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The Method of Repeated Application of a Quadrature Formula of Trapezoids and Rectangles to Determine the Values of Multiple Integrals

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The Method of Repeated Application of a Quadrature Formula of Trapezoids and Rectangles to Determine the Values of Multiple Integrals

1. 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2027 | Page Nazirov Sh. A. Tashkent University of Information Technologies, Tashkent, Uzbekistan Abstract: The work deals with the construction of multi-dimensional quadrature formulas based on the method of repeated application of the quadrature formulas of rectangles and trapezoids, to calculate the multiple integrals’ values. We’ve proved the correctness of the quadrature formulas. Keywords: multi-dimensional integrals, cubature formulas, re-use method. I. INTRODUCNION The paper is devoted to developing the multiple cubature formulas for the values of n-fold integrals calculation by means of methods of repeated application of, quadrature formulas of trapezoid and rectangles. According to mathematical analysis multiple integrals can be calculated by recalculating single integrals. The essence of this approach is the following. Let the domain of integration is limited to single-valued continuous curves [1] ))()(()(),( xxxyxy   and two vertical lines bxax  , (Fig.1). Placing the known rules in the double integral  )( ),( D dxdyyxfI (1) limits of integration, we obtain   )( )( )( .),(),( D x x b a dyyxfdxdxdyyxf   Let  )( )( .),()( x x dyyxfxF   (2) Then .)(),( )(   D b a dxxFdxdyyxf (3) Fig.1 Applying to a single integral in the right-hand side of (3), one of the quadrature formulas, we have    )( 1 ),(),(  n i ii xFAdxdyyxf (4) where - ),...,2,1(],[ nibaxi  and iA are some constant coefficients. In turn, the value y x0 а b )(xy  )(xy   The Method of Repeated Application of a Quadrature Formula of Trapezoids and Rectangles to Determine the Values of Multiple Integrals
2. 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2028 | Page  )( )( ),()( i i x x ii dyyxfxF   can also be found on some quadrature formulas ,),()( 1   im j jiiji yxfBxF where ijB iy - appropriate constants. From (4) we obtain     )( 1 1 ),,(),( D n i m j jiiji i yxfBAdxdyyxf (5) where iA and ijB iy - are known constants. Geometrically, this method is equivalent to the calculation of the volume I, given by the integral (2) by means of cross-sections. For cubature formulas of the type (3), the general comments, relating to the computation of simple integrals, are valid with appropriate modifications. In three dimensions (3) has the form      n i m j kji t k kijiji D i i zyxfCBAdxdydzzyxf 1 1 1 , )( ),,(),,( . Now these results are generalized to compute the values of multiple integrals ),,...,,(...... ...),,(... 21 )()3()2( 1 1 1 )1( 2121 )( 21321321 1 1 2 2 1 iiik k n k k iiiiiiiii n i n i n i i kk D xxxfAAAA dxdxdxxxxf         Where )( ... )2()1( 21211 ,...,, k iiiiii k AAA - are known constants. This approach can be applied to calculate the approximate value of multiple integrals using appropriate quadrature formulas. In this case, we apply the quadrature formula of trapezoid. II. ALGORITHM DISCRIPTION Theorem 1. Let the function ),...,,( 21 nxxxfy  Is continuous in a bounded domain D. Then the formula for the approximate values of the n-fold integral calculating   nn D n dxdxdxxxfJ ,...,),...,(... 211 )( , (6) based on the trapezoidal rule has the form                  1 0 1 0 1 0 1 0 ,...,, 1 0 0 1 01 1 2 2211 1 1 2 2 3 3 ...... n n n nn n p i i i ipipip n p n p n pi in fhJ Here D is n-dimensional domain of integration of the form niAxa iii ,1,  ; (7) 2 ii i aA h   and it is believed that each interval (5), is respectively, divided into nnnn ,...,, 21 parts. Proof. Approximate formula for calculating the values of the one-dimensional integral in this case is      1 0 101 2 )()( 2 )( 1 0 i i x x x x f h xfxf h dxxfJ , (8) where, n ab hx   , n - the number of the interval divisions into n parts. In (4) n = 1 . Now we define the approximate calculation formula for double integrals  )( 2 ),( D dxdyyxfJ (9)
