Journl of Physics: Conference Series PAPER OPEN ACCESS Modified Crout s method for n decomposition of n intervl mtrix To cite this rticle: T Nirml nd K Gnesn 2018 J. Phys.: Conf. Ser. 1000 012134 View the rticle online for updtes nd enhncements. Relted content - Improving performnce of multidimensionl Prticle-In-Cell codes for modelling of medium pressure plsm Z Pekárek, M hut nd R Hrch - ACA for RF coil simultion Shumin Wng, Jcco A de Zwrt nd Jeff H Duyn - est-squres reverse time migrtion in frequency domin using the djoint-stte method Horn Ren, Huzhong Wng nd Shengchng Chen This content ws downloded from IP ddress 148.251.232.83 on 19/08/2018 t 11:59
Modified Crout s method for n decomposition of n intervl mtrix Nirml T 1 nd Gnesn K 2 1,2 Deprtment of Mthemtics, Fculty of Engineering & Technology, SRM Institute of Science nd Technology, Kttnkulthur 603 203 nirml.ts@ktr.srmuniv.c.in, gnesn.k@ktr.srmuniv.c.in Abstrct. In this pper, we propose n lgorithm for computing decomposition of n intervl mtrix using modified Crout s method bsed on generlized intervl rithmetic on intervl numbers. Numericl exmples re lso provided to show the efficiency of the proposed lgorithm. Keywords: Generlized intervl rithmetic, intervl numbers, modified Crout s method AMS Subject Clssifiction: 15A09, 65F05, 65G30 1. Introduction Mtrix decomposition is the process of trnsformtion of the given mtrix into product of lower tringulr mtrix nd n upper tringulr mtrix. Mtrix decomposition is fundmentl concept in liner lgebr nd pplied sttistics which hs both scientific nd engineering significnce. Computtionl convenience nd nlytic simplicity re the two mjor spects of mtrix decomposition. In the rel world it is not esy for most of the mtrix computtion to be clculted in n optiml explicit wy such s mtrix inversion, mtrix determinnt, solving liner systems nd lest squre fitting. Thus to convert difficult mtrix computtion problem into severl esier problems such s solving tringulr or digonl systems will gretly simplify the clcultions. Also decomposition methods provide n importnt tool for nlyzing the numericl stbility of system.in literture there re severl uthors such s Alefeld nd Herzberger[1], Hnsen nd Smith[7], Neumier[12], Rohn[17]nd Gnesn nd Veermni[5]etc hve discussed intervl mtrices. Alexndre Goldsztejn nd Gilles Chbert[2] hve investigted generlized intervl decomposition nd its pplictions. Zhili Zho, Wei i, Chongyng Deng nd Huping Wng [19]proposed new intervl Cholesky decomposition method bsed on generlized intervls.in this pper we propose n lgorithm for modified Crout s method for n decomposition of n intervl mtrix. The rest of this pper is orgnized s follows: In section 2, we recll the bsic concepts nd the rnking of generlized intervl numbers nd their rithmetic opertions. In section 3, we recll the bsic concepts of intervl mtrices nd rithmetic opertions on intervl mtrices. In section 4, we propose n lgorithm for Content from this work my be used under the terms of the Cretive Commons Attribution 3.0 licence. Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl cittion nd DOI. Published under licence by td 1
