ed.

Tdbuteto S. Preslc

Papers Celebratlns hl. 6. IIrthclay

Instltut SANU

35

73 2001.

CIP -

012

TRIBUTE to S. Presic tribute to S. Presic: Papers celebrating his 65th birthday /

Aleksandar Krapez, ed. - Beograd : Matematicki institut SANU, 2001 (Zemun : Akademska stampa). - 122 str. ; 24

Tiraz 300. ISBN 86-80593-31-1

1. Gl. stv. nasl. 2. Krapez, Aleksandar 929:51

(1933- ) -

1 I i

CONTRIBUTION AND INFLUENCE OF S.B. PRESIC

NUMERICAL FACTORIZATION OF POLYNOMIALS

Gradimir V. Milovanovic

ABSTRACT. This paper devoted to contributions S.B. Presic numerical fac-torization algebraic polynomials, well to of his work this subject. Beside general factorization of polynomials, we consideI" important special

and point out some accelerated iterative

1. Introduction

The numerical factorization of algebraic polynomials is very important math-ematical subject. There are several methods for it in the literature, beginning witll the well-known methods of Bairstow [2] and of Lin [19-20]. Many of tllem are quadratically convergent, but most require sufficiently close starting va!ues for factorization. In their survey paper, Householder and Stewart [14] mentioned also the method of Graeffe and the qd algorithm, tllough they are not primarily for this assignment. number of methods related to an algorithm proposed

Sebastiao Silva [38]. Some generalizations of this algorithm were given Housellolder [11] in 1971 (see also [12], [41], [6]). In addition we also method of Samelson [36] from 1959, wllich generalizes the Bauer-Samelson iteration [3]. In his paper Samelson noted that his method is related to Bairsto"r's method. Taking monic algebraic polynomial over the field of complex numbers, with zeros Zl, Z2, .. · , Zn, i.e.,

(1.1) P(z) = zn + PIZn-l + ... + Pn-lZ + = (z - Zk), k=l

1991 Subject Classification. Primary Secondary Supported Ministry of Science and of Serbia, grant

47

48 MILOVANOVIC

Samelson seeks its factorization two factors

and v(z) = (z - Zm+1)(z - Zm+2)'" (Z - Zn)'

Let and q monic polynomials of degree and - approximating and v repectively. Then his quadratically convergent iterative procedure defines improved approximations and q* the formula

(1.2) p*q + q*p = + pq.

If and q are relatively prim, then and q* are uniquely defined (1.2) Samel-son's iteration was discovered independently Stewart who characterized pq* as the linear combination of q, zq, . .. ,zm-lq that is divisible Householder and Stewart gave the exact connection between these characterizations (see al-so and for another derivation of Samelson's method and the corresponding error bounds for the iteration, as well as the paper of Schr6der for connection with Newton's method).

In 1966 and 1968 S.B. Presic inspired only some results of D. Markovic gave iterative method for numerical factorization of algebraic polYllomials s (2 :::; s :::; factors. The purpose of the presellt paper is to show contributions of S. Presic, as well as to POillt out an of S. PresiC's work to this subject. The paper is orgallized as follows. In Section 2 we explain S. PresiC's approach to numerical factorizatioll of polynomials and give all example 2 - 2 factorization of polYllomial of fourth degree. Sections 3 and 4 are dedicated to an 1 - 1 - ... - 1 factorization alld some accelerated iterative formulas, respectively.

Later, in 1969 Dvorcuk [9] considered factorization into quadratic factors, alld ill 1971 Grau [10] used Newtoll-type of approximation for simultaneously improving complete set of approximate factors for givell polynomial. Recelltly, Carstellsell [4] alld Carstensen and Sakurai [5] gave some generalizations of this method.

Here, we mention also that ill the last period papers have lished factorization of polYllomials over finite fields, Oll factorization methods for multivariable polYllomials, as well as factorization of matrix polynomials.

