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LAGUERRE-LIKE METHODS WITH CORRECTIONS FOR THE INCLUSION OF POLYNOMIAL ZEROS

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Nov Sad J. Math. Vol. 34, No. 1, 2004, LAGUERRE-LIKE METHODS WITH CORRECTIONS FOR THE INCLUSION OF POLYNOMIAL ZEROS Modrag S. Petkovć 1, Dušan M. Mloševć 1 Abstract. Iteratve methods of Laguerre
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Nov Sad J. Math. Vol. 34, No. 1, 2004, LAGUERRE-LIKE METHODS WITH CORRECTIONS FOR THE INCLUSION OF POLYNOMIAL ZEROS Modrag S. Petkovć 1, Dušan M. Mloševć 1 Abstract. Iteratve methods of Laguerre s type for the smultaneous ncluson of all zeros of a polynomal are proposed. Usng Newton s and Halley s correctons, the order of convergence of the basc method s ncreased from 4 to 5 and 6, respectvely. Further mprovements are acheved by the Gauss-Sedel approach. Usng the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analyss of total-step and sngle-step methods. The suggested algorthms possess a great computatonal effcency snce the ncrease of the convergence rate s attaned wthout addtonal calculatons. The case of multple zeros s also studed. Two numercal examples are gven to demonstrate the convergence propertes of the proposed methods. AMS Mathematcs Subject Classfcaton 2000: 65H05, 65G20, 30C15 Key words and phrases: Zeros of polynomals, smultaneous methods, Laguerre s method, convergence rate, crcular arthmetc 1. Introducton Ths paper s devoted to the constructon of ncluson methods wth very hgh computatonal effcency for the smultaneous ncluson of polynomal zeros and presents the contnuaton of a research exposed recently n [8]. We recall that teratve methods for the smultaneous determnaton of polynomal zeros, realzed n nterval arthmetc, produce resultng real or complex ntervals dsks or rectangles contanng the wanted zeros. In ths manner the nformaton about upper error bounds of approxmatons to the zeros are provded see the books [1], [10], [14] for more detals. The presentaton of the paper s organzed as follows. The basc propertes of crcular complex arthmetc, necessary for the development and convergence analyss of the presented ncluson methods, are gven n the ntroducton. The basc Laguerre-lke total-step method of the fourth order, recently proposed n [8], s presented n short n Secton 2. The man goal of our study s to acheve remarkably faster convergence wth only few addtonal numercal operatons, whch sgnfcantly ncreases the computatonal effcency. For ths purpose, the modfed total-step methods wth the ncreased convergence speed s developed n Secton 3 usng Newton s and Halley s correcton. The convergence analyss 1 Faculty of Electronc Engneerng, Unversty of Nš, Beogradska 14, Nš, Serba and Montenegro 136 M. S. Petkovć, D. M. Mloševć of these mproved methods s gven n Secton 4. Some mportant tasks, as the constructon of Laguerre-lke methods wth correctons n sngle-step mode and modfed varants for the ncluson of multple zeros, are studed n Secton 5. Numercal results obtaned by the consdered methods are gven n Secton 6. The constructon of ncluson methods n crcular complex arthmetc and ther convergence analyss requre the basc propertes of crcular complex arthmetc. A crcular closed regon dsk Z := z : z c r wth the center c := md Z and radus r := rad Z we wll denote by parametrc notaton Z := c; r. If Z k := c k ; r k k = 1, 2, then Z 1 ± Z 2 = c 1 ± c 2 ; r 1 + r 2, Z 1 Z 2 = c 1 c 2 ; c 1 r 2 + c 2 r 1 + r 1 r 2. The addton and subtracton of dsks are exact operatons. The nverson of a non-zero dsk Z s defned by the Möbus transformaton, 1 Z 1 = c; r 1 = c; r c 2 r 2 c r,.e. 0 / Z. The nverson Z 1 s also an exact operaton, that s, Z 1 = z 1 : z Z. Besde the exact nverson Z 1 of a dsk Z, the so-called centered nverson Z I c 2 defned by Z I c = c; r I c 1 := c ; r Z 1 c c r 0 / Z s often used. Sometmes, we wll use the symbol INV to denote both nversons, that s INV 1, Ic. Havng n mnd 1 and 2 the dvson s defned by Z 1 : Z 2 = Z 1 INVZ 2 0 / Z 2, INV 1, I c. The square root of a dsk c; r n the centered form, where c = c e θ and c r, s defned as the unon of two dsjont dsks see [3]: c; r 1/2 := c e θ/2 ; R 3 c e θ/2 ; R, where R = r c + c r. In ths paper we wll use the followng obvous propertes: z c; r z c r, c 1 ; r 1 c 2 ; r 2 = c 1 c 2 r 1 + r 2, md Z rad Z z md Z + rad Z z Z. More detals about crcular arthmetc can be found n the books [1, Ch. 5] and [14, Ch. 2]. Throughout ths paper dsks n the complex plane wll be denoted by captal letters. Laguerre-lke methods wth correctons Total step method wthout correctons Let P z = z n + a n 1 z n a 1 z + a 0 be a monc polynomal wth smple zeros ζ 1,..., ζ n and let I n := 1,..., n be the ndex set. For the pont z = z I n let us ntroduce Σ k, = n j 1 z ζ j k k = 1, 2, q = nt 2, n n 1 T 1,, 2 δ 