modified Crout s method for n decomposition of n intervl mtrices. In section 5, Numericl exmples re lso provided to show the efficiency of the proposed lgorithm. 2. Preliminries et IR { [ 1, 2 ]: 12 nd 1,2R} be the set of ll proper intervls nd IR { [ 1, 2 ]:12 nd 1,2R} be the set of ll improper intervls on the rel line R. If 1 2, then [, ] is rel number (or degenerte intervl). We shll use the terms intervl nd intervl number interchngebly. The mid-point nd width (or hlf-width) of n intervl number [ 1, 2 ] re defined s 2 m() 1 1 nd w() 2. We denote the set of 2 2 generlized intervls (proper nd improper) by DIR IR {[ 1, 2]: 1,2R}. The set of generlized intervls D is group with respect to ddition nd multipliction opertions of zero free intervls, while mintining the inclusion monotonicity. The dul is n importnt mondic opertor proposed by Kucher [8] tht reverses the end-points of the intervls expresses n element to element symmetry between proper nd improper intervls in D. For [, 1 2 ] D, its dul is defined by dul () = dul[ 1, 2]= [ 2, 1 ]. The opposite of n intervl [, 1 2 ] is opp[, ] [, ] which is the dditive inverse of [ 1, 2] nd 1 2 1 2 the multiplictive inverse of [ 1, 2], provided 0 [, 1 2]. Tht is, 1 2 1 2 1 2 2 1 1 1 2 2 1 1, is 1 2 +( dul ) dul ( ) [, ] dul( [, ]) [, ] [, ], [ 0,0] 1 1 nd 1, 2] [ 1, 2] dul [ 1 l( ) dul([ 1, 2]) [ 2, 1] 1 1 [, ],, [1,1]. 1 2 1 2 1 2 1 2 2.1. Compring intervl numbers et be n extended order reltion between the intervl numbers [ 1, 2],b [b1, b2 ] in D, then for m() m(b), we construct premise ( b) which implies tht is inferior to b (or b is superior to ). An cceptbility function A :DD [0, ) is defined s: (m(b) (b) m()) A (,b) A ( b), where w(b) w() 0. (w(b) w()) 2
A my be interpreted s the grde of cceptbility of the first intervl number to be inferior to the second intervl number b. 2.2. A new intervl rithmetic Gnesn nd Veermni[5] proposed new intervl rithmetic on IR. We extend these rithmetic opertions to the set of generlized intervl numbers D nd incorporting the concept of dul. For =[, ], b=[b,b ] D,,,, we define 1 2 1 2 nd for where b = [m() m(b) - k, m() m(b) + k], b) -α β- (m() k = min (m() m(b) m(b) - α, β - (m() m(b), nd re the end points of the intervl under the existing intervl rithmetic. In prticulr for ny two =[, 1 2 ], b=[b 1,b 2 ] D, (i)addition + 2) - (b 1+ 1) k= (b b =[, 2 1 2]+[b 1 b]=[{m()+m(b)}-k, 2 {m()+m(b) }+k], where b=[, ]+[b,b ] =[{m()+m(b)}-k, m(b)}- {m()+m(b) m(b)}+ k], where k =. 2 (ii) Subtrction + 2) - (b 1+ 1) k= (b b =[, 2 1 2 ]-[b 1 b] 2 [{m()-m(b)}-k, {m()-m(b) }+k], where b=[, ]-[b,b ] [{m()-m(b)}-k, m(b)}- {m()-m(b) m(b)}+ k], where k =. 2 Also if =b i.e.[ 1, 2]= [b 1, b 2 ], then b= dul() ) =[ 1, ] 2 dul([ 1, 2]) [, ] [, ] [ -, ] = [0,0] (iii) Multipliction.b=[ ].[b, m(b)}- 1, 2 1 b 2]= {(m()( b)}-k, {(m()( m(b)}+k, where k min{(m() m(b)) -, (m()m(b))} m(b))} nd β=mxb,b,b,b α=min b,b,b,b (iv) Division: 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 1 2 1 2 2 1 1 1 2 2 1 1 1 1 1 2-1 1 2-1 1 = = -k, =min, +k, where k [, 1 2] m() m() 2 1 + 2 1 1 + 2 nd 0 [, ]. Alsoif =bi.e.[, ]=[b,b], then 1 2 1 2 3
