vol.3, no.3, 2013, 1-15
ISSN: 1792-7625 (print), 1792-8850 (online)
Scienpress Ltd, 2013
Calculating the Values of Multiple Integrals by
Replacing the Integrands Interpolation by
Interpolation Polynomial
Sh.A. Nazirov 1 and A.A. Abduazizov 2
Abstract
The work deals with the construction of multidimensional quadrature formulas for
computation of the multiple integrals’ values by replacing the integrands by
interpolation polynomial. We’ve proved the quadrature formulas’ correctness.
Mathematics Subject Classification: 41A05
Keywords: multi-dimensional cubature formulas; interpolation polynomials;
multiple integrals; the Cotes’ coefficients.Schwarz's
1 Introduction
In [1-4] we’ll consider the methods of constructing cubature formulas on the
1
2
Tashkent University of Information Technologies, Tashkent.
Tashkent.
Article Info: Received : April 17, 2013. Revised : June 9, 2013
Published online : September 15, 2013
2
Calculating the Values of Multiple Integrals…
basis of the method of repeated application of different, Simpson and Chebyshev.
In this paper, to construct cubature formulas for the n-fold integrals values
calculating, we used method of replacing of the integrands by interpolating
polynomial. For this purpose, let’s replace the f ( x1 , x 2 ,...x n ) integrand by the
following polynomial:
f ( x1 , x2 ,..., xn ) = L( x1 , x2 ,..., xn ) = f ( x1(1) , x2(1) ,..., xn(1) ) P1 ( x1 , x2 ,..., xn ) +
+ f ( x1( 2) , x2( 2) ,..., xn( 2) ) P2 ( x1 , x2 ,..., xn ) + ... +
+ f ( x1( n) , x2( n) ,..., xn( n) ) Pn ( x1 , x2 ,..., xn ).
Integrating (1) ,we have:
J ≡ ∫∫ ...∫ f ( x1 , x2 ,..., xn )dx1dx2 ...dxn ≈ ∫∫ ...∫ L( x1 , x2 ,..., xn )dx1dx2 ...dxn ≈
(Ω)
(Ω)
n
= ∑ f ( x1( i ) , x2( i ) ,..., xn( i ) ) ∫∫ ...∫ Pi ( x1 , x2 ,..., xn )dx1dx2 ...dxn =
(Ω)
i =1
n
= ∑ Ci f ( x1(i ) , x2(i ) ,..., xn(i ) ),
i =1
(2)
where
Ci =
In simple fields, the
∫∫( D) ∫
Pi ( x1 , x2 ,..., xn )dx1dx2 ...dxn .
Ci values calculation is not difficult because
Pi ( x1 , x2 ,..., xn ) is a polynomial. In complex areas to calculate the Сi values it
is appropriate to use the V.L. Rvachev’s R-functions [2,5]. This method
determines the points of integration.
For simplicity, let’s consider the n-dimensional rectangular region and some grid
lines on it, formed by:
xi=
a1 + i1h1
1
(3)
.....................
xi=
an + in hn
n
i1 = 0, n1; i2 = 0, n2 ;... in = 0, nn ; hi =
h−q
, i = 1, n.
ni
Sh.A. Nazirov and A.A. Abduazizov
3
In this case we use the ( xi1 , xi2 ,..., xin ) : interpolation formula, using the nodes:
n1
wn ( x1 ) wn ( x 2 )...wn ( x n )
