Computatonal Aspects of the Implementaton of Dsk Inversons Ivan Petkovć Faculty of Electronc Engneerng, Department of Computer Scence, Unversty of Nš, Serba van.petkovc@elfak.n.ac.rs Abstract More than three decades the mplementaton of teratve methods for the smultaneous ncluson of polynomal zeros n crcular complex nterval arthmetc s carred out usng the exact nverson of dsks. Based on theoretcal analyss and numercal examples, we show that the centered nverson gves smaller ncluson dsks. Ths surprsng result s the consequence of better convergence of the mdponts of produced dsks when the centered nverson s employed. Some examples of ncluson methods wth the centered and exact nverson, together wth numercal results, are gven. Keywords: Zeros of polynomals; smultaneous methods; ncluson methods; dsk nverson; crcular nterval arthmetc; convergence. AMS subject classfcatons: 65H05, 65G20, 30C5. Introducton A great mportance of numercal methods for determnng polynomal zeros n the theory and practce (for example, n solvng many problems of appled and fnance mathematcs, control theory, sgnal processng, nonlnear crcuts, boscence, and other dscplnes) has led to the development of a great number of zero-fndng methods n ths feld, see, e.g., the books [7], [2], [4]. These numercal methods have become practcally applcable together wth the rapd growth of dgtal computers some ffty years ago. However, the computed soluton of an algebrac equaton s only an approxmaton to the exact soluton due to the errors orgnatng from dscretzaton, truncaton and from roundng. Ths naturally leads to the queston what s the error n the result? Solvng polynomal equatons, apart from the work engaged n the procedure appled to mprove the approxmate result, a consderable amount of work s nvolved n determnng error bounds of the mproved approxmatons to the polynomal zeros. An Submtted: January 8, 2009; Accepted: January 4, 200. Ths work was supported by the Serban Mnstry of Scence under grant 74022. 8
82 Ivan Petkovć, Computatonal Aspects of Dsk Inversons effcent approach that overcomes the aforementoned problem and gves satsfactory results s based on the use of nterval arthmetc. Partcularly, t turns out that teratve nterval methods for the smultaneous ncluson of polynomal zeros, realzed n crcular complex nterval arthmetc, are effcent n the case of complex zeros. These methods produce dsks that contan the wanted zeros n each teraton. For ths reason, such methods can be regarded as a self-valdated numercal tool that provdes automatc computaton of rgorous error bounds (gven by rad of resultng ncluson dsks) to the approxmate solutons. Ths very useful (ncluson) property s the man advantage of nterval methods. The am of ths note s to pont to the effcent use of a proper nverson of a dsk n the mplementaton of a class of smultaneous ncluson root-fndng methods based on fxed pont relatons. Although nterval methods started beng developed snce the 970 s, they were realzed usng the nverson based on Möbus s transformaton of a dsk Z by the functon z /z, the so-called exact nverson, denoted by Z. In crcular nterval arthmetc (arthmetc whch deals wth dsks) ths operaton s the exact operaton snce the mage Z completely concdes wth the exact range {/z : z Z}. Note that only a couple of authors tred to deal wth some other type of nversons n order to obtan smaller ncluson dsks. The reason probably les n the fact that the exact nverson gves the smallest dsks compared wth other nversons so that t seemed that ts applcaton s qute reasonable. In ths paper we show that the sze of ncluson dsks depend heavly on some other (extra-arthmetcal) features, not only of the employed arthmetc. 