2. S. PresiC's approach to numerical factorization

Let mOllic algebraic polYllomial over the field of complex llumbers givell (1.1) and let it expressed ill factorized form

(2.1)

where Av(z) are monic polYllomials of degree

(2.2) ...

Av(z) = 2: aviZn ... - i , i=O

(v=I,2, ... ,s),

NUMERlCAL FACTORlZATION OF POLYNOMIALS 49

s and = The case 8 = 2 is mentioned in the previous sectio,n.

11=1 Assuming that zeros of (1.1) are simple, S.B. Presic gave iterative method

for numerical determination such factorization, so-called - - •.. - n s fac-torization, in which successive iterated monic factors

(2.3) = L znv-i, = 1 (v=1,2, ... ,8) i=O

are determined from the relation

Aik+1) ... + Aik) ... + ... + Aik) ...

- (8 - ... Aik) =

i.e.,

Taking the coefficients alli of polynomials (2.2) as coordinates of an n-dimen-sional vector

and (coefficients of iterated factors (2.3» as coordinates of the corresponding also n-dimensional vector a(k), S. Presic observed that (2.4) implies system of linear equations of the form

(2.5)

where is matrix depending only a(k), and is n-dimensional vector depending also a(k) and coeffi.cients of the polynomial (1.1), =

Further, he concluded that there exists neighbourhood V of such that (2.5) expressed in the following form

(2.6) (k = 0,1, ... ; a(k) V),

where V -+ V is enough times differentiable operator (in Frechet sense). Practically, S. Presic proved that = and is zero operator, so that

Thus, S. Presic's result sumarized as:

= lim a(k)). k-++oo

50 MILOVANOVIC

1.1. 18 ne1ghbourhood V 80 that arbitrary V, the 1terative (2.6) quadrat1cally converge8 to

Thus, (v = 1,2, ... ,s),

give the factorization (2.1). In his paper [35], S. Presic derived formulas for 2 - 2 - 2 factorization of

polynomial of degree 6. Here, as an illustration, we give simpler case wllen P(z) = Z4 + Plz3 + P2Z2 + + and when we seek its 2 - 2 factorization, with

A 1(z) = z2 + all Z + A 2(z) = z2 + a21Z + In that case the system (2.5) becomes

+ - b(k) - 1 ,

(k+l) _ b(k) - 2 ,

(k) (k+1) + (k) (k+1) I (k) (k+1) + (k) (k+1) _ b(k) - ,

+ (k) (k+1) _ b(k) - 4 ,

where

(k) . = (k) _ (k) (k)

- + , (k) _ (k) (k) (k) (k)

- + + , (k) _ (k) (k)

- + .

Solving this system we obtain an iterative procedure of the form (2.6). This case (s = 2) reduces to Samelson's iteration.

Using the previous idea polynomial factorization, Petric and S.B. Presic [32] treated problem of simultaneous determination of solutions of the system of algebraic equations

== + + + 2D1x + + Fr = == + + + 2D2x + + =

3. Factorization 1 - 1 - ... - 1

In the case s = i.e., nv = 1 (v = 1,2, ... the factors are linear

Av(z) = z + avO = z - Zv (v = 1,2, ...

and (2.4) reduces to

(z - - ... (z - L z - Zv (k) - + 1 == P(z). (k+1) )

v=l Z - Zv

NUMERICAL FACTORIZATION OF POLYNOMIALS

Then, the scalar form of (2.6) can obtained easily as

(3.1) (k»)

Z(k+1) = z(k) _ Zv v v (k) (k)

(zv - Zj )

j:f.v

(v = 1,2, ... , k = 0,1, ... ).

51

in this important S. PresiC's factorization approach leads to the Weier-strass' formulas (3.1) [44]), which were not well-known in that period.

were obtained several times in various ways authors. Weierstrass used them in constructive proof of fundamental theorem of algebra. In book numerical solution of algebraic equations from 1960, written French mathematician Durand [8], chapter was dedicated to iterative methods for simultaneous finding polynomial zeros, where the author obtained formulas (3.1) in implicit form. It that Bulgarial1 mathematician Docev [7] was the first who used these forl1lulas il1 their original form for numerical calculation who proved their quadratic converegence.