nn
n2
f ( x1 , x 2 ,..., x n ) = ∑ ∑ ...∑ f ( x i , x i ,..., x i )
i1 = 0 i 2 = 0
1
in = 0
n
2
1
n
2
( x1 − x i )( x 2 − x i )...( x n − x i ) wn′ ( x i )...wn′ ( x i )
1
n
2
1
1
+R
(4)
n
where
wn1 ( x1 ) = ( x1 − x1(1) )( x1 − x 2(1) )...( x1 − x n(1) ),
wn2 ( x 2 ) = ( x 2 − x1( 2 ) )( x 2 − x 2( 2 ) )...( x 2 − x n( 2 ) )
..................................................................
(5)
( xn x1( n ) )( xn − x2( n ) )...( xn − xn( nn ) )
wnn ( xn ) =−
Integrating it, we obtain
n1
J = ∫∫ ...∫ f ( x1 , x2 ,..., xn )dx1dx2 ...dxn = ∑
(Ω)
A1
×∫
a1
i1 =0
A2
wn1 ( x1 )
( x1 − x i1 ) w1n1 ( x i1 )
dx1 ∫
a2
( x 2 − x i2 ) w1n2 ( x i2 )
(Ω)
∑ ...∑ f ( xi1 , xi2 ,..., xin ) ×
i2 =0
in =0
An
wn2 ( x 2 )
∫∫
nn
n2
dx2 ... ∫
an
wnn ( x n )
( x n − x in ) w1nn ( x in )
dxn +
...∫ Rdx1dx2 ...dxn .
(6)
Here R is the remainder of interpolation formula (6).
In the last equation, introducing the notation,
Ak
Ci(l )
k
=
wnk ( x k )
∫ ( xk − xi
aK
) wn′ k ( xik )
k
dxk ,
k = 1, n; l = 1, n;,
(7)
we have
n1
J = ∫∫( Ω ) ...∫ f ( x1 , x2 ,..., xn )dx1 dx2 ,..., dxn = ∑
...C
(n)
in
i1 = 0
f ( xi , xi ,..., xi ) + ∫∫( D ) ∫ Rdx1 dx2 ...dxn .
1
2
n2
nn
i2 = 0
in = 0
∑ ...∑
Ci(1) Ci( 2 ) ...
1
2
(8)
n
Ci(lk ) are the coefficients of numerical integration formulas of homogeneous
integrals.
On this basis in the work [4] and introducing the notation
4
Calculating the Values of Multiple Integrals…
xk − x0( k )
,
qk =
hk
(9)
qk[ nk +1] =qk (qk − 1)...(qk − nk ),
k =1, 2,..., n
and coefficients (7) , we reduce to
Ci( l ) = ( Al − al ) ⋅ H i( l ) , i = 0,1,..., n; l = 1, n ,
H i( l ) =
(10)
1 (−1) n −i n q [ n +1]
dql , il = 0,1,2,..., nl ; l = 1, n.
∫
nl ik ! (nl − il )! 0 ql − il
l
l
l
l
(11)
Now, let’ consider the special cases. Calculating the double integrals’ values.
2 Double integrals’ values’ calculation algorithm’s description
Theorem 1. Suppose that the f ( x1 , x2 ) function is defined and continuous in a
given region of integration. Then the cubature formula for the double integral
values, obtained by replacing the integrand by a polynomial interpolation for
n1 = 1 and n2 = 1 has the form
J = ∫∫ f ( x1 , x2 )dx1dx2 =
( D)
hx h y
[ f ( x0 , y0 ) + f ( x0 , y1 ) + f ( x1 , y0 ) + f ( x1 , y1 )] =
2 2
hh
= x 2 [ f 0 , 0 + f 0 ,1 + f10 + f11 ].
h
Proof. In proving theorems we use equations (8) - (11). In this case, (8) and (11)
respectively have the form
J=
A1 A2
n1
n2
∫∫ f ( x , x )dx dx = ∫ ∫ f ( x, y)dxdy = ∑ ∑ C
1
( D)
2
1
2
a1 a 2
i1 = 0 i 2 = 0
(1)
i1
Ci(22 ) f ( xi1 , xi2 ),
(13)
Sh.A. Nazirov and A.A. Abduazizov
5
n

