COMPUTING QUATERNIONIC ROOTS BY NEWTON’S METHOD

Newton’s method for finding zeros is formally adapted to finding roots of Hamilton’s quaternions. Since a derivative in the sense of complex analysis does not exist for quaternion valued functions we compare the resulting formulas with the more classical formulas obtained by using the Jacobian matrix and the Gâteaux derivative. The latter case includes also the so-called damped Newton form. We investigate the convergence behavior and show that under one simple condition all cases introduced, produce the same iteration sequence and have thus the same convergence behavior, namely that of locally quadratic convergence. By introducing an analogue of Taylor’s formula for , we can show the local, quadratic convergence independently of the general theory. It will also be shown that the application of damping proves to be very useful. By applying Newton iterations backwards we detect all points for which the iteration (after a finite number of steps) must terminate. These points form a nice pattern. There are explicit formulas for roots of quaternions and also numerical examples. Key words. Roots of quaternions, Newton’s method applied to finding roots of quaternions. AMS subject classifications. 11R52, 12E15, 30G35, 65D15 1. Introduction. The newer literature on quaternions is in many cases concerned with algebraic problems. Let us mention in this context the survey paper by Zhang [15]. Here, for the first time we try to apply an analytic tool, namely Newton’s method, to finding roots of quaternions, numerically. Let be a given mapping with continuous partial derivatives. Then, the classical Newton form for finding solutions of is given by ! " $#% '&( " *) + -,. -/1032% 4 5#6) , (1.1) where & stands for the matrix of partial derivatives of , which is also called Jacobian matrix. The equation (1.1) has to be regarded as a linear system for ) with known . The further steps consist of iteratively solving this system with /1032 . In this paper we want to treat a special problem ! " 7 8 with 9 ;:< =: , where : denotes the (skew) field of quaternions. We use the letter : in honor of William Rowan : amilton (1805 – 1865), the inventor of quaternions. In this setting we will try also other forms of derivatives of than the matrix of partial derivatives. For illustration in this introduction, we use the simple equation ! " 7 > ? !@BA+C with C ,D E : . If we follow the real or complex case for defining derivatives, we have two possibilities because of the non commutativity of the multiplication in : , namely & " > GFIHKJ LNMPO1Q R ! " S#UT VA ! " D DTXW!Y[Z\ GFKHIJ LNMPO ] " S#UT^ @ A @ DTXW!Y 5#_FKHIJ LNMPO T T!WXY[, `&a " > GFIHKJ LNMPO Q T WXY " S#bT VAc D dZP GFKHIJ LNMPO T WXY ] " #UT @ A @ e S#_FIHIJ LNMPO T W!Y T$f If we put g L 4 hTT W!Y for any Tji then from later considerations we know that k g L kl mk Vk and g L Y n Y . Thus, g L fills the surface of a three dimensional ball and there is no unique limit. In other words, the above requirement for differentiability is too strong. One can even show that only the quaternion valued functions (op % > qC'o #srN,t (op % > uopC #mr[, C ,vr Ee: , respectively, are differentiable with respect to the two given definitions, Sudbery [13, Theorem 1]. w Received April 18, 2006. Accepted for publication October 18, 2006. Recommended by L. Reichel. Institute of Chemical Technology, Prague, Department of Mathematics, Technická 5, 166 28 Prague 6, Czech Republic (janovskd@vscht.cz). University of Hamburg, Faculty for Mathematics, Informatics, and Natural Sciences [MIN], Bundestraße 55, 20146 Hamburg, Germany (opfer@math.uni-hamburg.de). 82 ETNA Kent State University etna@mcs.kent.edu COMPUTING QUATERNIONIC ROOTS 83 In approximation theory and optimization a much weaker form of derivative is employed very successfully. It is the one sided directional derivative of j :x : in direction T or one sided Gâteaux 1 derivative of in direction T (for short only Gâteaux derivative) which for $,yTzE7: is defined as follows: & " {,vT | > }FKHIJ ~€‚ ~ ƒl ! " 5#b„VT …Ac „ }FKHIJ ~N† ~ ƒl S#%„…T @ A @ „ TS#bT$f (1.2) Let T E‡ ˆ Q `Z , then & " {,vT t 9‰ŠT and from (1.1) replacing & " with & " {,vT we obtain the damped Newton form /‹032 > hŒ% " | 4 e S#  ‰lT Ž $W!Y1CBA  if T‘  . For T  we obtain the common Newton form for square roots. If we work with partial derivatives, the equation ! " > @’A C implies `&a " > n‰ “”• Y A @ A –—A -˜ @ Y – Y ˜ Y ™1š › f (1.3) Matrices of this form are known as arrow matrices. They belong to a class of sparse matrices for which many interesting quantities can be computed explicitly, Reid [11], Walter, Lederbaum, and Schirmer [14], and Arbenz and Golub [1] for eigenvalue computations. The special cases C^,] zEz and C^,] zEzœ reduce immediately to the common Newton form /1032 4 Œb " >  ‰ Ž S# C  f The treatment of analytic problems in : goes back to Fueter [5]. A more recent overview including new results is given by Sudbery [13]. However, Gâteaux derivatives do not occur in this article. We start with some information on explicit formulas for roots of quaternions. Then we adjust the common Newton formula for the  -th root of a real (positive) or complex number to the case of quaternions. Because of the non commutativity of the multiplication we obtain two slightly different formulas. We will see that under a simple condition both formulas produce the same sequence. We see by examples that in this case the convergence is fast and we also see from various examples that in case the formulas produce different sequences, the convergence is slow or even not existing. Later we apply the Gâteaux derivative and the Jacobian matrix of the partial derivatives to formula (1.1) and show that under the same condition the same formulas can be derived which proves that the convergence is locally quadratic. The Gâteaux derivative gives also rise to the damped Newton form which turns out to be very successful and superior to the ordinary Newton technique. 2. Roots of quaternions. We start by describing a method for finding the solutions of ! " | > AjCS ,žCŸE7:5ˆ[ \,. Ez ¡,. ¢+‰`, (2.1) explicitly. The solutions of h will be called roots of C . We need some preparations. If CS m C Y ,vC @ ,]C – ,]C ˜ tE‘: we will also use the notation CS hC Y #%C @X£ #bC –l¤ #bC ˜X¥ , 1René Gâteaux, French mathematician (Vitry 1889 – [Verdun?] 1914) ETNA Kent State University etna@mcs.kent.edu 84 D. JANOVSKÁ AND G. OPFER where £ , ¤ , ¥ stand for the units ( ,  ,] -,] Š ¦,‹ ( ,] -,  ,] Š ¦,‹ ( ,] -,] -,  , respectively. DEFINITION 2.1. Two quaternions C ,vr are called equivalent, denoted by C ̈§©r , if there is TcEz:5ˆ Q `Z such that CŸ 9T W!Y r¦T (or T^CŸ 9r¦T ). The set of all quaternions equivalent to C is denoted by a CŠ« . Let C‡ 4 ¬ (C Y ,]C @ ,vC – ,]C ˜ ­E :®ˆ . We call C' ̄ > G -,]C @ ,vC – ,]C ˜ the vector part of C . By assumption C' ̄Ÿi . The complex number ° C > s C Y ,€± C @@ #bC @– #%C @ ˜ ,] -,] Š † 5 C Y #nk C' ̄`k £ (2.2) has the property that it is equivalent to C (cf. (2.3)) and it is the only equivalent complex number with positive imaginary part. We shall call this number ° C the complex equivalent of C . Because of CS hT WXY r¦T x2 T k TVk(3 W!Y r T k TVk there is no loss of generality if we assume that k TVk @ ́  . Since C%E+ commutes with all elements in : we have a CŠ« Q C^Z . In other words, for real numbers C the equivalence class a CŠ« consists only of the single element C . Let μ­E œ , then μ and the complex conjugate μ belong to the same class a μy« because of μ’ m ¤ WXY μ ¤ . LEMMA 2.2. The above notion of equivalence defines an equivalence relation. And we have C5§nr if and only if ¶ CS ¶ rN,·k C kŠ mk r k4f (2.3) Proof. Let T^C9 ̧r¦T for some T}i < . Then, the general rule k -gXk\ k VkIk gXk yields k C k; 1k r k . Let us put Ce oT WXY r¦T and apply another general rule ¶ " ^g Ÿ ¶ "g' . Then ¶ Cz ¶ aT WXY r1T^ ¶ ] (T W!Y r1 DT ¡ ¶ (T T WXY r¦ \ ¶ r . It remains to show that (2.3) implies the existence of an T i such that T^C5 nr¦T . Let C E‘ . Then (2.3) implies CS nr and hence, Tz  . Otherwise, (2.3) is equivalent to a real, linear, homogeneous »Ÿ1⁄4‘» system. It can be shown, that the rank of the corresponding matrix is two. There are situations where there are infinitely many roots. THEOREM 2.3. Let be defined as in (2.1) but with real C . If there exists a complex root of C which is not real, then there will be infinitely many quaternionic roots of C . Proof. Let > Y #6 @‹£ be a root of C with @ i h . We have ! " > + A C . Let TcE :®ˆ Q `Z . We multiply the last equation from the left by T W!Y and from the right by T and obtain T!WXY] ! " ]T ́ nT!WXYy T ̈A6T!WXYyC`T m aT!WXYy T^ AjCS h (2.4) since real numbers commute with quaternions. Therefore, ! (T W!Y T^ t n or, in other words, the whole equivalence class a -« of consists of roots. COROLLARY 2.4. Let Cji s be real. For U¢©1⁄2 there are always infinitely many roots of C . For ‡ h‰ there are infinitely many roots if CŸ3⁄4e . The finding of roots of quaternions is based on the following lemma. LEMMA 2.5. Let C©Em:®ˆ[ and let ° C be the corresponding complex equivalent of C where ° CS T WXY C`T for some T i + such that ¿ ° C U . Then, will be a root of C if and only if À ‡ 4 T W!Y T is a root of ° C . Proof. (i) Let be a root of C . By applying (2.4) we obtain À A ° C 9 . (ii) Let À be a root of ° C . I.e. we have À A ° CS h . Multiplying from the left by T and from the right by T WXY gives the desired result. This lemma yields the following steps for solving (2.1) for CŸ 4 m (C Y ,vC @ ,]C – ,vC ˜ ¡i Ez . ETNA Kent State University etna@mcs.kent.edu COMPUTING QUATERNIONIC ROOTS 85 (i) Compute ° CŸ 4 Á (C Y ,‹Â C @@ #%C @– #bC @ ˜ ,] -,] p … hC Y #hk C ̄ k £ E‘œ . (ii) Let À Ã5E‘œ be the roots of ° C Ezœ : À í mk C!k Y]Ä |ŦÆ`Ç £{ÈlÉ @ ÃdÊ d,vË ̈ h ,  ,1f‹f1fd,] ́A  , Ì1͊Π„j ?Ï1Ð Ñ Ï Ñ ,X„bE a ,]҂a . (iii) Find T7E‘: such that ° CŸ 4 hT W!Y C'TzE7œ . (iv) Then, the sought after roots are