3. 3. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2029 | Page where D is the two-dimensional area of integration. It is generally believed that the region of integration in this case is within the rectangle, that is, bxa  , dyc  . Then we rewrite the integral (7) in the form  d c b a dyyxfdxJ ),(2 . (10) Using the trapezoidal rule to calculate the inner integral, we obtain         b a b a y dxyxfdxyxf h J ),(),( 2 2 . (11) Now to each integral we use the trapezoid formulas and the result is       ,),( 4 ),(),(),(),( 4 ),(),( 2 ),(),( 22 1 0 1 0 11100100 111001002            i j ji yxyx xxy yxf hh yxfyxfyxfyxf hh yxfyxf h yxfyxf hh J (12) where m cd hy   . Similarly, for the approximate calculation of the values of the triple integral we obtain the following formula:   ,),,( 8 ),,(),(),(),,(),,(),,( ),,(),,(),,( 8 ),,( 1 0 1 0 1 0 111011011101001110 0101000003        i j k kji zyx zyx A a B b C c zyxf hhh zyxfzyxfzyxfzyxfzyxfzyxf zyxfzyxfzyxf hhh dxdydzzyxfJ (13) where p cC hz   . The four-dimensional integral is defined with 16 members:   ;),,,( 16 ),,,(),,,( ),,,(),,,(),,,(),,,(),,,( ),,,(),,,(),,,(),,,(),,,( ),,,(),,,(),,,( 16 ),,,( 1 0 1 0 1 0 1 0 11111011 00111101010110010001 11100110101000101100 0100100000004           i j k l lkji uzyx uzyx A a B b C c D d uzyxf hhhh uzyxfuzyxf uzyxfuzyxfuzyxfuzyxfuzyxf uzyxfuzyxfuzyxfuzyxfuzyxf uzyxfuzyxfuzyxf hhhh dxdydzduuzyxfJ (14) Here t dD hu   . Considering (8.10), we present formulas for the approximate calculation of the n-fold integral on the trapezoidal:           1 0 1 0 1 0 1 0 21 1 32121 1 2 3 22 1 1 2 2 3 3 ),...,,(... 2 1 ...),...,,(... i i i i niii n i in A a A a A a A a nnn n n n n xxxfhdxdxdxdxxxxfJ (15) Then for the formulas (6) and (8) - (11) we derive the general trapezoid formula. For this purpose, one-dimensional integral for integral intervals [a, b] is divided by n equal parts: xinn ihxxxxxx  0110 ],,[],...,,[ . The two-dimensional integral of the ],[],[ dcba  region of integration is divided, respectively, by n and m parts: ],[],...,,[];,[],...,,[ 110110 mmnn yyyyxxxx  . , 0 0 yj xi jhyy ihxx   (16) In the same way ,the three-, four-, etc. n-dimensional integration region is divided by the corresponding parts. In this approach, the formula (6) takes the form
4. 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2030 | Page . 2 )( 2 )( 2 )( 2 )( 1 0 0 1 1 1 0 1 1 0 11                 n i i n n i ii n i ii n i ii b a f ff ff h ff h yy h dxxfJ (17) The general trapezoid formula for double integrals calculating has the form   . 4 4 1 0 1 0 1 0 1 0 1 0 1 0 1,1,11,,2 1 2 21                n i m j i i ijii yx n i m j jijijiji yx f hh ffff hh J (18) The following is a general formula for the trapezoidal three-and four-fold integrals calculating   , 8 8 ),,( 1 0 1 0 1 0 1 0 1 0 1 0 ,, 1,1,1,1,1,1,11,,1,,11,1, 1 0 1 0 1 0 ,1,1,,,,3 1 2 3 321                         n i m j p k i i i ikijii zyx kjikjikjikjikjikji n i m j p k kjikjikji A a zyx B b C c f hhh ffffff fff hhh dxdydzzyxfJ (19)   . 16 16 ),,,( 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ,,, 1,1,1,1,1,1,11,,1,1,,1,11,1,,1,1,,1 ,,,1,,,11,1,1,,1,1,1,,1,,,1,1,1,, ,1,,1,,,,,,4 1 2 3 4 4321                      n i m j p k t l i i i i ilikijii uzyx lkjilkjilkjilkjilkjilkji lkjilkjilkjilkjilkjilkjilkji lkjilkjilkji A a uzyx B b C c D d f hhhh ffffff fffffff fff hhhh dudxdydzuzyxfJ (20) Based on the above formula, the approximate calculation of n-fold integral, using the formula (D), takes the form .......)( 2 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ,...,, 1 1 2 2 3 3 1 2 2211                  i n n n nn n p n p n p n p i i i ipipip n i iinn faAJ (21) Thus the theorem is completely proved. Theorem 2. Approximation formulas for determining the values of the double integral, obtained by the repeated use of the trapezoid quadrature formula is calculated as follows:     n i m j ijij D yx f hh dxdyyxf 1 1)( 4 ),(  , (22) where                      12...221 24...442 ........................... 24...442 24...442 12...421* ij . (23) Proof. Let’s use the formula (12). Revealing this amount and similar terms, we obtain the following formula:    nnnnnnn nnnnnnn nn nn yx n i m j jijiijij yx fffff fffff fffff fffff hh ffff hh ,1,3,2,1, ,11,13,12,11,1 ,21,2232221 ,111131211 1 0 1 0 1,1,11 2...2 .44...444 .............................................................. 44...442 2...22 4 4               (24) The (17) can form the matrix elements - ij in the form (16) and the result (15), indicating the proof of the theorem.