Introducing Q(z) = (z-zy»), formulas (3.1) represented in the form

(Newtonian type) (k»)

(3.2) z(k+l) = z(k) - (v = 1,2, ... , k = 1, ... ). v v QI(zv)

Beside the Q(z) we consider also polynomials Rv(z)

Q(z) (k) Rv(z) = (k) = (z - Zj ) Z - Zv

j:f.v

(v = 1,2, ... ,

Their forms are Q(z) = zn - <71 Zn- 1 + <72Zn-2 - ... + (-l)n<7n ,

R () n-1 (v) n-2 + (v) + ( 1)n-1 (v) v Z = Z - <71 Z <72 Z - • •• - <7n _ 1 ,

where <71, <72, .•. , <7n are elementary symmetric of Z1, Z2, . .. ,Zn [23, Sectiol1 1.3.1]). For the sake of simplicity, we omit the upper

(k+l) th t t· S··1 1 (v) (v) (v) 1 h lor zv we zv. ar <71 , <72 , ••. , <7n- 1 are functions that do 110t zv. It is to that QI(zv) = Rv(zv) (v 1 = {1, 2, ... , ). In the note [43], which was our first paper in mathematics inspired only the S,;" Presic paper [35], we showed: 11 zeros 01 Q(z) are simple, then the inverse matrix 01

[ 1

] (2)

<71 <71 <71 (3.3) w=

(1) (2) <7n_ 1 <7n- 1 <7n_ 1 /

-::.. ./

52 MILOVANOVIC

is given

(3.4) D n-1 D n-2 2Z2 - 2Z2

W- 1 =

...

with D" = l/Q'(z,,) (v 1).

(-1)n- 1D2

(_1)n-1 Dn

,

corresponding S. PresiC's form (2.6), i.e., vector form of (3.2) written as

(3.5) (k = 0,1, ... ),

where T(z) = z - e(z) and

'= [} e(z) = Q(z) = (z - Zj). )=1

Taking the system of Viete's formulas for polynomial P(z), given (1.1), j(z) = where the i-th coordinate in the vector j(z) is equal to O"i+( -1)i-1pi (i = 1,2, ... and applying the known iterative procedure of Newton-Kantorovic,

(3.5) (k = 0,1, ... ),

in of(ler to solve the previous system of nonlinear equations, we obtain (3.5). Here, matrix is exactly given (3.3) and its inverse (3.4). It seems that

Kerner [16] was the first who observed this fact. proof was sliglltly different fram ours.

Regarding to the iterative method (3.2), in 1980 Dirk Laurie stated the .',

following problem: 1/ I: Z" = prove that ,,=1

(3.6)

It is nice property of the method (3.2) and it was known earlier (see Docev [7]). Relation (3.6) holds regardless of the value of I: Z". We gave now proof of

,,=1 that as application of the residue method and it was published in the book [26, Indeed, since z" = Z" - P(z,,)/Q'(z,,), v = 1,2, ... we have

(3.7)

NUMERICAL FACTORIZATION OF POLYNOM1ALS 55

No doubt the S. PresiC's work this area is very important and that it has great influence the development of this field in our country. In the last thirty years several mathematicians in Serbia, those from the University of Nis and University of Novi Sad, have been very active this field. For the refer-ences for instance, [28] and [29].

References [1] G. Alefeld and Herzberger, convergence speed algorithms the simul-

taneous approximation polynomial roots, S1AM Numer. 2 (1974), 237-243.' [2] L. Bairstow, Investigations relating to the stability the aeroplane, Rep. & 154,

Advisory Committee for Aeronautics, 1914. [3) F.L. Bauer and Samelson, Polynomkerne und Iterationsver/ahren, Math. Z. 67 (1957),

93-98. [4) Carstensen, method /actorization polynomials, S1AM

Numer. 29 (1992), 601-613. [5) Carstensen and Sakurai, Simultaneous /actorization polynomial rational

proximation, Comput. Math. 61 (1995), 165-178. [6] S.P. Chung, Generaliztion and acceleration algorithm Sebasiiiio Silva and its

duals, Numer. Math. 25 (1976), 365-377. [7] Videoizmenen metod Newton za edinovremenno priblizitel'no presmyatane

vsichki koreni dadeno algebrichno uravnenie, Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 5 (2) (1962), 136-139.

