Newton Method In The Context Of Quaternion Analysis

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CMATCentro de Matemática da Universidade do MinhoUniversidade do MinhoCampus de Gualtar 4710-057 Braga Portugalwww.cmat.uminho.ptUniversidade do MinhoEscola de CiênciasCentro de MatemáticaNewton Method in the Context of QuaternionAnalysisM.I. FalcãoDepartamento de Matemática e Aplicações and CMATUniversidade do Minho, PortugalInformationAbstractKeywords:Newton method; quaternions; radial derivative.In this paper we propose a version ofNewton method for finding zeros ofa quaternion function of a quaternionvariable, based on the concept of quaternion radial derivative. Several numerical examples involving elementary functions are presented.Original publication:Applied Mathematics and Computation 236, (2014), 458-470DOI: 1IntroductionSince 1928 R. Fueter, one of the founders of quaternion analysis ([1, 2]), tried todevelop a function theory to generalize the theory of holomorphic functions ofone complex variable, by considering quaternion valued functions of a quaternion variable. Nowadays, this well known and developed theory is recognizedas a powerful tool for modeling and solving problems in both theoretical andapplied mathematics. For a survey on quaternion analysis and a list of references we refer to the book [3]; a historical perspective of the subject and severalapplications can be found in [4].In this work we revisit the classical Newton method for finding roots (orzeros) of a complex function and propose a quaternion analogue, based on theconcept and properties of quaternion radially holomorphic functions. We showthat for a certain class of functions (including simple polynomials and otherelementary functions) this method produces the same sequence as the classicalNewton method for vector valued functions. In this way, we can obtain, withless computational effort, local quadratic convergence for a class of quaternionfunctions.

This idea was already considered by Janovská and Opfer in [5], where theauthors formally adapted, for the first time, Newton method for finding roots ofHamilton quaternions, by considering the quaternion equation xn a 0. Morerecently, Kalantari in [6], using algebraic-combinatorial arguments, proposed aNewton method for finding roots of special quaternion polynomials. Workingin the framework of quaternion analysis, we can provide a motivation for thetechniques used in those works and simultaneously extend the applicability ofthe method.The paper is organized as follows. In Section 2 we introduce the basicnotations and the results that are needed for our work in Section 3.Section 3 contains the main results of the paper. Here, by making use of thetheory given in Section 2, and after establishing new properties on the radialderivative of a special class of functions, we propose a Newton method in theframework of quaternion analysis.Finally, in Section 4 several numerical examples illustrating the applicabilityof the aforementioned methods are presented.2Quaternion AnalysisWe start by first recalling some basic results concerning Hamilton quaternionalgebra H, which can be found in classic books on this subject. For resultsconcerning quaternion analysis we refer to [7, 8, 3].Let {1, i, j, k} be an orthonormal basis of the Euclidean vector space R4 witha product given according to the multiplication rulesi2 j2 k2 1, ij ji k.This non-commutative product generates the well known algebra of realquaternions H. The real vector space R4 will be embedded in H by identifyingthe element x (x0 , x1 , x2 , x3 ) R4 (or the column vector in R4 1 , x (x0 x1 x2 x3 )T ) with the element x x0 x1 i x2 j x3 k H. Throughoutthis paper, we will also use the symbol x to represent an element in R4 , wheneverwe need to distinguish the structure of H from R4 .The real or scalar part of a quaternion x x0 x1 i x2 j x3 k is denoted bySc x and is equal to x0 , the vector part of x is x : x1 i x2 j x3 k and thereforex can be written as x x0 x. The conjugate of x is x : x0 x1 i x2 j x3 k x0 x. The mapping x 7 x is called conjugation and has the property xy y x,for all x, y H. The norm x of x is defined by x 2 xx xx and coincideswith the corresponding Euclidean norm of x, as a vector in R4 . It follows that xy x y and each non-zero x H has an inverse given by x 1 x x 2 .Moreover, x 1 x 1 .In this work we are going to consider also the representation of the quaternionx x0 x1 i x2 j x3 k by means of the real matrix in R4 4 x0 x1 x2 x3 x1 x0 x3 x2 .Lx : (1) x2 x3x0 x1 x3 x2 x1x02

