* This work has been supported by NIMP, University of Plovdiv under contract No MU-1.In this paper we present methods, theoretical basis of algorithms, and computer tools, which we 
have used for constructing our Web-based equation solver

Iliev, Anton

Kyurkchiev, Nikolay

Todorov, Todor

Bulgarian Digital Mathematics Library at IMI-BAS

International Journal "Information Theories & Applications" Vol.10  468 WEB-BASED SIMULTANEOUS EQUATION SOLVER12 A. Iliev, N. Kyurkchiev, T. Todorov  Abstract: In this paper we present methods, theoretical basis of algorithms, and computer tools, which we have used for constructing our Web-based equation solver. Keywords: automatic equation solver, Web interface, simultaneous extraction of all roots, simultaneous methods, parallel processors, algebraic equations 2000 Mathematics Subject Classification: 68Q22, 65Y05  Introduction Many industrial and optimization tasks lead to the problem of finding all roots of (1) or arbitrary their part. One of branches for solving polynomial equations is parallel methods for simultaneous determination of all roots. With our solver automatically we can search simultaneously all or only one part of all roots of (1) (real, complex, lying in given area).   Iteration methods Let us consider algebraic polynomial   ( )( )( )( )A x x a x a x ax a x a x a x ann n nnn n= + + + + == + + + + +− −−11221 2 1...... ... . (1) The approximations of the k th iteration to zeroes x x xw1 2, , ... ,  of (1) are denoted by x x xk kwk1 2[ ] [ ] [ ], , ... ,  and their multiplicities by    α α α1 2, ,..., w  1 1 11≤ ≤ − + = =⎛⎝⎜⎞⎠⎟=∑α αi iiwn w i w n, , , . Classical methods for individual searching of multiple roots of (1) can be written in this general way   ( )x x F x a a ai w kikikiki n[ ] [ ] [ ] , , , , ... , ,, , , , , ... .+ = −= =11 21 0 1 2α (2) Other approach is given by methods for simultaneous extraction of all multiple roots and we can write them as    ( )x x F x x x a a ai w kikik k kwkw n[ ] [ ] [ ] [ ] [ ], , ... , , , , ... , , , , ... , ,, , , , , ... .+ = −= =11 2 1 2 1 21 0 1 2α α α (3) Methods (3) are steadier and also they have a larger domain of convergence with comparison with methods (2). This is the main reason because these methods are object of detailed investigations in last twenty years. For natural reasons we want to find simultaneously only one part of all roots of (1). Namely, we want to find simultaneously ( )p w≤  different roots with multiplicities p21 α,...,α,α                                                            12 This work has been supported by NIMP, University of Plovdiv under contract No MU-1. International Journal "Information Theories & Applications" Vol.10  469( )p,1j,1α1pmn,mnα...αα jp21 =≥≥+−−−=+++ . In our Web-based equation solver we use [Iliev, Kyurkchiev, 2002a, 2002b, 2003], [Kyurkchiev, Iliev, 2002] type methods, which in common can be written as ( )x x F x x x a a ai p kikik k kpkp n[ ] [ ] [ ] [ ] [ ], , ... , , , , ... , , , , ... , ,, , , , , ... .+ = −= =11 2 1 2 1 21 0 1 2α α α (4)  Theoretical basis of our methods Polynomial (1) can be presented in this way ( )A x Q x T xn n m m= − ( ) ( ) , (5) where Q xn m− ( )  is polynomial, whose zeroes we seek and ( )T xm  is polynomial, whose zeroes we ignore. Respectively   ( )Q x x b x b x b x bT x x c x c x c x cn mn m n m n msn m sn mmm m mpm pm−− − − − − − −−− − −= + + + + + += + + + + + +( ) ... ...... ... .11221122  Between the coefficients of polynomial (1) and the coefficients of polynomials (5) there exist the following relations 11221111222111......bcbcbcbcabcbcabcallllll −−− +++++=++=+= ...  ..........1123121112211mnmnmmmmmmmmmmmbcabbcbcbcabcbcbcbca−+−−+−−−=++++=+++++= We want to find simultaneously ( )p w≤  different roots with multiplicities    pααα ,...,, 21  ( )pjpmnmn jp ,1,11,...21 =≥≥+−−−=+++ αααα  and we set ( )( ) .......,......)(][][2][21][1][][][2][21][11][][kmpmkpmkmkmkmkmnsmnksmnkmnkmnpjkjkmncxcxcxcxxTbxbxbxbxxxxQ j++++++=++++++==−=−−−−−−−−−−−=− ∏ α (6) From (6) it follows   ∑=−=pjkjjk xb1][][1 α    ( ) ( )∑∑ ∑=−= +=−+⎥⎦⎤⎢⎣⎡=pjkjjjpjpjsksskjjk xxxb12][11 1][][][2 21αααα    ...   ( ) ( )b xn mk n m jkjpj−−== − ∏[ ] [ ]11α. International Journal "Information Theories & Applications" Vol.10  470 Combinative algorithms can be used for finding coefficientsb b bk k n mk1 2[ ] [ ] [ ], ,... , − . We define c j mjk[ ] , ,= 1  using formulae   ( )c a bc a b a b b a b c bc F a a a b b b a b c bk kk k k k k k kmkmk kmkm mkjkm jkjm1 1 12 2 2 1 1 1 2 2 1 11 2 1 211[ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ]...( , ,..., , , ,..., ) .= −= − − − = − −= = − − −=−∑ For simultaneous searching of roots of Q xn m− ( )  from (5) we give the following iteration algorithm   ( )( ) ( ) ( )( )x xA xA x A xx xT xT xi p kikik i n ikn ikn ik jikjkj j ipmkikmkik[ ] [ ][ ][ ] [ ][ ] [ ],[ ] [ ][ ] [ ],, , , , ,... .+= ≠= −′ − − +′⎡⎣⎢⎢⎤⎦⎥⎥= =∑111 012αα (7) When m n= −1 and α α α1 2 1= = = =... n  method (7) coincides with the classical Obreshkoff’s method [Obreshkoff, 1963] for individual searching of one simple zero and if p w=  (7) is method for finding all roots of (1). Theorem. Let d x xi j i jdefmin≠ − , c > 0  and 1 0> >q  be real numbers such that   d c> 2 ,    ( ) ( ) ( )[ ]2 22 2 2 1 1 2c n m d c gP yP Gi i− − − + + <−α α/ , i p= 1, , where P P G g1 2 1, , ,  and y  are appropriate positive constants. If initial approximations x x xp1020 0[ ] [ ] [ ], , ...,  to the real roots of (1) satisfy inequalities x x cq i pi i[ ] , ,0 1− < =  then for every k N∈  the inequalities   x x cq i pikik[ ] , ,− < =3 1  are satisfied. From this theorem [Iliev, Kyurkchiev, 2003] it follows that iteration method (7) holds cubic convergence.  Localization technique for automatic determination of multiplicity of the roots and their initial approximations For applying in practice in Obreshkoff’s monograph [Obreshkoff, 1963] is given that Fujiwara prove that the circle with centre origin and radius R a ap n n p np= ≤ ≤ −2 11max // in the complex plane contains all zeroes of polynomial (1). We will use presentation   ( ) ( )( ) ( )A x x x x x x x e e en n n i i i n= − − − =1 2 1 2 1 2... ... ...ρ ρ ρ ϕ ϕ ϕ , where ρp  are modulo of complex numbers x x p np− =, ,1  and ϕp p n, ,= 1  are their arguments.  After one pass of contour with appropriate step in counter-clockwise direction every arguments of the roots in domain will be changed with 2π  and every arguments out of contour will not be changed. Using Cauchy approach [Obreshkoff, 1963] if the argument change of ( )z A xn= , where x  with appropriate step in International Journal "Information Theories & Applications" Vol.10  471counter-clockwise direction are different points from passed contour, is 2πs , [ ]s n∈ 1, , it follows that in explored domain there are exactly s  roots.  