Relationship formula between nonlinear polynomial equations and the

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1 Relatiohip formula betwee oliear polyomial equatio ad the correpodig Jacobia matrix W. Che Preet mail addre (a a JSPS Potdoctoral Reearch Fellow): Apt.4, Wet 1 t floor, Himawari-o, 316-2, Wakaato-kitaichi, Nagao-city, Nagao-ke, , JAPAN Permaet affiliatio ad mail addre: Dr. We CHEN, P. O. Box , Jiaghu Uiverity of Sciece & Techology, Zhejiag City, Jiagu Provice , P. R. Chia Preet chew@homer.hihu-u.ac.jp Permaet Abtract: Thi paper provide a geeral proof of a relatiohip theorem betwee oliear aalogue polyomial equatio ad the correpodig Jacobia matrix, preeted recetly by the preet author. Thi theorem i alo verified geerally effective for all oliear polyomial algebraic ytem of equatio. A two particular applicatio of thi theorem, we gave a Newto formula without requirig the evaluatio of oliear fuctio vector a well a a imple formula to etimate the relative error of the approximate Jacobia matrix. Fially, ome poible applicatio of thi theorem i oliear ytem aalyi are dicued. 1

2 Key word. Noliear computatio, method of weighted reidual, Jacobia matrix, oliear polyomial equatio. AMS ubject claificatio. 47H17, 65J15 1. Itroductio: Recetly, the preet author [1] proved a relatiohip theorem a tated below Theorem 1. If N ( m ) ( U ) ad J ( m ) ( U ) are defied a oliear umerical aalogue of the m order oliear differetial operator ad it correpodig Jacobia matrix, repectively, the N ( m) 1 U m J ( m) ( ) = ( U ) U i alway atified irrepective if which umerical techique i employed to dicretize. Some igificat applicatio of the above theorem were give i [1]. However, that paper failed to provide a geeral proof of thi theorem icludig fractioal order oliear problem. The objective of thi paper i to verify the effectivee of thi theorem for geeral oliear polyomial fuctio vector. The theorem wa alo ued to derive a Newto formula without requirig the evaluatio of oliear fuctio vector a well a a imple formula to etimate the relative error of the approximate Jacobia matrix. 2

3 2. Geeral proof The method of weighted reidual (MWR) i recogized the origi of almot all popular umerical techique [2, 3]. Coider the differetial equatio of the form ψ {} u f = 0, i (1) with the followig boudary coditio u = u, o u (2a) q = q, o q (2b) where i the outward ormal to the boudary, u q, the upper bar idicate kow value o boudary, ad q = u. More complex boudary coditio ca be eaily icluded but they will ot be coidered here for the ake of brevity. I the MWR, the deired fuctio u i the differetial goverig equatio i firt approximated by a et of liearly idepedet bai fuctio k (x), uch that u = uˆ = c k φ k, (3) k = 1 where c k Õ are the ukow parameter. I the Galerki ad FE method, the bai fuctio are uually choe o a to atify certai give coditio uch a the boudary coditio ad the cotiuity. I additio, thee bai fuctio hould be complete. Subtitutig Eq. (3) ito Eq. (1) produce a error, which i called the reidual, amely, ψ {} û f = R 0. (4) 3

4 Thi error or reidual R i forced to be zero i the average ee by ettig weighted itegral of the reidual equal to zero, amely, Ω [ ψ {} û f] Wd j Ω= RWd j Ω= 0, j=1, 2, É, N, (5) Ω where W j Õ are weightig fuctio. The ue of differet weightig fuctio ad bai fuctio give rie to differet umerical techique uch that the Galerki, Leat quare, fiite elemet, boudary elemet, momet, pectral method, fiite differece ad collocatio method. Thi paper place it emphai o the oliear computatio. Let u coider (1+)- order oliear operator of geeral form: pu ( )[ ru ( )] + Lu ( )= f, (6) where p(u), r(u) ad L(u) are liear differetial operator, f deote the cotat, ad ay real umber. The cheme of weighted reidual approximatio Eq. (6) i give by [ ] = Wi pu ( ˆ) [ ru ( ˆ) ] + Lu ( ˆ) fwd qˆ qwd uˆ u d i Ω ( ) i Γ ( ) Γ, j=1,2,é,n. (7) 2 1 Ω Γ Γ Subtitutig equatio (3) ito the above equatio (7) yield ψ()= c ckp( φk) ckr φk ckl φk f Wd i k ( ) + ( ) Ω Ω = 1 k = 1 k = 1 Γ2 φk u W q Wd i Γ + ckφk u i d Γ = ck 0 Γ k = 1 1 k = 1 j=1, 2, É, N. (8) The above equatio (8) ca be retated a ( ) ()= + ()+ = 1+ ψ c Dc N c b 0, (9) where c i a vector compried of the ukow c k, D i the liear coefficiet matrix, ad 4