This representation is called matrix left representation of x and can be associatedwith the product of quaternionsxy (x0 x1 i x2 j x3 k)(y0 y1 i y2 j y3 k) (x0 y0 x1 y1 x2 y2 x3 y3 ) (x1 y0 x0 y1 x3 y2 x2 y3 )i (x2 y0 x3 y1 x0 y2 x1 y3 )j (x3 y0 x2 y1 x1 y2 x0 y3 )k,(2)through the identificationz xy z Lx y,(3)where y is the (column) vector in R4 corresponding to the quaternion y.Any arbitrary nonreal quaternion x can also be written in the so-calledcomplex-like form(4)x x0 ω(x) x ,whereω(x) : x x belongs to the unit sphere in R3 . Since ω(x) ω(x) ω(x), it followsimmediately that ω(x)2 ω(x)ω(x) ω(x) 2 1. In other words, wecan consider that ω behaves like the imaginary unit and therefore the complexlike form (4) is similar to the complex form a ib. We use the conventionω(x) : 0, for real quaternions x. The following properties play an importantrole in the present work.Proposition 1 If x x0 x1 i x2 j x3 k and y y0 y1 i y2 j y3 k arequaternions, the following statements are equivalent:(i) xy yx;(ii) x1 y2 x2 y1andx1 y3 x3 y1andx2 y3 x3 y2 ;(iii) ω(x)ω(y) ω(y)ω(x);(iv) ω(x) ω(y).Proof: The equivalence between the first three statements follows at once fromthe multiplication rules (cf. (2)). If ω(x) ω(y), clearly ω(x)ω(y) 1 ω(y)ω(x) and we conclude that (iv) implies (iii).Now, we prove that (ii) implies (iv) for the case of nonreal quaternions. Ify is nonreal, we can assume that, for example, y1 6 0. Using (ii) one canwrite x2 x1 yy21 and x3 x1 yy31 and this, in turn, implies that x x1 i 1 x2 j x3 k (y1 i y2 j y3 k) xy11 . This means that x y x y1 and the resultω(x) x x y x1 y1 y x1 y1 ω(y) follows. Corollary 1 If x and y are quaternions such that ω(x) and ω(y) commute then:(i) xy yx;(ii) xy 1 y 1 x.3

Proof: It is sufficient to note that ω(y 1 ) ω(y) ω(y) and apply the resultsof Proposition 1. In what follows, we consider complex-like functions f : Ω R4 H R4of the formf (x) u(x0 , x ) ω(x)v(x0 , x ) ,(5)where x 6 0 and u and v are real valued functions. Continuity, differentiabilityor integrability are defined coordinate-wisely.R. Fueter proposed a generalization of complex analyticity to the quaternioncase ([1, 2]) which leads to close analogues of several important results from classical complex function theory. In this framework the analogue of holomorphicfunctions, usually refer to as monogenic functions, is obtained as the set of nullsolutions of a generalized Cauchy-Riemann system. In addition, Sudbery, in [9],defined a regular quaternion valued function by the existence of its quaternionderivative. Unfortunately, contrary to the complex case, neither the canonicalquaternion variable x nor any of its nonnegative integer powers xn are monogenic functions and therefore, the quaternion derivative of xn does not exist.This means, among other things, that based on the concept of monogenicityone can not generalize Newton method to finding roots of simple quaternionpolynomials, i.e. polynomials with quaternion coefficients located on only oneside of the powers.In order to introduce a notion of regular function that fits our purpose, wefollow [8, 3, 10] and adopt the following usual definition.Definition 1 Let f be a complex-like function of the form (5) and let h h0 ω(x)hr H. Such function f is called radially holomorphic or radiallyregular in Ω iflim (f (x h) f (x)) h 1h 0exists. In the case of existence, this limit is called the radial derivative of f atx and is denoted by f 0 (x). (This notation will be justified later).Defining on the set C 1 (Ω, H) the radial operators rad : 12 ( 0 ω(x) r ), rad : 21 ( 0 ω(x) r ),where r : x , 0 : results. x0and r : r ,(6)one can prove the following essentialTheorem 1 ([10, Corollary 5.2]) If f is a radially holomorphic function,thenf 0 (x) rad f (x).Theorem 2 ([3, Corollary 11.30]) A function f of the form (5) is radiallyholomorphic if and only if rad f 0,which is a Cauchy-Riemann type differential equation( 0 u r v, 0 v r u.4