After first pass of localization of the roots with presented here algorithm in different domains we explore these domains, which have more than one root. For every such area arises the question whether or not in it there are localized one or several roots, which are sufficiently close. Confirmation or rejection of found multiplicity in "near" neighborhood could be made with Schröder’s method [Schröder, 1870]. It will have quadratic convergence only when the multiplicity of the root is exact. Exactly we have in mind ε  discernible roots (zeroes). If the roots are not multiple we will repeat Cauchy algorithm procedure. This is because we need fine localization of roots only in these areas, which contain more than one different roots.  Program description The main modules are realized on Pascal program language [Krushkov, Iliev, 2002], using Delphi 5 environment. We realized specialized program units for complex numbers, multiple precision, input polynomial analyzer, specialized methods for finding a part of all roots [Iliev, Kyurkchiev, 2002a, 2002b, 2003], [Kyurkchiev, Iliev, 2002]. Also we have used dynamic structures for the economy of memory and for faster program code optimization. For Web-based input-output form interface for users is developed.  Conclusion With this equation solver we try to give practical cover of theoretical improvements from classic, advanced techniques and iteration algorithms from last several years. It can be used simultaneously from many users with Internet connection without influence of distance.  Bibliography [Iliev, Kyurkchiev, 2002a] A. Iliev, N. Kyurkchiev. On a Generalization of Weierstrass-Dochev Method for Simultaneous Extraction of only a Part of All Roots of Algebraic Polynomials: Part I. C. R. Acad. Bulg. Sci., 2002, 55, № 2, 23–26. [Iliev, Kyurkchiev, 2002b] A. Iliev, N. Kyurkchiev. Some Methods for Simultaneous Extraction of only a Part of all Roots of Algebraic Polynomials Part II. C. R. Acad. Bulg. Sci., 2002, 55, № 9, 17–22. [Kyurkchiev, Iliev, 2002] N. Kyurkchiev, A. Iliev. On the R-order of Convergence of a Family of Methods for Simultaneous Extraction of Part of All Roots of Algebraic Polynomials. BIT, 2002, 42, № 4, 879–885. [Iliev, Kyurkchiev, 2003] A. Iliev, N. Kyurkchiev. On a Generalization of Obreshkoff-Ehrlich Method for Simultaneous Inclusion of only Part of All Multiple Roots of Algebraic Polynomials. Comput. Math. Math. Phys., 2003, 43, № 4, 507–513. [Obreshkoff, 1963] N. Obreshkoff. On the numerical solving equations. Ann. of the Sofia University, 1963, 56, 73–83. [Obreshkoff, 1963] N. Obreshkoff. Polynomial zeroes. Acad. Bulg. Sci., Sofia, 1963. [Schröder, 1870] E. Schröder. Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 1870, 317–365. [Krushkov, Iliev, 2002] H. Krushkov, A. Iliev. Practical programming guide in Pascal, Parts I and II. Anima, 2002.  Authors information Anton Iliev - University of Plovdiv, Faculty of Mathematics and Informatics, Department of Computer Science, 24, Tsar Assen Str., Plovdiv-4000, Bulgaria; e-mail:  aii@pu.acad.bg, http://anton.iliev.tripod.com Nikolay Kyurkchiev - Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 8, Sofia-1113, Bulgaria; e-mail: nkyurk@math.bas.bg Todor Todorov - University of Plovdiv "Paisii Hilendarski", Faculty of Mathematics and Informatics, Department of Computer Science, 24, Tsar Assen Str., Plovdiv-4000, Bulgaria; e-mail:  tod_bul@hotmail.com 

Web-based Simultaneous Equation Solver

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Web-based Simultaneous Equation Solver

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