1 (−1) n1 − i1 1 q1[ n1 +1]
dq
,
i
=
0
,
1
,
2
,...,
n
;

1
1
1
n1 i1!(n1 − i1 )! ∫0 q1 − i1


n 2 − i 2 n 2 [ n1 +1]
1 (−1)
q1

=
dq2 , i2 = 0,1,2,..., n2 .
∫
n2 i2!(n2 − i2 )! 0 q2 − i2

H i(11) =
H i(22 )
Calculate the values of H i1 and H i2 , for n1 = 1 and n2 = 1 respectively:
1
1− 0 1
q1 (q1 − 1)
(q − 1) 2 1
(1) 1 ( −1)
H0 = ⋅
dq1 = − (q 1 − 1)dq1 = − 1
=
0
q1 − 0
1 0!(1 − 0)!
2
0
0
∫
∫
=−
1
[0 − 1] = 1 .
2
2
1
1−1 1
q1 (q1 − 1)
q12 1 1
(1) 1 ( −1)
H1 = ⋅
dq1 = q1dq1 =
= .
q1 − 1
1 1!0!
2 0 2
0
0
∫
∫
In the same way we calculate the values H 0( 2) and H1( 2) :
1
1
H 0(2 ) = , H1(2 ) = .
2
2
On the basis of H 0(1) , H 1( 2 ) we compute C0(1) and C1(1) , as well as on the H 0( 2)
and H1( 2) base ,the C 0( 2 ) and C1( 2) value and taking into account (10):
hy
h
C0(1) = C1(1) = x ; C0( 2) = C1( 2) =
.
2
2
Here
(15)
f i, j = f ( xi , x j ), i, j = 0,1.
Substituting this result in (13), we obtain (12).
This result coincides with the result, obtained by repeated application of the
trapezoidal quadrature rule, which shows the equivalence of the method of the
integrand interpolation replacing by polynomial method and re-use of quadrature
formulas.
Theorem 2. Suppose that the f ( x1 , x 2 ) function is defined and continuous in the
closed area of integration. Then the cubature formula for the double integral
6
Calculating the Values of Multiple Integrals…
values, obtained by replacing the integrand by a polynomial interpolation for
n1 = 2, n2 = 2 has the form
J=
∫∫
f ( x1, x2 )dx1dx2 =
hx h y
9
[( f00 + f 20 + f02 + f 22 ) + 4( f1,0 + f0,1 + f 2,1 + f1,2 ) + 16 f11].
(16)
( D)
Proof. For this purpose, we calculate the Ci(1) and Ci( 2) values now for n1 = 2
1
2
and n2 = 2 :
2
2−0 2
( q1 − 1)
11
(1) 1 ( −1)
H0 =
dq1 =
(q1 − 1)(q1 − 2)dq1 =
2 0!(2 − 0)! q1 − 0
22
0
0
2
2
3
2


q
q
1
1
18
 1
(q12 − 3q1 + 2)dq1 =  1 − 3 1 + 2q1  =  − 6 + 4  = ,
=

4
4  3
2
 6
0 4  3
0
∫
∫
∫
2
H1(1)
2
(−1) 2 −1 q1 (q1 − 1)(q1 − 2)
1 1
dq1 = − ⋅ ⋅ q1 ⋅ (q1 − 2)dq1 =
=
q1 − 1
1!(2 − 1)!
2 1
∫
∫
0
0
2
2
1
1  q3
18
 2
(q12 − 2q1 )dq1 = −  1 − q12  = −  − 4  = ,
=−