1 1 1 1 1 2, 2] [ 1, 2] [ 1, 2],, [1,1] b dul() [ 1 ) dul([ 1, 2) [ 2, 1] 1 2 1 2 λ 1, λ 2, for λ 0 From iii, it is cler tht λ = λ 2, λ 1, for λ < 0 3. Min results An intervl mtrix A is mtrix whose elements re intervl numbers. An intervl mtrix A will be 11... 1n written s A......... ( ij ), where ech m,1 j n 1 i ij [ ij, ] (or) A [A, A] ij m1... mn for some A, A stisfying A A. We use D m n to denote the set of ll (m n) intervl mtrices. The midpoint of n intervl mtrix A is the mtrix of midpoints of its intervl elements defined s m( 11 )... m( 1n ) m(a).......... The width of n intervl mtrix A is the mtrix of widths of its m( m1 )... m( mn ) w( 11)... w( 1n ) intervl elements defined s w(a)......... w(... m1) w( mn ) 0... 0 to denote the null mtrix......... 0... 0 1... 0 use I to denote the identity mtrix... 1... 0... 1 which is lwys nonnegtive. We use O 0... 0 nd O to denote the null intervl mtrix.......... Also we 0... 0 nd I to denote the identity intervl mtrix 1... 0... 1.... If m(a) m(b), then the intervl mtrices A nd B re sid to be equivlent nd is 0... 1 denoted by A B. In prticulr if m(a) m(b) nd w(a) w(b), then A B. If m(a) O, then we sy tht A is zero intervl mtrix nd is denoted by O. In prticulr if m(a) O nd 4
[0,0]... [0,0] w(a) O,then A.......... Also if m(a) O nd w(a) O, then [0,0]... [0,0] 0... 0 A......... O 0... 0 intervl mtrix. If m(a) I then we sy tht A is n identity intervl mtrix nd is denoted by [1,1]... [0,0] prticulr if m(a) I nd w(a) O,then A... [1,1].... Also, if m(a) I nd [0,0]... [1, 1] 1... 0 w(a) O, then... 1... I. 0... 1 3.1 Arithmetic Opertions on Intervl Mtrices We define rithmetic opertions on intervl mtrices s follows: If A, B D m n, x D n nd D, then (i) A ( ij ) 1 i m, 1 j n (ii) (A B) ( ij b ij ) 1 i m, 1 j n. If A O (i.e. A is not equivlent to O ), then A is sid to be non-zero ( b ij ) 1 i m, 1 j n, if (iii) (A A B B) ij A dul(a) 0 0, if A B (iv) AB n ik b kj k 1 1 i m, 1 j n (v) j1 n Ax ij x j 1 i m I. In 4. Algorithm for modified Crout s method for decomposition of n intervl mtrix The following steps to write the given intervl mtrix A into lower tringulr intervl mtrix nd upper tringulr intervl mtrix. For simplicity, consider four-by-four squre intervl mtrix for Crout s method s 5
A 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 [ 11, 11] [12, 12] [13, 13] [14, 14] [ 21, 21] [22, 22] [23, 23] [24, 24] 31, 31] [32, [ 32] [33, 33] [34 34] [ 41, 41] [42, 42] [43, 43] [44, 44] 11 0 0 0 11 12 13 14 21 22 0 0 0 1 23 24 31 32 33 0 0 0 0 1 34 41 42 43 44 0 0 0 1 Step: 1 Mke the pivot element s unity, i. e. 11 1. For this purpose, if required, use elementry mtrix row opertions.the reduced intervl mtrix is 11 1 12 13 14 22 23 24 A 21 31 32 33 34 Step: 2 41 42 43 44 The first column of the mtrix ij below 11 1 is the sme s tht of the mtrix A, i.e., i1 i1, for i1,2,3,4. 1 0 0 0 22 0 0 21 33 0 31 32 43 44 41 42 Step: 3 The first row of the mtrix ij fter 11 1 is the sme s tht of the mtrix A, i.e., 1j 1j, for j1,2,3,4. 1 12 13 14 23 0 0 1 24 0 0 1 34 0 0 0 1 6