5. 5. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2031 | Page Theorem 3. Approximate formula for calculating the value of triple integrals, obtained by the repeated use of the trapezoid quadrature formula, is calculated as follows:      n i m j S k ijkijk zyx D f hhh dxdydzzyxf 1 1 1)( 8 ),,(  , (25) where .1,2, 244...442 248...884 .............................. 248...884 244...442 ,;1, 12...2221 24...4442 .............................. 24...4442 24...4442 12...2221                                         ni ni ijk ijk   (26) Proof. Let’s use (13). Revealing this amount and similar terms, we obtain .4...222 2...4...4442 .............................................................................................................................. 24...4442 2...222 24...4442 48...8884 ............................................................................................................ 48...8884 24...4442 .................................................................................................. 24...4442 48...8884 ............................................................................................... 48...8884 24...4442 2...222 24...4442 ....................................................................................... 24...4442 2...222 ,,1,,4,,3,,2,,1,, ,1,1,1,4,1,3,1,2,1,1,1, ,2,1,2,4,2,3,2,2,2,1,2, ,1,1,1,4,1,3,1,2,1,1,1, ,,11,,1,,1,,12,,11,, ,1,11,1,14,1,13,1,12,1,11,1,1 ,2,11,2,14,2,13,2,12,2,11,2,1 ,1,11,1,14,1,13,1,12,1,11,1,1 ,11,21,11,24,11,23,11,22,11,21,11,2 ,10,21,10,24,10,23,10,22,10,21,10,2 ,2,21,2,24,2,23,2,22,2,21,2,2 ,1,21,1,24,1,23,1,22,1,21,1,2 ,,11,,14,,13,,12,,11,,1 ,1,11,1,14,1,13,1,12,1,11,1,1 ,2,11,1,14,2,13,2,12,2,11,2,1 ,1,11,1,14,1,13,1,12,1,11,1,1 11111 nnnnnnnnnnnnnn nnnnnnnnnnnnnn nnnnnnnn nnnnnnnn nnnnnnnnnnnnnnnnn nnnnnnnnnnnnnn nnnnnnnn nnnnnnnn nn nn nn nn nnnnnnnn nnnnnnnn nn nn ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff ffffff nnnn                                  (27) From (19) we form the ijk elements, resulting in the relation (18), which shows the proof of the theorem. Theorem 4. Approximation formulas for the values determining of the quadruple integrals, obtained by the repeated use of the trapezoid quadrature formula, is calculated as follows:            1 0 1 0 1 0 1 0)( 16 ),,,( n i m j l k s l ijklijkl uzyx D f hhhh dxdydzduuzyxf  , (28)
6. 6. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2032 | Page                       nn nn nn nn n n n n nn nn A A A A A A A A AAAA AAAA ,1 1, 1,1 2, 2,1 1, 1,1 21,22221 11,11211 ... ... ....... ... ... ,                  1 2 . 2 1 2...2 4...4 2 4 2 4 1 2 ........ 4...4442 2...2221 11 nnnn AAA , 1,1, 14... 48... 4 8 4 8 2 4 ........ 48...884 24...442 ,1,                   njAA jjn , 1,...,3,2 ,1,...,3,2 , 4 8 8 16 ... ... 8 16 4 8 ....... 816...168 48...84 ,1,2, 2 4 4 8 ... ... 4 8 2 4 ....... 48...84 24...42 1                                     nj ni AniAA ijini Proof. Disclose the amount (14) and obtain the following relations: nnnnnnn nnnnnnnn nn nn nn ffffff ffffff ffffff ffffff ffffff ,1111,11411311211111 ,1,111,1,114,1,113,1,112,1,111,1,11 ,1131,1131134113311321131 ,1121,1121124112311221121 ,11111111114111311121111 2...222 24...4442 ..................................................................................... 24...4442 24...4442 2...222           ...................................................................................................... 