5 N (1+) (c) mea the (1+)-order oliear vector term. For example, it i the quadratic oliear fuctio vector whe =1 ad 3/2 order oliear oe whe =1/2. b repreet the cotact vector. The Jacobia matrix of oliear vector N (1+) (c) i give by J ( 1+ ) ij ( ) 1+ N = c j () c = p( φj ) ckr( φk) + Ω k = 1 1 r( j ) ckr( k) ckp k Wd i k = 1 ( ) φ φ φ Ω k = 1 i, j=1, É, N. (10) By uig equatio (8), (9) ad (10), we ca eaily etablih N 1 c J ( ) ()= (). 1 c c (11) + ( 1+ ) 1+ The above equatio (11) i geerally effective for umerical aalogue of both iteger ad fractioal order oliear differetial or itegral operator. A metioed earlier, early all popular umerical techique uch a the fiite elemet, boudary elemet, fiite differece, Galerki, pectral, leat quare, momet ad collocatio method ad their variat ca be derived from the weighted reidual methodology. The oly differece amog a variety of umerical method lie i the ue of differet weightig ad bai fuctio. From the foregoig deductio, it i oted that equatio (11) ca be obtaied o matter what weightig ad bai fuctio we ue i the method of weighted reidual. Therefore, it i traightforward that the formula (11) i geerally effective for all umerical method which ca be derived from the weighted reidual method. 5

6 I what follow, we will provide aother traightforward proof of theorem 1 for oliear algebraic equatio with explicit expreio. At firt, the cocept of the Hadamard product ad power are defied a i [4, 5]. Defiitio 2.1. Let matrice A=[a ij ] ad B=[b ij ] C N M, the Hadamard product of matrice i defied a A B= [a ij b ij ] C N M. where C N M deote the et of N M real matrice. Defiitio 2.2. If matrix A=[a ij ] C N M, the A q =[a q ij ] C N M i defied a the Hadamard power of matrix A, where q i a real umber. Epecially, if a ij 0, A (-1) =[1/a ij ] C N M i defied a the Hadamard ivere of matrix A. A 0 =11 i defied a the Hadamard uit matrix i which all elemet are equal to uity. Theorem 2.1: lettig A, B ad C C N M, the 1. A B=B A (12a) 2. k(a B)=(kA) B, where k i a calar. (12b) 3. (A+B) C=A C+B C (12c) Coider the oliear pu ( )[ ru ( )] i equatio (6) agai, the correpodig umerical aalogue by uig a poit-wie approximatio techique ca be expreed [ ] = ( ) i ( ) ( ) { } ( ) o { } = i ( ) ( r ) = ( 1 A ) p ( ) + pu ru pu ru U o AU N U, i=1,2,é,n, (13) 6