Step: 4 The digonl elements of the mtrix ij re kept s unity i.e., ii 1, fori 1,2,3,4. 1 12 13 14 0 0 1 23 24 0 0 1 34 0 0 0 1 Tht is, we decomposen intervl mtrix A into the equivlent intervl tringulr form A s A 0 0 0 11 1 12 13 14 1 0 0 0 11 12 13 14 0 0 21 22 23 24 21 22 0 1 23 24 31 32 33 34 31 32 33 0 0 0 1 34 44 0 0 0 1 41 42 43 44 41 42 43 44. Step: 5 Compute the unknowns in mtrices nd using A. 5. Numericl exmples Exmple: 5.1 Consider n exmple discussed by Zhili Zho, Wei i, Chongyng Deng nd HupingWng [19] [2,5] A [1,2] [, ] [2,5] [6,15] By pplying the proposed lgorithm, divide the first row of the given intervl mtrix A by the element 11, i.e. [1, 2] [2,5] A [1, 2] [1,1] [0.66735,3.9996] [1, 2] [2,5] [6,15] [2,5] [6,15] The new intervl mtrix A cn be expressed s lower nd upper tringulr intervl mtrices nd, 11 1 12 11 1 0 11 1 12 1 21 22 21 22 0 7
Tht is [1,1] [0.66735,3.9996] [1,1] [0, 0] [1,1] [0.66735,3.9996] [2,5] [6,15] [2,5] [0,0] [1,1] 22 0 [1,1] [0.66735,3.9996] [1,1] [0.66735,3.9996] [2,5] [6,15] [2,5] [2,5][0.66735,3.9996] 22 Equting we get, [2,5][0.66735,3.9996] [6,15] 22 [1.3347,14.9999] [6,15] 22 22 [6,15] [1.3347,14.9999] [ 8.9999,13.6653] 22 [ Therefore, [1,1] [0.66735,3.9996] [1,1] [0,0] [1,1] [0.66735,3.9996] A [,. [2,5] [6,15] [2,5] [ 8.9999,13.6653] [0, 0] [1,1] We rrive [0.66735,3.9996] [1,1] [,] [0,0] [1,1] [,]. [2,5] [ 8.9999,13.6653] nd [0,0] [1,1] Exmple: 5. 2 Consider nother exmple discussed by Zhili Zho, Wei i, Chongyng Dengnd Huping Wng [19] [2, 2] A [1, [, 4]. [2,2] [5,5] Applying the proposed method, we get [1,1] [ 0.4, 2] [1,1] [0, 0] [1,1] [ 0.4, 2] A [,]. [2,2] [5,5] [2,2] [1,5.8] [0,0] [1,1] We hve [1,1] [,] [0,0]. [2, 2] [1,5.8] nd [1,1] [,] [ 0.4, 2] [0, 0] [1,1] Exmple: 5. 3 Consider nother exmple discussed by Alexndre Goldsztejn nd Gilles Chbert[2] [9,11] [ 1,1] [ 1,1] A [ 11,11] [8,12] [ 2, 2] [ 11,11] [ 12,12] [7,13] Divide the first row by 11, the new intervl mtrix cn be expressed s 8
[9,11] [ 1,1] [ 1,1] [9,11] [9,11] [9,11] A [ 11,11] [8,12] [ 2, 2] [ 11,11] [ 12,12] [7,13] [ 1,1] [ 1,1] [1,1] [9,11] [9,11] A [ 11,11] [8,12] [ 2,2] [ 11,11] [ 12,12] [7,13] The new intervl mtrix A cn be expressed s lower nd upper tringulr intervl mtrix nd, 11 1 12 13 11 1 0 0 11 1 12 13 A 21 22 23 21 22 0 0 1 23 0 1 31 32 33 31 32 33 0 [1,1] [0,0] [0,0] [1,1] [0,0] [0,0] [1,1] [0,0] [0,0] [ 11,11] [8,12] [ 2, 2] [ 11,11] [0,0] [1,1] 22 [0,0] 0] 0] 1] 23 [ 11,11] [ 12,12] [7,13] [ 11,11] [0,0] [1,1] 32 33 [0,0] 0] [1,1] [0,0] [0,0] [1,1] [0,0] [0,0] [ 11,11] [8,12] [ 2, 2] [ 11,11] 22 22 23. [ 11,11] [ 12,12] [7,13] [ 11,11] 33 32 32 23 By equting, we get 22 [8,12]; 32 [ 12,12]; 22 23 [ 2, 2]; 32 23 33 [7,13]; 22 [8,12]; 23 [0, 0]; 32 [ 12,12]; 12]; 33 [7,13]. The intervl mtrix A cn be expressed s lower nd upper tringulr intervl mtrices nds [1,1] [0,0] [0,0] [1,1] [0,0] [0,0] [ 11,11] 1111] 11] [812] [8,12] [00] [0,0] 0] nd [00] [0,0] [11] [1,1] [00] [0,0]. [ 11,11] [ 12,12] [7,13] [0,0] [0,0] [1,1] Exmple: 5. 4 Consider nother exmple discussed by Alexndre Goldsztejn nd Gilles Chbert[2] [2,3] [1,1] [0,0] A [1,1] 1] [2, 3] [1,1] [0,0] [1,1] [2,3] By pplying the proposed lgorithm, the given intervl mtrix A cn be expressed s lower nd upper tringulr intervl mtrices nds 9