2 Crcular complex nterval arthmetc We start wth a short revew of the basc operatons n crcular nterval arthmetcs. For more detals see the books [2] and [2]. A crcular closed regon (dsk) Z := {z : z c r} wth center c := md Z and radus r := rad Z we wll denote n ths paper by parametrc notaton Z := {c; r}. Usng the Möbus transformaton we ntroduce the exact nverson { {c; r} = c c 2 r 2 ; r c 2 r 2 } = {/z : z {c; r} (0 / {c; r}). () As we wll see n ths paper, n some applcatons t s more convenent to take an nverse dsk {c; r} Ic := {/c; ρ} whose center s just /c, where c s the center of the orgnal dsk Z = {c; r}. Denote the crcumference of such nverse dsk wth D c, and let D e = { z : z c c 2 r 2 = r c 2 r 2 be the crcumference of the exact nverse dsk {c; r} gven by (). Snce {c; r} s the exact range, t has to be {c; r} Ic := D c ntd c {c; r}. Accordng to ths and Fg., the radus ρ = rad {c; r} Ic s equal to ρ = max w {c;r} c w = max θ [0,2π) c + r exp(θ) c c 2 r 2 = r max c θ [0,2π) r + cexp(θ) c 2 r 2 = r max r + c exp(α) c α [0,2π) c 2 r 2 = r c ( c r). }
Relable Computng 5, 20 83 Ths formula, often used by Rokne, Wu, Ratschek, Rump and others, can be also derved usng a general approach to crcular centered forms of elementary complex functons, see [] and [3]. Fg. The exact and centered nverson It s necessary to check that the dsk {c; r} Ic = {/c; ρ} completely contans the exact range {/z : z C} = {c; r}, n other words, we have to prove the nequalty md D c md D e rad Dc radd e, that s, c c c 2 r 2 r c ( c r) r c 2 r 2, whch reduces to the equalty. Ths means that the crcle D e touches (nsde) D c (see Fg. ). Therefore, the so-called centered nverson s gven by { } {c; r} Ic = c ; r {c; r} (0 / {c; r}). (2) c ( c r) Actually, the nverson defned n ths way concdes wth the Taylor form of nverson derved n []. One observes that the centered nverson always produces larger dsks than the exact nverson (). If Z k = {c k ; r k }(k =,2), then Z ± Z 2 = {c ± c 2; r + r 2}, w Z = {w md Z; w rad Z} (w C), Z Z 2 = {c c 2; c r 2 + c 2 r + r r 2}, Z : Z 2 = Z INVZ 2 (0 / Z 2, INV {(),() Ic }). The addton, subtracton and nverson Z are exact operatons, that s, Z Z 2 = {z z 2 : z Z, z 2 Z 2}, {+,,() }. We wll use the abbrevaton INV to denote the nverson of a dsk. 3 Smultaneous ncluson of polynomal complex zeros Let P be a monc polynomal of degree n wth smple real or complex zeros ζ,..., ζ n and assume that we have found an array of n dsks Z = (Z,..., Z n) such that
84 Ivan Petkovć, Computatonal Aspects of Dsk Inversons ζ Z ( I n := {,..., n}). Denote by ζ = (ζ,..., ζ n) and z = (z,..., z n) the vectors of the exact zeros of P and the centers of dsks, z = md Z, and let us represent a fxed pont relaton n a general form ζ = F (z, ζ) ( I n). (3) Let N(z) = P(z)/P (z) denote Newton s correcton. Our study wll be carred out n partcular cases of the followng two examples of the fxed pont relatons ζ = z ζ = z P(z ) n (z ζ j) j= j N(z ) n j= j ( I n), (4) z ζ j whch can be easly obtaned from the factorzaton P(z) = n (z ζ j) j= ( I n), (5) applyng the logarthmc dervatve n the case of (5). Substtutng the zeros on the rght sde of (3) by ther ncluson dsks and usng the ncluson property, we obtan the ncluson ζ Ẑ := F (z, Z) ( I n). (6) Under sutable ntal condtons (takng nto account the sze of ntal dsks and ther dstrbuton), the set Ẑ s a new contracted dsk contanng the zero ζ. In general, we wll use the symbol to denote quanttes n the subsequent teraton. Settng (Z,..., Z n) =: (Z (0),..., Z(0) n ), from (6) we can construct the followng teratve methods for the smultaneous ncluson of all smple zeros of the polynomal P : Z (m+) = F (z (m), Z (m) ) (m = 0,,... ; I n). (7) Let Z (m) such that r (m) := {c (m) ; r (m) } be ncluson dsks produced by the teratve method (7) 0 ( I n) when m, and let r (m) = max. If there exsts a real number k and a nonzero constant γ such that r (m+) (r (m) ) k γ, n r(m) then k s called the order of convergence of the teratve nterval method (7). In practce, for small enough r (m) t s suffcent to show that r (m+) = O ) k ) ((r (m), where O s the Landau symbol. Ths defnton of the order of convergence s satsfactory for the class of nterval methods consdered n ths paper. A more general defnton of the convergence speed, expressed by the so-called R-order, can be found n [2]. Havng n mnd (6) and (7), we start from the fxed pont relatons (4) and (5) and construct the followng partcular methods for the smultaneous ncluson of all smple zeros of the polynomal P :