[8] Durand, Solution Numerique des Equations Algebraique (tome 1), et Compagnie, Paris, 1960.

[9] Factorization polynomial into quadratic /actors Newton method, Mat. 14 (1969), 54-80.

[10] Grau, The simultaneaus Newton improvement complete set approximate /actors polynomial, S1AM Numer. 8 (1971), 425-438.

[11] A.S. Househo!der, Generalizations algorithm Sebastiiio Silva, Numer. Math. 13 (1971), 38-46.

[12] A.S. Householder, Postscript to: "Generalizations algorithm Sebastiiio Silva", Numer. Math. 20 (1972/73), 205-207.

[13] A.S. Householder and G.\V. Stewart, Comments "Some iterations /actoring polyno-mials", Numer. Math. 13 (1969), 470-471.

[14) A.S. Householder and G.W. Stewart, The numerical/actorization polynomial, S1AM Rev. 13 (1971), 38-46.

[15] method obtaining iterative /ormulas higher order, Mat. Vesnik 9 (1972), 365-369.

[16] 1.0. Kerner, GesamtschrittverJahren zur Berechnung der Nullstellen von Polynommen, Numer. 8 (1966), 290-294.

[17) N. Kjurkcblev, Some remarks Weierstrass root-finding method, C.R. Acad. Bulgare Sci. 46 (8) (1993), 17-20.

[18] D.P. Laurie, The Fibonacci Quarterly 18 (1980), 190. [19] Shin-nge Lin, method successive approximations evaluating the real and complex roots

and higher-order equations, Math. 1nst. Tech. 20 (1941), 231-242. [20) Shin-nge Lin, method finding roots algebraic equations, Math. 1llst.

22 (1943), 60-77. [21] D. Markovic, /aktorizaciji Vesnik Drus. Mat. Fiz. NRS, 8 (1955),

53-58. [22) G.V. Milovanovic, method to accelerate iterative Banach Univ.

grad. Electrotehn. Fak. Ser. Mat. Fiz. No 461 - No 495 (1974), 67-71.

56

[23] G.V. Milovanovic, D.S. Mitrinovic, Th.M. Rassias, Polynomials: Extremal Prob-Inequalities, Zeros, World Scientific, Singapore - New - London -Hong Kong,

1994. [24] G.V. and M.S. Petkovic, the convergence order modified method

simultaneous finding polynomial zeros, Cornputing 30 (1983), 171-178. [25] G.V. and M.S. Petkovic, computational the iterative methods

the simultaneous approximation polynomial zeros, Math. Software 12 (1986), 295-306.

[26] D.S. Mitrinovic and J.D. The Cauchy Method Residues - Theory and Applica-tions, Mathernatics and Applications (East Series), 9, D. Reidel Publishing

Dordrecht - Boston - Lancaster, 1984. [27] Ortega and W.C. Rheinboldt, Iterative Solution Nonlinear Equations in Several

Variables, Acadernic New York, 1970. [28] M.S. Petkovic, Iterative Methods Simultaneous Inc!usion Polynomia! Zeros, Springer-

1989. [29] M.S. Petkovic, D. Herceg, S. Point Estimation Theory and its Applications, Il1stitute

of Mathernatics, Novi Sad, 1997. [30] M.S. Petkovic al1d G.V. note some improvements the simultaneous

methods polynomial zeros, Cornput. Appl. Math. 9 (1983), 65-69. [31] M.S. Petkovic, G.V. L.V. Stefanovic, some higher-order methods the si-

multaneous approximation multip!e polynomial zeros, Cornput. Math. Appl. 9 (1) (1986), 951-962.