24...4442 48...8884 .......................................................................................... 48...8884 48...8884 24...4442 ,121,12412312212112 ,1121,1124,1123,1122,1121,112 ,1231,1231234123312321231 ,1221,1221224122312221221 ,1211,1211214121312121211 nnnnnnnn nnnnnnnn nn nn nn ffffff ffffff ffffff ffffff ffffff           nnnnnnnnnn nnnnnnnnnn nnnnnn nnnnnn ffff ffff ffff ffff ,1112111 ,111,1,12,111,1,1 2,112,11221211 ,11,11121111 2...2 24...42 ........................................................... 24...42 2...2         nnnnnnn nnnnnnn nn nn fffff fffff fffff fffff ,211,21321221121 ,1211,1213,1212,1211,121 ,2121,212212321222121 ,2111,211211321122111 24...442 48...884 ......................................................................... 48...884 24...44        
7. 7. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2033 | Page ............................................................................................ 48...884 816...16168 ......................................................................... 816...16168 816...16168 48...884 ,221,22322222122 ,1221,1223,1222,1221,122 ,2231,223223322322231 ,2221,222222322222221 ,2211,221221322122211 nnnnnnn nnnnnnn nn nn nn fffff fffff fffff fffff fffff           ...................................................................................................... 816...16168 48...884 ,2,,11,2,,123,122,121,1 ,1,,11,1,,113,112,111,1 nnnnnnnnnnnn nnnnnnnnnnnn fffff fffff     nnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnn nnnnnnnnnnnn nnnnnnnnnnnn fffff fffff fffff fffff ,,1,1,,1,3,,1,2,,1,1,,1, ,1,1,1,1,1,3,1,1,2,1,1,1,1,1, ,2,1,1,2,1,23,1,22,1,21,1, ,1,1,1,1,1,13,1,12,1,11,1, 24...442 48...884 .............................................................................................. 48...884 24...442         nnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnn fffff fffff ,,,11,,,13,,,12,,,11,,,1 ,1,,11,1,,13,1,,12,1,,11,1,,1 88...884 816...16168     nnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnn nnnnnnnnnnnn nnnnnnnnnnnn fffff fffff fffff fffff ,1321 ,1,1,1,3,12,11,1 ,21,2232221 11,1131211 2...22 44...442 ............................................................................ 44...442 2...22         From these relations the formula (20) is formed, which fully demonstrates the proof of the theorem. Now, we calculate the values of integrals by repeated application of quadrature formulas, based on the formula of rectangles. The rectangle formula to calculate the values of the integrals is quite simple. In fact, here i is taken as ],[ 1 ii xx  midpoints. For a uniform grid ),1,( nihhi  we obtained formulas of the form      n i i b a f n ab dxxfJ 1 2/11 )( , (29) where bxax h xff nii  ,), 2 ( 02/1 . Double integral in this case is calculated by the formula     , ,, ),(),( 1 2 1 22 1 1 2/1,2/1 21 1 1 2/12/1 211 2/1 2 2                          n i n j ji n i n j ji A a n j j B b A a A a B b f n bB n aA yxf n bB n aA dxyxf n bB dyyxfdxdxdyyxfJ (30) where ;,, 2 212/1,2/1 yx y j x iji h bB n h aA n h y n h xff           xh - Step on the OX axis, yh y - a step on the OY axis. In the same way we define cubature formulas for calculating the values of the three-and four-fold integrals