Relable Computng 5, 20 85 Weerstrass-lke method [2], [5], the convergence order 2: Ẑ = z P(z ) n INV(z Z j) ( I n). (8) j= j Gargantn-Henrc s method [5], the convergence order 3: Ẑ = z INV 2 /N(z ) j)) n INV (z Z ( I n). (9) j= j Here we assume that INV,INV,INV 2 {(),() Ic }. The subscrpt ndces and 2 n (9) pont to the order of applcaton of nversons. The nterval method (9) (wth the exact nversons) was proposed n [5] so that t s often refereed to as Gargantn- Henrc s method. Let us note that orgnal methods (8) and (9) presented n the papers cted above, as many other smlar methods based on fxed pont relatons, used only the exact nverson, that s, INV = INV = INV 2 = (). We could consder some other fxed pont relatons and correspondng nterval methods, but the conclusons are entrely the same as n the case of nterval methods (8) and (9). Remark The man advantage of nterval methods (8) and (9) s the ncluson property; namely, n each teraton these nterval methods produce the array of dsks Z (m),..., Z n (m) such that ζ Z (m) (m = 0,, 2,... ; I n). In ths way the automatc control of error s provded snce md Z (m) ζ rad Z (m), takng the mdponts of dsks to be approxmatons to the zeros. From the convergence analyss of nterval methods (8) and (9) we can fnd that ˆr = rad Ẑ = O( P(z) r), r = max r (0) n for the Weerstrass-lke method (8) and ˆr = rad Ẑ = O( P(z) 2 r) () for the Gargantn-Henrc method (9), see the book [2] for detals. Obvously, snce P(z ) = z ζ z ζ j = O(r), j from (0) and () we conclude that the convergence order of the methods (8) and (9) s two and three, respectvely. Besdes the study of smultaneous ncluson methods, let us consder teratve methods for the smultaneous determnaton of complex zeros realzed n ordnary complex arthmetc. Wthout loss of generalty, we wll restrct out attenton to the methods correspondng to the ncluson methods (8) and (9). If we start from the fxed pont relatons (4) and (5) and substtute the exact zeros ζ,..., ζ n by ther ( pont ) approxmatons z,..., z n, then we obtan the followng two methods for the smultaneous approxmaton of polynomal zeros:
86 Ivan Petkovć, Computatonal Aspects of Dsk Inversons Weerstrass-Durand-Kerner method [6], the convergence order 2: ẑ = z P(z ) n (z z j) j= j Ehrlch-Aberth s method [], [4], the convergence order 3: ẑ = z N(z ) n j= j z z j ( I n); (2) ( I n). (3) For more detals on these methods see the recent book [7]. From (0) and () we nfer that the convergence of rad strongly depends on the centers of ncluson dsks; when the centers are closer to the zeros, the convergence of rad s faster. Let us examne now the convergence behavor of the centers of dsks Ẑ produced by the ncluson methods (8) and (9) dstngushng two cases: () the exact nverson () s appled; () the centered nverson (2) s appled. Ths behavor can be smply examned consderng the resultng dsks obtaned by the nversons () and (2). Startng from () we fnd (assumng that r s suffcently small and c > r). { } {c; r} c = c 2 r ; r 2 c 2 r 2 { ( ) } = c + (r/ c )2 + (r/ c ) 2 (r/ c ) 4 r + ;. (4) c c 2 r 2 }{{} Bearng n mnd (2) and the mappng functon z /z, we note that the centered nverson preserves the property of centerng, whle the exact nverson does not. Ths means that the centers of dsks produced by the ncluson methods (8) and (9) concde wth the teratve methods (2) and (3), respectvely, when the centered nverson s appled. On the other hand, applyng the exact nverson, we observe by comparng () and (4) that the centers of dsks obtaned by the methods (8) and (9) are removed (for underlned part) and wll not concde wth (2) and (3). Therefore, the convergence of centers wll be spoled when the exact nverson s employed. Consequently, takng nto account the estmaton (0) and (), the ncluson methods (8) and (9) show faster convergence when the centered nverson s appled. At frst sght ths s a paradox snce the centered nverson always