2
2  3
23
 3
0
0
∫
H 2(1) =
2
2
1  q13 q12 
1 1
1 (−1) 2− 2 q1 (q1 − 1)(q1 − 2)
2
 −  =
(
)
⋅
−
=
=
⋅
dq
q
q
dq
1
1
1
1
4  3
2 
2 2 ∫0
2 2!(2 − 2)! ∫0
q1 − 2
1 8
 1
= ∫  − 2 = .
4 03
 6
2
In the same way, for H i( 2 ) (i = 0,1,2) we have
2
1
1
H 0( 2) = , H1( 2) = , H 2( 2) = .
6
6
3
Then ,based on the H i(11) and H i(22 ) values,we calculate C i(11) and C i(22 ) . Then,
substituting the C i(11) and C i(22 ) values in (13), we obtain (16).
Equation (16) coincides with the result, obtained, using the re-use of Simpson’s
quadrature formulas. This fact confirms the equivalence of double integrals,
Sh.A. Nazirov and A.A. Abduazizov
7
obtained by the repeated application of quadrature formulas and methods of
replacing the integrand by interpolating polynomial ,computing.
Let’s rewrite (12) and (16) ,respectively, in the form of a component as:
hx h y
J T = ∫∫ f ( x1 , x2 )dx1dx2 =
4
D
J S = ∫∫ f ( x1 , x2 )dx1dx2 =
D
hx h y
9
2
1
1
i =0
i =0
∑ ∑f
2
∑ ∑ (C
i1 = 0
i2 = 0
i1
2
,
ij
(17)
) 2 (C 2i ) 2 f ij .
(18)
2
To calculate the n-fold integrals, using (7), we can derive the corresponding
formulas.
Theorem 3. Let the f ( x1 , x 2 ,..., x n ) functions be defined and continuous in the
closed domain of integration Ω n . . Then n-dimensional cubature formulas for
f ( x1 , x 2 ,..., x n ) for n1 = n2 = ... = nn = 1 and n1 = n2 = ... = nn = 2 approximate
calculations will relatively take the form
∫∫ ...∫ f ( x1 , x2 ,..., xn )dx1 dx2 ...dxn =
J=
(D)
h1 h2 ...hn 1
∑
2 n i =0
1
1
1
i1 = 0
in = 0
∑ ...∑
f i i ...i , (19)
n
1 2
where hi = Ai − ai ;
J=
∫∫ ...∫ f ( x , x
1
2
,..., x n )dx1 dx 2 ...dx n =
(D)
Here
h1 h2 ...hn 2
∑
3 4 i =0
1
hi =
2
∑
i1 = 0
2
...∑ (C 2i ) 2 (C 2i ) 2 ...(C 2( i ) ) f i i ...i , (20)
2
1
2
in = 0
2
1 2
n
Ai − ai
.
2
The proof of this theorem can also be made by induction.
(19) and (20), respectively, coincide with the formulas ,obtained by the repeated
application of the quadrature trapezoidal and Simpson, and are given in [2].
Based on these results, let’s make the following conclusions.
1. If we put n1 = n2 = ... = nn = 1 to (7) , we obtain cubature formulas of the form
(19), which coincides with the formula, obtained by repeated application of a
quadrature trapeze formula to calculate values of multiple integrals.
8
Calculating the Values of Multiple Integrals…
2. If we put n1 = n2 = ... = nn = 2 to (7), we obtain formulas for the approximate
calculation of n-fold integrals of the form (20), which coincides with the formula,
obtained by repeated application of the Simpson quadrature formula to compute
the values of multiple integrals.
3. If we put n1 = n2 = ... = nn = k (k > 2) to (7), we obtain the Newton-Cotes high
order formulas for the approximate calculation of n-fold integrals, and thus
provide a high order of accuracy of n-fold cubature formulas.
4.If we put
n1 = S1 , n2 = S 2 ,..., nn = S n to (7)
(21)
we ‘ll get the n-fold composite cubature formulas for n-fold integrals.
The essence of (21) means that for each xik (k = 1, n) we use different
Newton-Cotes’ formulas. This is of interest to the construction of n-dimensional
cubature formulas.
Theorem 4. Let f ( x1 , x2 ) be defined and continuous in the closed area of
integration. Then the cubature formula for the approximate calculation of values
by replacing the integrand interpolation by polynomial obtained when
n1 = n2 = 3 , has the form
∫∫
D
3
3
(h1 )(h2 )
f ( x1 , x2 )dx1dx2 =
dx =
Ci(1)
Ci(2)
f ( xi1 , xi2 ) =
{[ f ( x0 , y0 ) + f ( x0 , y3 ) +
∑
∑
1
2
64
=
i1 0=
i2 0
+ f ( x3 , y0 ) + f ( x3 , y2 )] + 3[ f ( x0 , y1 ) + f ( x0 , y2 ) + f ( x1 , y0 ) + f ( x1 , y3 ) + f ( x2 , y0 ) +
(22)
+ f ( x1 , y3 ) + f ( x3 , y2 ) + f ( x3 , y2 )] + 9[ f ( x1 , y1 ) + f ( x1 , y2 ) + f ( x2 , y2 ) + f ( x2 , y2 )]} =
=
where hi =
3
3
9
(h1h2 )∑ ∑ C3i1 C3i2 f ( xi1 , xi2 ),
64
=
i1 0=
i2 0
Ai − ai
, i = 1,2.
3
Sh.A. Nazirov and A.A. Abduazizov
9
Proof. For this purpose, in accordance with paragraph 3, we derive cubature
formulas to calculate values for double integrals for n1 = n2 = 3 in (12). In this
case, the H i(11) and H i(22 ) values, according to (13), take the form
q1[3+1]
1 (−1)3− i1
1 (−1)3− i1 q1 (q1 − 1)(q1 − 2)(q1 − 3)
H i(11) = ⋅
d
dq 1 , i1 = 0,1,2,3
=
q
q1 − i1
3 i1!(3 − i1 )! ∫0 q1 − i1 1 3 i1 (3 − i1 )! ∫0
3
3
q2 (q2 − 2)(q2 − 2)(q2 − 3)
1 (−1)3− i2
H i(22 ) = ⋅
dq 2 , i2 = 0,1,2,3,
∫
q − i2
3 i2!(3 − i2 )! 0
3
(23)
1 ( −1)3− 0 q1 (q1 − 1)(q1 − 2)(q1 − 3)
11
(q1 − 1)(q1 − 2)(q1 − 3)dq1 =
H 0(1) = ⋅
d q1 = −
∫
3 0!(3 − 0)! 0
3 3! ∫0
q1 − 0
3
3
3 1
1 1  q4
q2
= ⋅  1 − 2 ⋅ q13 + 11 ⋅ 1 − 6q1  = ,
3 3!  4
2
0 8
1 (−1)3−1 q1 (q1 − 1)(q1 − 2)(q1 − 3)
11
(q1 − 2)(q1 − 3)q1dq1 =
= ⋅
d q1 =
∫
3 1!(3 − 1)! 0
3 2! ∫0
q1 − 0
3
H
(1)
1
3
1 1  q14
q13
q12  3 3