7 where i idexe the umber of dicrete poit; A p ad A r repreet the coefficiet matrice of operator p(u) ad r(u), repectively, depedet o pecified umerical dicretizatio cheme. The Hadamard product i exploited to expre oliear dicretizatio term i the above equatio. It i oted that we ue the deired fuctio value vector U here itead of the ukow parameter vector C i equatio (11). I fact, both are equivalet. The above explicit matrix formulatio (13) i obtaied i a traightforward ad ituitive way. For more detail ee [6, 7]. The poit-wie approximatig umerical techique iclude the fiite differece, fiite volume, collocatio method ad their variat uch a differetial quadrature ad peudopectral method. I additio, the umerical techique baed o radial bai fuctio ca alo expre their aalogue of oliear differetial operator i the Hadamard product form. O the other had, it i worth treig that all explicit oliear polyomial equatio which may ot be origiated from the umerical approximatio ca be expreed i the Hadamard product form. The SJT product wa itroduced by the preet author [6, 7] to efficietly compute aalytical olutio of the Jacobia derivative matrix. Defiitio 2.3. If matrix A=[a ij ] C N M, vector U={u j } C N 1, the A U=[a ij u i ] C N M i defied a the potmultiplyig SJT product of matrix A ad vector U, where repreet the SJT product. If M=1, A B=A B. 7

8 Defiitio 2.4. If matrix A=[a ij ] C N M, vector V={v j } C M 1, the V T A=[a ij v j ] C N M i defied a the SJT premultiplyig product of matrix A ad vector V. Coiderig the Hadamard oliear expreio (13), we have J { }= ( ) + ( ) ( ) ( ) ( U)= ( AU p ) o ( AU r ) Ap AU r Ar AU r AU p. (14) U ( 1+ ) 1 Formula (14) produce the aalytical Jacobia matrix through imple algebraic computatio. The SJT premultiplyig product i related to the Jacobia matrix of the oliear formulatio uch a ( ) ( )= = ( ) k k k qu AU N U, (15) where q i a liear operator, ad m i ay real umber, ad A i the umerical coefficiet matrix of operator q( ). The correpodig Jacobia matrix i give by J o T ( 1) ( U)= { AU }= ( ku ) A. (16) U ( k ) k k It i oberved from the above formula (15) ad (16) that the Jacobia matrix of the oliear algebraic equatio of the Hadamard product ad power expreio ca be calculated by uig the SJT product i the chai rule imilar to thoe i differetiatio of a calar fuctio. The computatioal effort of a SJT product i oly 2 calar multiplicatio. The fiite differece method i ofte employed to calculate the approximate olutio of the Jacobia matrix ad alo require O( 2 ) calar multiplicatio. I fact, the SJT product approach require 2 ad 5 2 le multiplicatio operatio tha oe ad two 8

9 order fiite differece, repectively. Moreover, the SJT product produce the aalytic olutio of the Jacobia matrix. I cotrat, the approximate Jacobia matrix yielded by the fiite differece method affect the accuracy ad covergece rate of the Newto- Rapho method, epecially for highly oliear problem. The efficiecy ad accuracy of the SJT product approach were umerically demotrated i [6-8]. We otice the followig fact that the SJT product i cloely related with the ordiary product of matrice, amely, if matrix A=[a ij ] C N M, vector U={u i } C N 1, the the potmultiplyig SJT product of matrix A ad vector U atifie A U=diag{u 1, u 2,..., u N }A, (17) where matrix diag{u 1, u 2,..., u N } C N N i a diagoal matrix. Similarly, for the SJT premultiplyig product, we have V T A = A diag{v 1, v 2,..., v M }, (18) where vector V={v j } C M 1. By uig equatio (13), (14), (17) ad (18), we ca eaily cotitute N 1 U J ( ) ( )= ( 1 U ) U. (19) + ( 1+ ) 1+ Similarly, by uig equatio (15), (16), (17) ad (18), we have N 1 U k J k ( )= ( U ) U. (20) ( k) ( ) The Hadamard expreio a well a the previou MWR approximatio of oliear 9