We note that when rad f 0, the first radial operator in (6) simplifiesto rad f 0 f and Theorem 1 together with Theorem 2 imply that if f is aradially holomorphic function, thenf 0 (x) 0 f (x) 0 u(x0 , r) ω(x) 0 v(x0 , r),(7)like in the complex case, which justifies the use of the same notation for thecomplex derivative of a complex holomorphic function and the radial derivative of a quaternion radially holomorphic function. Next straightforward resultsupports these ideas.Proposition 2 If f and g are radially holomorphic functions of the formf (x) u(x0 , x ) ω(x)v(x0 , x )andg(x) u (x0 , x ) ω(x)v (x0 , x ),where u, v, u and v are real valued functions, then1. f g is radially holomorphic and (f g)0 (x) f 0 (x) g 0 (x);2. if α R, then αf is radially holomorphic and (αf )0 (x) αf 0 (x);3. f g is radially holomorphic and (f g)0 (x) f 0 (x)g(x) g 0 (x)f (x).Example 1 The following functions are radially holomorphic functions: monomials of the form xα , α R; the exponential function ex : Xxkk 0α 0α 1In addition, (x ) αxk! ex0 (cos r ω(x) sin r).and (ex )0 ex .We point out that usually when α / R, αf can not be written in the complexlike form (5), since, in general, αf (x) 6 f (x)α. For more details about radiallyholomorphic functions as well as other examples, see e.g. [8, 3, 10]. For a recentsurvey on elementary functions in the context of quaternion analysis, see [11].3Newton method: from C to HThe well known Newton method for finding a zero x of a holomorphic complexfunction f of one complex variable, consists on approximating x by means ofthe iterative processP1 : zk 1 zk f (zk ), k 0, 1, . . . ,f 0 (zk )(8)with z0 sufficiently close to x and f 0 (zk ) 6 0.Writing f (z) f (x iy) u(x, y) iv(x, y), where u and v are real valuedfunctions and identifying the complex number z x iy C with the vectorz (x, y) R2 , we can interpret f (z) 0 as a system of two equations intwo real unknowns, namely f (z) (u(z), v(z)) (0, 0) and apply the classical2D-Newton methodP2 : z k 1 z k (Jf (z k )) 1 f (z k ), k 0, 1, . . . ,5(9)

provided that the Jacobian matrix Jf (z k ) is nonsingular.We recall that, if f is holomorphic, due to the Cauchy-Riemann equations,we can write 1ux vxux vxJf ,andJf 1 2vx uxux vx2 vx uxwhere ux : u/ x, uy : u/ y, etc. Moreover, the determinant of theJacobian matrix satisfies Jf (x) f 0 (x) 2 .(10)Therefore, we can conclude, after simplifications, the well known fact that,in such cases, the 2D-Newton method (9) is identical to the complex Newtonmethod (8), i.e. P1 and P2 produce the same sequence.The problem concerning finding zeros of a quaternion function is, as expected, more difficult than the root finding problem in the complex plane. Incontrast to the complex case, the zeros of a quaternion function are not necessarily isolated, and its range is not necessarily open (see e.g. [9]). Since thewell known work of Niven [12], several authors gave contributions to this subject, in particular in connection to the study of zeros of quaternion polynomials[13, 14, 5, 15, 16, 17] (see also [18, 19] and [6, Section 4] and the referencestherein).In this section, we are going to show that the use of the radial derivativeleads to an iterative process analogue to the usual 4D-Newton process which canbe extended to a class of functions wider than the class of radially holomorphicfunctions.Let f be a radially holomorphic complex-like function f of the form (5), i.e.f (x) f (x0 x) u(x0 , r) ω(x)v(x0 , r),pwhere r x x21 x22 x23 , with radial derivative given by (7), i.e.f 0 (x) 0 u(x0 , r) ω(x) 0 v(x0 , r).Consider an iterative process of the formP1 :zk 1 zk f (zk )(f 0 (zk )) 1 , k 0, 1, . . . ,z0 c,where c H and f 0 (zk ) 6 0, for all k 0, 1, . . . . Observe that due to thestructure of f , this process produces exactly the same sequence asP2 :z̃k 1 z̃k (f 0 (z̃k )) 1 f (z̃k ), k 0, 1, . . . ,z̃0 c,with f 0 (z̃k ) 6 0, since the use of Proposition 1 and Corollary 1 allows to conclude that f and (f 0 ) 1 commute. Our objective now is to compare these twoequivalent processes with the classical Newton method for vector valued functionsP3 :z k 1 z k (Jf (z k )) 1 f (z k ), k 0, 1, . . . ,z 0 c,6