= − ⋅  − 5 ⋅ + 6 ⋅  = ,
3 2!  4
3
2 0 8
1 1
1 1 1
4
3 
3
q1 (q1 − 1)(q1 − 3)dq1 = − ⋅  q14 + q13 + q12  =
∫
3 2!1! 0
3 2!  4
3
2 0 8
3
H 2(1) = −
H 3(1) =
3
3
3
1 1
1 1  q14
1
q13 q12 


(
−
1
)(
−
2
)
=
⋅
−
3
⋅
+
= .
q
q
q
dq
1 1
1
1
∫


3 3!0! 0
3 3!  4
3
2 0 8
In the same way, for H i( 2 )
(i = 0,1,2,3) we have
1
3
1
H 0( 2 ) = , H 1( 2 ) = H 2( 2 ) = , H 3( 2 ) = .
8
8
8
Let’s rewrite the formula (23) in the form of:
∫∫
A1 A2
f ( x1 , x2 )dx1dx2 =
( D)
∫∫
a1 a 2
1
where C3 =
i
3!
,
i1!(3 − i1 )!
9
f ( x1 , x2 )dx1dx2 = h1h2
64
C3i =
2
3!
.
i2 !(3 − i2 )!
3
3
∑ ∑ C3i C3i
1
i1 = 0 i 2 = 0
2
f i1i2
, (24)
10
Calculating the Values of Multiple Integrals…
Theorem 5. Let f ( x1 , x 2 ,..., x n ) be defined and continuous in n-dimensional
integration region Ω n . Then cubature formula for the approximate computation
of the values of the f ( x1 , x 2 ,..., x n ) integrand by interpolation polynomial ,
obtained for n1 = n2 = ... = nn = 3 ,, has the form:
∫∫ ∫
f ( x1 , x2 ,..., xn )dx1dx2 ,..., dxn =
( D)
3
= 
8
where
C3ik =
n n
3
3
A1 A2
An
a1 a 2
an
∫∫ ∫
f ( x1 , x2 ,..., xn )dx1dx2 ,..., dxn =
3
∏ hi ∑ ∑ ...∑ C3i1 C3i2 ...C3in fi1i2 ,...in ,
i =1
i = 0 i2 = 0
i1 = 0
(25)
3!
, k = 1, n.
ik !(3 − ik )!
Proof. In the equations to calculate the values of double integrals (17) and (18),
respectively, and produced by n1 = n2 = 1 and n1 = n2 = 2 , and the values of
n-fold integrals (19) and (20), respectively, and produced by n1 = n2 = ... = nn = 1
и n1 = n2 = ... = nn = 2, and (24) to calculate the values of n-fold integrals as
n1 = n2 = ... = nn = 3 , are useful when the size of the region is very small. If the
Ai − ai dimensions are large, then to increase the accuracy of the n-fold cubature
formulas of each [ai , Ai ] , we ‘ll divide the system under the
[ai , Ai ] = [ai = ai( 0 ) , ai(1) ]  [ai(1)  ai( 2 ) ]  ...  [ai( k −1) , aik = Ai ]
integrals, and apply to each the appropriate cubature formulas.
To understand this approach fully, we’ll first consider the double integrals. Here
the two [a1 , A1 ] and [a 2 , A2 ] interpolations correspond to directions along
the ox1 and ox 2 . axes The [a1 , A1 ] intervals are divided by n1 , and the [a 2 , A2 ]
interval – by n2 parts. Next , for n1 = n2 = 2 , we have
h2 = h1 =
A1 − a1
A −b
, h1 = h2 = 2 2
n1
n2
(26)
Sh.A. Nazirov and A.A. Abduazizov
11
And for n1 = n2 = 2