10 algebraic equatio ecompae all implicit ad explicit polyomial oliear ytem of equatio. Accordig to equatio (11), (19) ad (20), we ca geeralize theorem 1 to ay oliear polyomial fuctio vector a tated below: Theorem 2.2. If N ( m ) ( U ) ad J ( m ) ( U ) are repectively defied a the m-order oliear polyomial fuctio vector ad it correpodig Jacobia matrix, the N ( m) 1 U m J ( m) ( ) = ( U ) U i alway atified. The oliear polyomial fuctio vector here idicate the ytem i which all ukow variable are icluded oly i polyomial fuctio. A a example of theorem 2.2, let u coider the typical quadratic oliear term ϕ( X)= XAX (21) ofte ecoutered i cotrol egieerig, where X ad A are rectagular matrice. It Jacobia matrix i give by [5] J ϕ X T I AX AX I = = ( )+ ( ) It i eaily validated. (22) ϕ( X)= 1 v JX, 2 (23) where v X i a vector by tackig the row of matrix X. The above reult i i agreemet with theorem

11 3. Newto iterative formula without evaluatio of oliear fuctio vector The Newto method i of vital importace i oliear algebraic computatio of variou umerical algebraic aalogoue equatio. The time-coumig effort iclude the repeated evaluatio of the oliear fuctio vector, Jacobia matrix ad it ivere [9]. It i recogized that the repeated umerical itegratio of force vector, i other word, oliear fuctio vector, i oe of factor which ifluece the efficiecy of ome popular umerical techique [10]. Although the computatioal burde i thi apect doe ot take a crucial part of oliear olutio, the relative cot i abolutely otrivial i olutio of a large oliear ytem epecially for the fiite elemet cheme. By uig theorem 2.2, the preet author [11] derived a Newto iteratio formula for oliear equatio (9) c c 1 J c ( Dc ( 1 ) b ), (24) 1 1 = + + ( ) + + k + 1 k 1 k k where u with upercript k+1 ad k mea repectively the iteratio olutio at the (k+1)-th ad k-th tep. It i oted that the iterative formula (24) doe ot require the evaluatio of the oliear fuctio vector yet i i fact equivalet to the tadard Newto iterative formula. The implicity ad effectivee of the formula (24) wa verified i the olutio of the tatic Navier-Stoke equatio of the drive cavity problem. The preet iterative formula may be epecially ueful to improve efficiecy of variou modified Newto method, i which the repeated evaluatio of the Jacobia ad it ivere ha bee reduced or avoided. The cot i the repeated calculatio of oliear fuctio vector occupie a larger part of the total computatioal effort [9]. 11

12 4. Error etimator of approximate Jacobia The repeated calculatio of the Jacobia matrix i required i geeral oliear computatio. Fuctio differece i oe of covetioal umerical method ofte ued i practice for thi tak. However, the method uffer from poible iaccuracy, particularly if the problem i highly oliear. Therefore, it i practically importat to detect the accuracy of the approximate Jaocbia matrix by the fiite differece method. By mea of theorem 2.2, we here give a imple error etimator of the Jacobia matrix. Coider oliear equatio (6), we have err = ψ() c -J ˆ () c c ψ() c, (25) where ĴU ( ) i the approximate Jacobia matrix of ψ(c). ψ () c i defied ()= + ( + ) ( ) ( )+ 1+ ψ c Dc 1 N c b, (26) which i differet from ψ(c) of equatio (9) i that the oliear vector N (1+) (c) i multiplied by the oliear order umber 1+. The above formula (25) provide a practically igificat approach to examie the relative deviatio error betwee the approximate ad exact Jacobia matrice by vector orm. 4. Some remark The kow vo Karma equatio of geometrically oliear bedig, bucklig ad vibratio of plate ad hell are a mixed quadratic ad cubic oliear differetial ytem of equatio. The reultig umerical dicretizatio of vo Karma equatio by 12