where c R4 is the vector corresponding to the quaternion c in processes P1and P2 , f is the vector valued function associated with the quaternion valuedfunction f , i.e. x1x2x3f (x) f (x0 , x1 , x2 , x3 ) u(x0 , r), v(x0 , r), v(x0 , r), v(x0 , r) (11)rrrand Jf (z k ) is the Jacobian matrix of f at z k which we assume is nonsingular.Next results reveal new important and interesting relationships between theradial derivative of a function f and the derivative of f .Proposition 3 Let f be the vector valued function (11) associated with a radially holomorphic complex-like function f of the form (5). The Jacobian determinant of f is related to the derivative of f by means ofv(x0 , r)2.r2Proof: The Jacobian matrix of f can be written as A B(x1 ) B(x2 ) B(x3 ) C(x1 , x2 )C(x1 , x3 ) B(x1 ) D(x1 , x2 , x3 ) ,Jf (x) C(x1 , x2 )D(x2 , x3 , x1 )C(x2 , x3 ) B(x2 ) B(x3 )C(x1 , x3 )C(x2 , x3 )D(x3 , x1 , x2 ) Jf (x) f 0 (x) 2(12)whereA 0 u(x0 , r);xB(x) 0 v(x0 , r);rxyxyC(x, y) 2 0 u(x0 , r) 3 v(x0 , r);rrx2y2 z2D(x, y, z) 2 0 u(x0 , r) v(x0 , r).rr3The result follows by cumbersome, but straightforward calculations. Remark 1 We underline the fact that, when x H, Jf (x) can be singularwhile f 0 (x) 6 0. This situation is quite different from the complex case wherethe equations Jf (x) 0 and f 0 (x) 0 are equivalent, for x C (cf. (10)).Proposition 4 If f is a radially holomorphic quaternion valued function anda H is such that ω(x) commutes with ω(a) thenJf (x)a af 0 (x) f 0 (x)a.(13)Proof: If a a0 a1 i a2 j a3 k, we can use Proposition 3 to write a0 0 u a1 x1 a2rx2 a3 x3 0 v a1 (x22 x23 ) a2 x1 x2 a3 x1 x3 v a1 x21 a2 x1 x2 a3 x1 x3 u a0 x1 v 00 r3r2r Jf (x)a a2 (x21 x23 ) a1 x1 x2 a3 x2 x3 ,a2 x22 a1 x1 x2 a3 x2 x3a0 x2 v 0 u r 0 v r3r2 a (x2 x2 ) a x x a x x a3 x23 a2 x3 x2 a1 x1 x3a0 x332 3 21 1 321v 0 u r 0 vr3r27

where all functions are considered at (x0 , r). Since ω(x) commutes with ω(a),we have from Proposition 1 thatx1 a2 x2 a1 ,x1 a3 x3 a1 ,and therefore Jf (x)a x2 a3 x3 a3(14) a1 x1 a2 x2 a3 x3 0 vr a1 0 u a0rx1 0 v .a0 x2a2 0 u r 0 v a0 0 u a3 0 u a0 x3r 0 v On the other hand, from (1) and (3), we have a0 a1af 0 (x) a2a3 a1a0a3 a2a0 0 u a2 a3a0a1 0 u a3 x1 0 v r a2 x2 a1 r 0 v x3a0r 0 v a1 x1 a2 x2 a3 x3 0 vra0 x1 a3 x2 a2 x3 0 vr a1 0 u a2 0 u a3 x1 a0rx2 a1 x3 0 v a3 0 u a2 x1 ar1 x2 a0 x3 0 vand the first part of the result is proved, using (14). The second part of (13)comes once more from Proposition 1. We can also establish the link between the Jacobian matrix and the corresponding determinant of the quaternion functions f and αf β, α, β H.Proposition 5 If f and g are quaternion valued functions defined on the setC 1 (Ω, H) such that g(x) αf (x) β, α, β H and α 6 0, thenJg(x) Lα Jf (x),(15)where Lα is the matrix left representation (1) of the quaternion α. Therefore Jg(x) α 4 Jf (x) .(16)Proof: Writing f (x) f0 (x) f1 (x)i f2 (x)j f3 (x)k, α α0 α1 i α2 j α3 kand β β0 β1 i β2 j β3 k and recalling (1) it follows that α0 f0 (x) α1 f1 (x) α2 f2 (x) α3 f3 (x) β0 α1 f0 (x) α0 f1 (x) α3 f2 (x) α2 f3 (x) β1 g(x) Lα f (x) β α2 f0 (x) α3 f1 (x) α0 f2 (x) α1 f3 (x) β2 .α3 f0 (x) α2 f1 (x) α1 f2 (x) α0 f3 (x) β3Straightforward calculations lead to result (15). Result (16) follows from thefact that det Lα α 4 . We are now in position to prove one of our main results.8