8. 8. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 www.ijmer.com 2034 | Page        1 2 3 1 1 1 2/1,2/1,2/1 321 3 ),,( n i n j n k kji A a B b C c f n cC n bB n aA dxdydzzyxfJ , (31) Where        2 , 2 , 2 2/1,2/1,2/1 z k y j x ikji h z h y h xff ,         1 2 3 4 1 1 1 1 2/1,2/1,2/1,2/1 4321 4 ),,,( n i n j n k n l lkji A a B b C c D d f n dD n cC n bB n aA dxdydzduuzyxfJ ; (32) Here        2 , 2 , 2 , 2 2/1,2/1,2/1,2/1 u l z k y j x ilkji h u h z h y h xff . On the basis of approximate formulas for calculating the values of one-, two-, three-and four-fold integrals, respectively, expressed by (21) - (24), the formula for calculating the values of n-fold integrals is given . Theorem 5. Let the ),...,,( 21 nxxxfy  function is defined and continuous in a n-dimensional bounded n integration domain. Then the cubature formula, obtained by repeated application of the rectangles’ formula, has the form           1 1 2 2 21 1 1 1 2/1,...,2/1,2/1 1 2121 ......),...,,(... n i n i n i iii n i i ii nn D n n n f n aA dxdxdxxxxf , (33) where niAax iii ,1],,[  ,        2 ,..., 2 , 2 21 2/1,...,2/1,2/1 2121 n iiiiii h x h x h xff nn . Proof. The proof is given by induction. From (21) - (24) we see that (25) holds for 4;3;2;1n . We assume that (25) is valid for kn  :           1 1 2 2 21 1 1 1 2/1,...,2/1,2/1 1 2121 ......),...,,(... k i k i k i iii k i i ii kk D k k k f k aA dxdxdxxxxf . Now we show fairness at 1 kn : .... 2 , 2 ,..., 2 , 2 ... , 2 ,..., 2 , 2 ... ...),,...,,(... 1 1 2 2 1 1 121 21 1 1 2 2 1 1 1 1 2 2 21 1 1 1 1 2/1,2/1,...,2/1,2/1 1 1 1 1 21 1 1 1 111 11 1 1 1 11 21 1 121121 )(                                                         n i n i n i n i iiii k i i ii k k k iii n i n i n i n i k i i ii k kk n i n i n i kk k iii A a k i i ii kkkk D k k k k kk k k k k k k k k f n aA h x h x h x h xf n aA n aA dxx h x h x h xf n aA dxdxdxdxxxxxf The obtained result proofs the theory completely. III. CONCLUSIONS Thus we have constructed the multi-dimensional cubature formulas to approximate the value of multiple integrals by repeated application of the quadrature formula of trapezoids and rectangles to determine the values of multiple integrals, which are easy to implement on a computer. REFERENCES [1]. Demidovich B.L, Maron I.A. “Foundations of Computational Mathematics”. - Moscow: Nauka, 1960. - 664 p. [2]. Rvachev V.L. “Theory of R-functions and some of its applications”. - Naukova. Dumka, 1982. - 552 p. [3]. Rvachev V.L. “An analytical description of some geometric objects”. Dokl. Kiev, 1963. - 153, № 4. 765-767pp. [4]. Nazirov S.A. “Constructive definition of algorithmic tools for solving three-dimensional problems of mathematical physics in the areas of complex configuration” / Iss. Comput. and glue. Mathematics. Tashkent .Scientific works set. IMIT Uzbek Academy of Sciences, 2009. - No. 121. - 15-44pp. [5]. Kabulov V.K., Faizullaev A., Nazirov S.A. “Al-Khorazmiy algorithm, algorithmization”. - Tashkent: FAN, 2006. -664 p. [6]. Gmurman V.E. “Guide to solving problems in the theory of probability and mathematical statistics”. Manual. book for students of technical colleges. - 3rd ed., Rev. and add. - M.: Higher. School, 1979. [7]. Ermakov S.M. “Monte Carlo methods and related issues”. - Moscow: Nauka, 1971. [8]. Sevastyanov B.A. “The course in the theory of probability and mathematical statistics”. - Moscow: Nauka, 1982. [9]. Mathematics. Great Encyclopedic Dictionary/ Ch. Ed. V. Prokhorov. - Moscow: Great Russian Encyclopedia, 1999. [10]. Gmurman E. “Probability theory and mathematical statistics.” Textbook for technical colleges. 5th ed., revised. and add. - M., Higher. School, 1977.