produces larger dsks than the centered nverson (see (2)). However, we have shown that the convergence speed of nterval methods depends not only of the dsk sze but also of the convergence behavor of centers of dsks. Accordng to the presented analyss t follows that the better convergence of centers of resultng ncluson dsks provdes the faster convergence. Hence, the followng natural queston could be posed: can the mprovement of convergence of centers accelerate the convergence speed of nterval methods? The answer s yes, whch was demonstrated for the frst tme n [3] where the followng ncluson method of Gargantn-Henrc s type wth Newton s correctons N(z j) = P(z j)/p (z j) was stated: Ẑ = z INV 2 (/N(z ) j)) n INV (z Z j + N(z ( I n). (5) j= j
Relable Computng 5, 20 87 We note that the centers md (Z j N(z j)) = z j N(z j) behave as the approxmatons obtaned by Newton s method, { that} eventually provdes the acceleraton of convergence of the sequences of rad rad Z (m). The followng statement was proved n [3]: Theorem If ntal ncluson dsks Z (0),..., Z(0) n are reasonable small, then the R-order of convergence O R(5) of the nterval method (5) s gven by O R(5) { (3 + 7)/2 = 3.562 f INV = (), 4 f INV = () Ic. In essence, the ncrease of the convergence rate s the result of the accelerated convergence of the centers of the dsks Ẑ calculated by (5). { In partcular, } when the centered nverson s appled n (5), then the sequences md Ẑ(m) behave as the sequences of approxmatons defned by the fourth-order Nouren s method [9] ẑ = z N(z ) n j j= z z j + N(z j) ( I n). In general, t s desrable to accelerate the convergence of centers of dsks appearng n teratve nterval formulas. The applcaton of the centered nverson moves the center of the mproved dsk Ẑ very close to the zero ζ. Further mprovement of the convergence rate of the nterval methods (9) and (5) can be acheved by applyng more rapd method nstead of Newton s method. The followng teratve method for solvng algebrac equaton P(z) = 0, proposed by Ostrowsk [0], s convenent n the acceleraton of convergence of nterval methods: P(z N(z)) P(z) ẑ = z N(z) 2P(z N(z)) P(z) = z g(z), P(z) N(z) = P (z). (6) The order of convergence of the Ostrowsk method (6) s four. The term g(z) n (6) s called Ostrowsk s correcton. Let us note that the teratve method (6) can be also appled to arbtrary (real or complex) functon. In a smlar way as n the constructon of the nterval method (5), we can derve the Gargantn-Henrc method wth Ostrowsk s correctons g(z j) n crcular complex arthmetc: j)) n Ẑ = z INV 2 /u(z ) INV (z Z j + g(z ( I n). (7) j= j Applyng crcular arthmetc operatons, t can be proved that the choce INV = INV 2 = () Ic n (7) produces the dsks Ẑ whose centers behave as the approxmatons obtaned by the smultaneous method ẑ = z N(z ) n j= j z z j + g(z j) ( I n). (8)
88 Ivan Petkovć, Computatonal Aspects of Dsk Inversons The convergence order of the method (8) s sx so that we can expect very fast convergence of the nterval method (7) snce Ostrowsk s approxmaton z j g(z j) s very close to the exact zero ζ j. More precsely, we can state the followng result: Theorem 2 Let (Z (0),..., Z(0) n ) := (Z,..., Z n) be an array of dsjont ntal dsks contanng the zeros ζ,..., ζ n of P. If the mdponts of ntal dsks are close enough to the zeros of P, then the R-order of convergence of the teratve method (7) s gven by { (3 + 7)/2 = 3.562 f INV = (), O R(7) 6 f INV = () Ic. We note that the R-order of the nterval method (3) s not ncreased when INV = () although the correctons of the method of hgher order s appled. Numercal examples confrms ths fact, see, e.g., Table 2. Detaled theoretcal explanaton of ths phenomenon s gven n [8] and [2]. 