[1,1] [0,0] [0,0] [1,1] [0.33,0.47] [0,0] [1,1] [1.53, 2.67] [0,0] nd [0,0] [1,1] [0.37,0.58] 1]. [0,0] [1,1] [1.42, 2.63] [0,0] [0,0] [1,1] 6. Conclusion By using new set of rithmetic opertions on generlized intervl numbers, we hve proposed n lgorithm for modified Crout s method for decomposing n intervl mtrix A into lower tringulr intervl mtrix nd n upper tringulr intervl mtrix.for simplicity, we hve considered the lgorithm for (44) intervl mtrices lone. But this lgorithm is true for ny (n n) intervl mtrices. It is to be noted tht for decomposing ny (nn) intervl mtrix A into lower tringulr intervl mtrix nd n upper tringulr intervl mtrix, the existing decomposition methods requires evlution of 2 totl of n number of unknown elements of the mtrices nd. But the proposed modified Crout s 2 method requires evlution of only (n 1) number of such unknowns.this difference becomes significnt for intervl mtrices of higher orders. Hence by pplying the proposed method, significnt mount of computtionl time nd efforts cn be reduced. REFERENCES [1] Alefeld G nd Herzberger J, 1983 Introduction to Intervl Computtions, Acdemic Press, New York. [2] Alexndre Goldsztejn nd Gilles Chbert 2007 A Generlized Intervl decomposition for the solution of Intervl iner Systems, Springer- Verlg Berlin Heidelberg, pp 312-319. [3] AtnuSengupt nd Tpn Kumr Pl, 2000 Theory nd Methodology: On compring intervl numbers, Europen Journl of Opertionl Reserch, vol 127, pp 28 43. [4] Burden R nd Fires J D, 2005 Numericl Anlysis Seventh Edition, Brooks/Cole. [5] Gnesn K nd Veermni P, 2005 On Arithmetic Opertions of Intervl Numbers, Interntionl Journl of ncertinty, Fuzziness nd Knowledge Bsed Systems, 13(6), pp 619 631. [6] Gnesn K, 2007 On some properties of intervl mtrices Interntionl Journl of Computtionl nd Mthemticl Sciences, 1(1), pp 25-32. [7] Hnsen E.R nd Smith R.R, 1967 Intervl rithmetic in mtrix computtions, Prt 2, SIAM, Journl of Numericl Anlysis, 4(1), pp 1-9. [8] Kucher E 1980 Intervl nlysis in the extended intervl spce IR, Computing, Suppl, 2, pp 33-49. [9] uc Julin, Michel Kieffer, Olivier Didrit nd Eric Wlter, 2001 Applied Intervl Anlysis, SpringerVerlg, ondon. [10] Moor R.E, 1979 Methods nd Applictions of Intervl Anlysis, SIAM, Phildelphi. [11] Moore R E, Kerfottnd R B nd Cloud M, 2009 Introduction to Intervl Anlysis, SIAM, Phildelphi. [12] Neumier A, 1990 Intervl Methods for Systems of Equtions, Cmbridge niversitypress, Cmbridge. [13] Nirml T, Dtt D, Kushwh H S nd Gnesn K, 2011 Inverse intervl mtrix: A newpproch, Applied Mthemticl Sciences, 5(13), pp 607-624. 10
[14] Nirml T, Dtt D, Kushwh H S nd Gnesn K, 2013 The determinnt of n intervl mtrix using Gussin Elimintion Method Interntionl Journl of Pure nd Applied Mthemtics, 88(1), pp 15-34. [15] Nirml T, Dtt D, Kushwh H S nd Gnesn K,2016 fctoriztion of n invertible intervl mtrix using Gussin elimintion method, Globl Journl of Pure nd Applied Mthemtics, 12(1), pp 414-423. [16] Rfique M, 2015 The Solution of System of iner Equtions by n Improved - Decomposition Method, Americn Journl of Scientific nd Industril Reserch, Science Huβ. [17] Rohn J, 1993 Intervl mtrices: singulrity nd releigen vlues, SIAM, Journl of Mtrix Anlysis nd Applictions, 14(1), pp 82-91. [18] Rohn J, 1993 Inverse intervl mtrix, SIAM, Journl of Numericl Anlysis, vol.3, pp 864 870. [19] Zhili Zho, Wei i, Chongyng Deng nd Huping Wng, 2012 A Generlized CholeskyDecomposition for Intervl Mtrix, Vol 479-481, pp 825-828. 11