[32] Petric and S.B. algorithm the solution 2 2 system nonlinear algebraic equations, 1l1st. Math. (Beograd) (N.S.) 12 (26) (1971),85-94.

[33] Jedan iterativni postupak za odret!ivanje k korena polinoma, Doktorska disertaci-Beograd, 1971.

[34] S.B. procede iteratiJ la Jactorisation polynomes, R. Acad. Sci. 262 (1966), 862-863.

[35] S.B. Jedan iterativni postupak za Jaktorizaciju polinoma, Mat. Vesnik 5 (20) (1968),

[36] Sarnelson, Factorisierung von Polynomes durch Junktionale Iteration, Akad. Wiss. Math. Natur. Abh. 95 (1959), 1-26.

[37] Factorization polynomials generalized Newton procedures, Constructive Aspects of the Fundarnental Theorem of Algebra Dejon and eds.), John Wiley, New York, 1969, 295-320.

[38] Sebastiiio Silva, Sur methode d'approximation semblable d celle de Portugal. Math. 2 (1941), 271-279.

[39] G.W. Stewart, Some iterations Jactoring polynomial, Math. 13 (1969), 458-470.

[40] G.W. Stewart, Samelson's iteration Jactoring polynomials, Numer. Math. 15 (1970), 306-314.

[41] G.W. Stewart, the convergence Sebastiiio Silva's method finding zero polynomial, SIAM Rev. 12 (1970), 458-460.

[42] Behaviour the sequences around multiple zeros generated some simulta-neous methods solving algebrac equations, Tech. Rep. Inf. Numer. Anal. 4-2 (1983), 1-6

[43] D.D. and G.V. application Newton's method to simultaneous de-termination zeros polynomial, Ul1iv. Beograd. Electrotehn. Fak. Mat. Fiz. No 412 - No 460 (1973), 175-177.

[44] Weierstrass, Neuer Beweis des Satzes, dass jede ganze rational Veriinder-lichen dargestellt werden kann als Product aus Linearen darselben Veriider-lichen, In: Gesammelten Werke, 3 (1903), Johnson, New York, 1967, 251-269.

Contents

Foreword ..... ............ ... ............... ... ....... .... ...... .......... 5

Invited papers

Zarko Mijajlovic: the scientific work of Slavisa Presic

Kapetanovic: Work of Slavisa Presic in artifical inteligence

Jovan D. Keckic: Contribution of Professor S. Presic to functional equations

Dragic Bankovic:

9

15

20

S. PresiC's work in reproductive equations ............................... 33

Ljubomir Ciric: S. PreSic's type generalization of contraction mapping principle 43

Gradimir V. Milovanovic: Contribution and influence of S. Presic to numerical factorization of polynomials ............................................................. 47

Vera V. Kovacevic-Contribution of S. Presic to the field of applied mathematics

Slobodan Vujic: Educational and tutorial work of Professor Slavisa Presic

Research papers

Zoran Petric: Cut elimination in category-like sequent systems

Miodrag RaSkovic, Zoran Ognjanovic, Vladimir Petrovic and Uros Majstorovic:

57

69

73

automated theorem prover for the probability logic LPP* ............... 79

Smile Markovski, Ana Sokolova, and Lidija Ilieva: the functional equation = in the variety of groupoids .... 84

Krapez: genera1ization of rectangular loops 89

Stojan Bogdanovic, Miroslav Ciric and Nebojsa Stevanovic: Semigroups in which any proper ideal is semilattice indecomposable 94 Kosta Dosen:

note the set-theoretic representation of arbitrary lattices 99

S. Jovanovic: Note the methods for numerical solution of equations ................... 102

Appendices

1 Stavrev: R.aznice - Varia ........................................................... 109

Bibliography of Slavisa Presic

List of Ph. D. Students of S. Presic

115

119

List of Reviewers ................................................ :......... 120