Theorem 3 Let g(x) αf (x) β be a function defined on the set C 1 (Ω, H)such that f is a radially holomorphic function in Ω, α, β H and α 6 0.Formulaszk 1 P1 (zk )z0 cz̃k 1 P2 (z̃k )z̃0 cz k 1 P3 (z k )z0 c(17)where c H and Pi , i 1, 2, 3 are the iterative functionsP1 (z) z g(z)(αf 0 (z)) 1 ,0 1P2 (z) z (αf (z))P3 (z) z (Jg(z)) 1(18)g(z),(19)g(z)(20)produce the same sequence (k 0, 1, . . . ), if ω(c) commutes with ω(α) and ω(β)and Jf (z k ) is nonsingular.Proof: Observe that under the assumption that ω(c) commutes with ω(α) andω(β) we can conclude that α and β commute, as well as ᾱ and β (see Proposition 1 and Corollary 1). In addition, using the fact that f is a radially holomorphic function of the form (5), we can also establish the commutativity betweenf (c) and f 0 (c), α and f 0 (c), f and α and, finally, β and f 0 (c). Thereforez1 c (af (c) β)(af 0 (c)) 1 c c 1 af 0 (c) 21(af (c) β)(af 0 (c)) af 0 (c) 2(af (c) β)(f 0 (c)α) c 10 (c) βαf 0 (c)af(c)αf a 2 f 0 (c) 2 c βα f 0 (c)f (c)f 0 (c) 2 002 f (c) α f (c) 2andz̃1 c (af 0 (c)) 1 (af (c) β) c 1 af 0 (c) 2(af 0 (c))(af (c) β) c 10 (c) α 2 f (c) f 0 (c)αβf a 2 f 0 (c) 2 c f (c)f 0 (c)βα f 0 (c) 2 0.02 f (c) α f (c) 2This proves that if ω(c) commutes with ω(α) and ω(β), then z1 z̃1 . Moreover,using similar arguments we can conclude that ω(z 1 ) commutes with ω(α) andω(β) and therefore z2 z̃2 . Using induction one can, in fact, prove thatzk z̃k , k 1, 2, . . . .We proceed now by examining formula (20). We point out that, by (16), Jg(xk )is nonsingular if and only if Jf (xk ) is nonsingular. In addition, using Proposition 5 we obtain(Jg(c)) 1 g(c) (Lα Jf (c)) 1 (Lα f (c) β) (Jf (c)) 1 L 1α (Lα f (c) β) (Jf (c)) 1 f (c) (Jf (c)) 1 L 1α β.9

Since c c0 ω(c) c commutes with f (c) u(c0 , c ) ω(c)v(c0 , c ) and also,under the assumptions of the theorem, with α and β and therefore with α 1 β(see Proposition 1 and Corollary 1), it follows from Proposition 4 that(Jg(c)) 1 g(c) (f 0 (c)) 1 f (c) (f 0 (c)) 1 α 1 β (f 0 (c)) 1 α 1 αf (c) (αf 0 (c)) 1 β (αf 0 (c)) 1 (αf (c) β) (αf 0 (c)) 1 g(c).This proves that z 1 z̃1 . Repeating the same arguments we can prove byinduction that z k z̃k , k 1, 2, . . . , When α and β are real numbers, the function g in Theorem 3 is radiallyholomorphic (see Proposition 2) and the result reads as follows:Corollary 2 If f is a radially holomorphic function defined on the set C 1 (Ω, H),then the processes (17) with the iterative functionsP1 (z) z f (z)(f 0 (z)) 1 ,0P2 (z) z (f (z)) 1P3 (z) z (Jf (z))f (z), 1f (z),(21)(22)(23)produce the same sequence, for all c H, if J(f (z k )) is nonsingular.We observe that the equivalence of the processes (21)-(23) does not dependon the choice of the initial value c, exactly as in the case of the complex Newton method (8) and the 2D-Newton method (9). This comes from the factthat for radially holomorphic functions, f (x) always commutes with x and with(f 0 (x)) 1 .Once the equivalence of the process P3 and P1 (and P2 ) is established, thelocal quadratic convergence of the method (18) (and (19)) can be established,provided that the initial guess c is chosen suffi

Analysis M.I. Falc ao Departamento de Matem atica e Aplica c oes and CMAT Universidade do Minho, Portugal Information Newton method; quaternions; ra-dial derivative. Original publication: Applied Mathematics and Compu-tation 236, (2014), 458-470 DOI: 10.1016/j.amc.2014.03.050 www.journals.el

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