h2 = h1 =
h − a1
A − b2
, h y = h2 = 2
2n1
2 n2 .
(27)
Taking into account (25) and (26), the (17) and (18) can be respectively rewritten
as
A1
A2
a1
a2
∫∫
f ( x1 , x 2 )dx1 dx 2 = ∫
∫∫
f ( x1 , x 2 )dx1 dx 2 = ∫
D
D
A1
a1
A2
∫
α
∫
f ( x1 , x 2 ) dx1 dx 2 =
h1 h2 n n 1 1
∑ ∑ ∑ ∑ f i + i ,i + i ,
4 i =0 i =0 j =0 j =0
1
1
i
f ( x1 , x 2 ) dx1 dx 2 =
2
2
2
1
1 2
(28)
2
2
h1 h2 n1 n2 2 2
∑ ∑ ∑ ∑ (C 2i1 ) 2 (C 2i2 ) 2 f i +i ,i2 +i . (29)
1
1
2
9 ii =0 i2 =0 j1 =0 j2 =0
Reasoning by analogy (28) and (29), the formulas (19), (20) and (24) respectively,
are rewritten as following:
A1
A2
An
a1
a2
an
1
2
∫∫ ...∫ f ( x1 , x 2 ,..., x n )dx1 dx 2 ...dx n = ∫
D
n
=∏
i =1
∫ ... ∫ f ( x1 , x 2 ,..., x n ) dx1 dx 2 ...dx n =
h
1
1
1
h  h h
 i  ∑ ∑ ... ∑ ∑ ∑ ... ∑ f i
 2  i =0 i =0 i =0 j =0 j =0 j =0
1
i
n
2
(30)
n
2
n
1
+ j 1 , i 2 + j 2 ,..., in + j n
,
A1 A2 An
=
f
x
x
x
dx
dx
dx
...
(
,
,...,
)
...
∫∫ ∫ 1 2 n 1 2 n ∫ ∫ ... ∫ f ( x1, x2 ,..., xn ) dx1dx2 ...dxn =
D
a1 a 2 a n
(31)
2n1 2n 2 2n n 2
n
2
2
a 
i
i
i
= ∏  i  ∑ ∑ ... ∑ ∑ ∑ ... ∑ (C 1 ) 2 (C 2 ) 2 ...(C n ) 2 f i j , i j ,..., i j ,
n+ n
1+ 1
2+ 2
2
2
2
3
i =1   ii = 0 i 2 = 0 i n = 0 j1 = 0 j 2 = 0 j n = 0
∫∫ ...∫
( D)
A1
A2
An
a1
a2
an
f ( x1 , x2 ,..., xn )dx1dx2 ...dxn = ∫
h
n1
n2
nn
3
3
∫ ... ∫
f ( x1 , x2 ,..., xn ) dx1dx2 ...dxn =
3
= ∏ hi ∑ ∑ ... ∑ ∑ ∑ ... ∑ C3i1 C3i2 ...C3in f i1+ j1 ,i2+ j2 ,...,in+ jn .
i =1
ii =0 i2 =0
in =0 j1 =0 j2 =0
jn =0
It should be noted that in the calculation of multiple n1 = n2 = ... = nn = 2
integrals, C 2ik will be present under the sum, and when n1 = n2 = ... = nn = 3 C 3ik etc. Note that this rule, since n1 = n2 = ... = nn = 4 is violated. For clarity,
we present below the Cotes coefficients (Table 1).
12
Calculating the Values of Multiple Integrals…
According to the values of numbers, standing in rows and columns (see Table 1),
the Cotes coefficients are increasing rapidly with the increasing of values and,
since n = 8 ,become negative. Therefore, we have developed a program in Maple
11 mathematical system for the coefficients’ and constructed square Newton Cotes formulas’ calculating. We now give a quadrature formulas, constructed by
using this Maple procedure.
In Table 1 (for convenience), according to [4], the Coates recording rate for each
H
Ĥ1 is presented in the form of fractions:
Hi = i
N
n
is clear ,that