4 Numercal examples The presented analyss wll be llustrated by the followng examples. Example We have appled Weerstrass-lke method (8) wth INV = () and INV = () Ic for the ncluson of zeros of the polynomal P(z) = z 7 + z 5 0z 4 z 3 z + 0. We have started wth the ntal dsks Z (0) = {z (0) ;0.3} that contan the exact zeros 2, ±, ±, ±2. The maxmal rad of the obtaned dsks are gven n Table, where A( q) means A 0 q. Methods max r () max r (2) max r (3) max r (4) max r (5) max r (6) (8) wth () 0.39 0.22.77( 2) 3.2( 5) 6.8( ).7( 22) (8) wth () Ic 0.44 0.28 7.84( 3).7( 6) 8.84( 5) 3.77( 3) Table : Weerstrass-lke method (8) wth exact and centered nverson Example 2 We have appled two versons of the Gargantn-Henrc method (9) for the ncluson of zeros of the followng polynomal of the 25th degree wth ntal dsks wth the same rad r (0) = 0.3, P(z) = (z 4)(z 4 )(z 4 8)(z 2 8z + 7)(z 2 6z + 3)(z 2 4z + 5)(z 2 2z + 5) (z 2 4z + 3)(z 2 + 2z + 5)(z 2 + 4z + 5)(z 2 + 4z + 3). The maxmal rad are dsplayed n Table 2. Methods max r () max r (2) max r (3) max r (4) (9) wth () 9.27( 2) 6.96( 4).99( 2).42( 39) (9) wth () Ic.70( ).08( 4).8( 5) 8.99( 50) Table 2: Gargantn-Henrc method (9) wth exact and centered nverson
Relable Computng 5, 20 89 Example 3 Apart from the Gargantn-Henrc method (9), we have also appled the accelerated methods (5) and (7) wth INV = INV 2 = () and INV = INV 2 = () Ic for ncluson of the zeros of the polynomal P(z) = z 9 + 3z 8 3z 7 9z 6 + 3x 5 + 9z 4 + 99z 3 + 297z 2 00z 300. We have started wth the ntal dsks Z (0) = {z (0) ;0.3} that contan the zeros 3, ±, ±2, ±2 ±. The maxmal rad of dsks are gven n Table 3. Methods max r () max r (2) max r (3) max r (4) (9) wth () 6.20( 2) 8.3( 5) 4.45( 5).47( 46) (9) wth () Ic.0( ) 5.73( 5) 6.2( 6).52( 50) (5) wth () 6.20( 2) 5.65( 5).2( 7) 5.05( 62) (5) wth () Ic.0( ) 4.57( 5) 2.6( 9) 3.0( 76) (7) wth () 6.0( 2).78( 5) 2.0( 8) 3.90( 64) (7) wth () Ic.0( ) 6.40( 6).70( 3) 6.9( 89) Table 3: The methods (9), (5) and (7) wth exact and centered nverson From Tables, 2 and 3 we observe that the centered nverson (2) gves smaller dsks n the case of all tested methods (8), (9), (5) and (7). The mprovement s especally stressed when the methods (5) and (7) wth correcton s appled. Slghtly larger dsks n the frst teraton, when the centered nverson s appled, s the results of relatvely slow convergence of the centers at the begnnng of teratve process. References [] O. Aberth, Iteraton methods for fndng all zeros of a polynomal smultaneously, Math. Comp., vol. 27, pp. 39 344, 973. [2] G. Alefeld, J. Herzberger, Introducton to Interval Computaton, Academc Press, New York, 983. [3] C. Carstensen, M. S. Petkovć, An mprovement of Gargantn s smultaneous ncluson method for polynomal roots by Schroeder s correcton, Appl. Numer. Math., vol. 25, pp. 59 67, 993. [4] L.W. Ehrlch, A modfed Newton method for polynomals, Comm. ACM, vol. 0, pp. 07 08, 967. [5] I. Gargantn, P. Henrc, Crcular arthmetc and the determnaton of polynomal zeros, Numer. Math., vol. 8, pp. 305 320, 972. [6] I. O. Kerner, En Gesamtschrttverfahren zur Berechnung der Nullstellen von Polynomen, Numer. Math., vol. 8, pp. 290 294, 966. [7] J. M. McNamee, Numercal Methods for Roots of Polynomals, Part I, Elsever, Amsterdam, 2007. [8] D. Mloševć, Iteratve methods for the smultaneous nlcuson of polynomal zeros, Ph. D. Theses, Unversty of Nš, Nš, 2005. [9] A. W. M. Nouren, An mprovement on two teraton methods for smultanneously determnaton of the zeros of a polynomal, Internat. J. Comput. Math., vol. 6, pp. 24 252, 977.
90 Ivan Petkovć, Computatonal Aspects of Dsk Inversons [0] A. M. Ostrowsk, Solutons of Equatons and System of Equatons, Academc Press, New York, 966. [] Lj. D. Petkovć, M. S. Petkovć, The representaton of complex crcular functons usng Taylor seres, Z. Angew. Math. Mech., vol. 6, pp. 66 662, 98. [2] M. S. Petkovć, Iteratve Methods for Smultaneous Incluson of Polynomal Zeros, Sprnger-Verlag, Berln-Hedelberg-New York, 989. [3] H. Ratschek, J. Rokne, Computer Methods for the Range of Functons, Ells Horwood, Chchester, 984. [4] B. Sendov, A. Andreev, N. Kyurkchev, Numercal Soluton of Polynomal Equatons (Handbook of Numercal Analyss), Vol. VIII, Elsever Scence, New York, 994. [5] X. Wang, S. Zheng, The quas-newton method n parallel crcular teraton, J. Comput. Math., vol. 4, pp. 305 309, 984.