13 ay umerical techique ca be writte a ( ) ( ) 2 3 Lu + N ( u)+ N ( u)= f, (27) where u i the deired diplacemet vector, L i the liear coefficiet matrix, N (2) (u) ad N (3) (u) mea repectively the quadratic ad cubic oliear vector term, ad f repreet the force vector. By uig theorem 2.2, we have 1 ( 2) 1 ( 3) Lu + J ( u) u + J ( u) u = f, (28) 2 3 where J (2) (u) ad J (3) (u) deote the Jacobia matrice of N (2) (u) ad N (3) (u), repectively. Therefore, we ca write equatio (27) i liear like form Kuu ( ) = f, (29) where K(u) i defiite phyical tiffe matrix of ytem rather the commo taget tiffe matrix ( ) ( ) = + ( )+ ( ) K L J 2 u J 3 u (30) reultig from a Newto liearizatio procedure. The preet K(u) ca reflect better the origial phyical characteritic of oliear ytem. The above aalyi how that theorem 2.2 ca brig a direct aalyi of oliear ytem without uig liearizatio procedure uch a the Newto method. I fact, all complex problem i phyic ad egieerig uch a hock, chaotic ad olito wave ivolve oliear ytem of goverig equatio. By applyig theorem 2.2 to umerical dicretizatio of thee equatio, we ca get explicit ytem matrix. It i expected that pecial tructure feature of ytem matrix have cloe relatio to the ytem defiite 13

14 behavior. We ugget a itace (ample) aalyi approach i which the aalyi of ytem matrix i doe at certai time ad pace itace. The tadard liear algebraic aalyi uch eigevalue, coditio umber ad circulat tructure, etc. ca be ued to detect uch ytem matrix. Thi may lead to a eible algebraic udertadig of mathematical phyic feature of the oliear ytem. Theorem 2.2 wa applied i [1] to tability aalyi of oliear iitial value problem ad direct olutio of oliear algebraic equatio by uig the liear iterative method uch a the Jacobi, Gau-Seidel ad SOR method without the liearizatio procedure. I particular, it hould be poited out that by applyig theorem 2.2, we ca get the explicit ytem matrix of oliear equatio ad thu ca aalyze tability behavior of oliear iitial value problem by the exitig effective approach for liear varyigcoefficiet problem [12]. Fially, it i worth treig that a very large cla of real-world oliear problem ca be modeled or umerically dicretized polyomial algebraic ytem of equatio. Theorem 2.2 i i geeral applicable for all thee problem. Therefore, thi work i of practical igificace i broad phyical ad egieerig area. Referece: 1. Che W., Jacobia matrix: a bridge betwee liear ad oliear polyomial-oly problem, (ubmitted). Alo publihed i Computig Reearch Repoitory (CoRR): 14

15 (Abtract ad full text ca be foud there). 2. Filayo, B. A., The Method of Weighted Reidual ad Variatioal Priciple. Academic Pre, New York (1972). 3. Lapidu L. & Pider G.F., Numerical Solutio of Partial Differetial Equatio i Sciece ad Egieerig. Joh Wiley & So Ic., New York (1982). 4. Hor R.A., The Hadamard product, i Matrix Theory ad Applicatio, Serie: Proc. of Sympoia i applied mathematic, V. 40, , Joho C.R. Ed., America Mathematical Society, Providece, Rhode Ilad (1990). 5. Ni G., Ordiary Matrix Theory ad Method (i Chiee). Shaghai Sci. & Tech. Pre, Shaghai (1984). 6. Che W. & Zhog T., The tudy o oliear computatio of the DQ ad DC method, Numer. Method for P. D. E., 13, (1997). 7. Che W., Differetial Quadrature Method ad It Applicatio i Egieerig Applyig Special Matrix Product to Noliear Computatio. Ph.D. diertatio, Shaghai Jiao Tog Uiverity, Shaghai, Dec Che W., Shu C., He S., The applicatio of pecial matrix product to differetial quadrature olutio of geometrically oliear bedig of orthotropic rectagular palte, Computer & Structure (i pre). 9. Ortega J.M., & Rheiboldt W.C., Iterative Solutio of Noliear Equatio i Several Variable. Academic Pre, New York (1970). 10. Liu W.K., Belychko T., Og J.S., Law S.E., The ue of tabilizatio matrice i oliear aalyi, i "Iovative Method for Noliear Problem" (Liu W.K., 15

16 Belychko T., Park P.C., Ed), , Pieridge Pre, Swaea, U. K (1984). 11. Che W., A Newto method without evaluatio of oliear fuctio value, (ubmitted). 12. Gutafo B., Krei H., Oliger J., Time depedet problem ad differece method. Joh Wiley & So (1995). 16