∑H
i =n
i
with a common factor N. It
= N . According Table 1, with large values,the results may be
negative.
Now, the value of this integral is calculated approximately by the Newton-Cotes,
when n = 6 (seven ordinates)
Example. Let’s calculate
1 1
Y = ∫∫ y⋅
0 0
1
dxdy .
x +1
Table 1
n
^
H1
^
H2
^
H3
^
H4
^
H5
^
H6
1
1
2
4
1
3
3
3
1
4
32
12
32
7
5
75
50
50
75
19
6
216
27
272
27
216
41
7
3577
1323
2989
2989
1313
3572
8
5888
-928
10469
-4540
10496
-928
Sh.A. Nazirov and A.A. Abduazizov
13
9
15741
1080
19344
5778
5778
19344
10
106300
-48525
272400
260550
472368
-260550
11
13486539
-3237113
25226685
-9595542
15493566
15493566
12
9903168
-7587864
35725120
-5149130
87516288
87797136
Continuation of table 1
^
H7
^
^
H8
^
H9
^
H 10
H 11
^
H 12
^
^
^
H 13 H 14 H 15
Value
objects
2
6
8
90
288
840
751
17280
5888
989
28350
1080
15741
2857
272400
106300
106300
16067
-9595542
25226685
-3237113
1348653
87516288
-5149130
35725120
89600
598732
87091200
2171465
9
-7587864
9903168
1364651
The exact value of the integral is the following:
1
1
1
y2
dx
1
=
+
n
x
IT = ∫
ydy

2
x + 1 ∫0
0
0
h1 = hx =
1
0
= ln 2 ⋅
1 ln 2 0,7935
=
= 0,3467,
=
2
2
2
1− 0 1
1− 0 1
=
= ; h2 = h y =
6
6
6
6
Let’s draw up a table of values (Table 2), assuming that H i = 840 Ai .
63063000
14
Calculating the Values of Multiple Integrals…
Table 2:. Calculating values.
y
f 2 = yi
Hi
H i f1 ( xi )
H i f 2 ( yi )
0
0
0
41
41
0
1
6
6
7
1
6
1
6
1
3
3
4
1
3
1
3
1
2
2
3
1
2
1
2
2
3
3
5
2
3
1
3
5
6
6
11
5
6
1
6
i
xi
f1 ( xi ) =
0
0
1
2
3
4
5
1
1 + xi
216
27
272
27
216
185,14257
26
= 36
72
20,25
27 3
= =9
18 2
181,3333
272
= 136
4
16,2
327 ⋅ y 2
= 19
189
117,81918
25 ⋅ 26
= 180
72
2
6
7
1
2
1
Σ
2
41
20,25
581,99437
41
= 41
2
420
2
Now, based on this table we calculate the value of double integral:
I пр =
1
420
1
⋅ 591,994372 ⋅
= 0,6933 ⋅ = 0,3466
2
940
840
despite the fact that the values I T (precision) and I пр (approximate) are close
enough. Along with this, we note that the n of Cotes coefficient are negative, large
(in absolute value) that are very different from each other. Therefore, the Newton Cotes is not recommended for large n.
Sh.A. Nazirov and A.A. Abduazizov
15
Instead, when n = 1 or n = 2 or n = 3 each integration interval should be divided
into n parts and use the appropriate Newton-Cotes in each of them.
3 Conclusions
Thus we have constructed the multi-dimensional cubature formulas to
approximate the values of multiple integrals by replacing the integrands by
interpolation polynomial, which can be easily implemented on a computer.
References
[1] S.A. Nazirov, Algorithm for computing the multiple integrals values with a
combination of Monte Carlo and R-functions, Iss. High. and glue,
Mathematics, Sat scientific, Tashkent.
[2] S.A. Nazirov, Approximate calculation of integrals by the brief knowledge of
quadrature Simpson formulas re-use, High and glue, Mathematics, Sat
scientific, Tashkent.
[3] S.A.Nazirov, The method of re-use of quadrature formulas for trapeziums
and rectangles to determine the values of multiple integrals, High and glue.
Mathematics, Sat scientific, Tashkent.
[4] B.P. Demidovich and
I.A. Maron, Foundations of Computational
Mathematics, AN: Science, (1966), p. 664.
[5] V.L. Rvachev, Theory of R - functions and some applications, Kiev.
Naukova Dumka, (1952), p. 502.