1, = P z P z, δ 2, = P z 2 P z P z P z 2, ε = z ζ. The followng dentty 7 nδ 2, δ1, 2 q = 1 n 2 δ 1, n 1 ε was proved n [8]. From 7 we obtan the fxed pont relaton 8 n ζ = z I n, δ 1, ± n 1nδ 2, δ1, 2 q whch s the base for the constructon of ncluson methods of Laguerre s type. To smplfy the notaton, let us ntroduce the followng vectors of dsks Z m = Z m 1,..., Z n m ncluson dsks, Z m N = Z m N,1,..., Zm m N,n, Z N, = Zm Z m H = Z m H,1,..., Zm m H,n, Z H, = Zm where m = 0, 1, 2... s the teraton ndex and N z m Newton s dsks, Halley s dsks, H z m Nz = P z P Newton s correcton, z [ P z Hz = P z P ] 1 z 2P Halley s correcton. z For brevty, we wll wrte sometmes z, r, ẑ, ˆr, Z, Ẑ, Z N,, Z H, nstead of z m, r m, z m+1, r m+1, Z m, Z m+1, Z m N,, Zm H,. In what follows we wll wrte w 1 w 2 or w 1 = O M w 2 the same order of magntude for two complex numbers w 1 and w 2 that satsfy w 1 = O w 2. Let us defne the dsk 1 k n k, 9 S k, X, W := INV 1 z X j + INV 1 z W j j=+1 138 M. S. Petkovć, D. M. Mloševć for k = 1, 2, where X = X 1,..., X n and W = W 1,..., W n are vectors whose components are dsks and INV 1 1, I c, and defne the dsk Q X, W = ns 2, X, W n n 1 S2 1,X, W. Then, usng 9 and the defnton of q, accordng to the ncluson sotoncty we have q Q X, W. Let Z 0 1,..., Z0 n be ntal dsjont dsks contanng the zeros ζ 1,..., ζ n, that s, ζ Z 0 for I n. Takng ncluson dsks Z m 1,..., Z n m nstead of these zeros n 8, we defne the dsk A m = δ m 1, + [ n 1 nδ m 2, δ m 2 1, Q Z m, Z m] 1/2 and state the followng total-step method for the smultaneous ncluson of all zeros of P, 10 where z m Z m+1 = z m n INV 2 A m I n, = md Z m, INV 2 1, I c. In the realzaton of the teratve formula 10 we frst apply the nverson INV 1 to the sums 9, and then the nverson INV 2 n the fnal step. The nterval Laguerre-lke method 10 was recently stated n [8]. Accordng to 3, the square root of a dsk n 10 produces two dsks; the symbol ndcates that one of the two dsks has to be chosen. That dsk wll be called a proper dsk. From 7 and the ncluson q Q we conclude that the proper dsk s one whch contans n/ε δ 1,. Takng nto account 3, we have n 1 nδ 2, δ 2 1, Q 1/2 = G1, G 2,, md G k, = g k,, g 1, = g 2, for I n, k = 1, 2. The crteron for the choce of a proper dsk s consdered n [3] see also [9] and reads: If the dsks Z 1,..., Z n are reasonably small, then we have to choose that dsk between G 1, and G 2,, whose center mnmzes P z /P z g k, k = 1, 2. The teratve method 10 wth INV 1, INV 2 = 1 or I c has the order of convergence equal to four see [8]. The convergence of ths method can be accelerated usng already calculated dsks n the current teraton Gauss-Sedel approach. In ths manner we obtan the sngle-step method 11 where B m Z m+1 = z m n INV 2 B m I n, [ = δ m 1, + n 1 nδ m 2, δ m 2 1, Q Z m+1, Z m] 1/2. Laguerre-lke methods wth correctons The R-order of convergence of the sngle-step method 11 s at least 3 + x n, where x n 1 s the unque postve root of the equaton x n x 3 = 0 see [10]. 3. Laguerre-lke methods wth correctons Let us ntroduce the abbrevatons r m = max 1 n rm, ρ m = mn 1,j n j ε m = z m ζ, ɛ m = max 1 n z m ε m z m j r m j for I n, m = 0, 1,.... Further ncrease of the convergence speed of the teratve methods 10 and 11 can be acheved usng Newton s or Halley s correcton n the smlar way as n [2], [11] and [12]. In ths constructon we assume that ntal ncluson dsks Z 0 1,..., Z0 n, contanng the zeros ζ 1,..., ζ n, have been chosen n such a way that each dsk Z 0 N md Z 0 0 or Z H md Z 0 also contans the zero ζ I n. Ths pont s the subject of the followng asserton where, for smplcty, the teraton ndces are omtted. Lemma 1. Let Z 1,..., Z n be ncluson dsks for the zeros ζ 1,..., ζ n, ζ Z, and let z = md Z, r = rad Z. If the ncluson dsks Z 1,..., Z n are chosen so that the nequalty, 12 ρ 3n 1r s satsfed, then for I n we have the mplcatons: ζ Z ζ Z N, := Z Nz ; ζ Z ζ Z H, := Z Hz. Ths lemma can be proved n a smlar way as n [13] so that we omt the proof. Startng from the fxed-pont relaton 8 we can construct the total-step Laguerre-lke ncluson methods wth Newton s and Halley s correctons. We wll study the convergence rate of these methods smultaneously, usng a unform approach. For ths purpose we ndcate these methods wth the addtonal superscrpt ndces λ = 1 for Newton s correcton and λ = 2 for Halley s correcton. Consequently, we denote the correspondng vectors of dsk approxmatons as follows: Z 1 = Z 1 1,..., Z1 n = ZN,1,..., Z N,n Z 2 = Z 2 1,..., Z2 n = ZH,1,..., Z H,n. 140 M. S. Petkovć, D. M. Mloševć Both correctons Nz and Hz wll be also denoted by C 1 z and C 2 z, respectvely. For smplcty, we wll omt the teraton ndex for all quanttes at the m-th teraton, whle the quanttes at the m + 1-st teraton wll be denoted wth the addtonal symbol ˆ hat. Now we can wrte both methods n the unque form as Ẑ = z ninv 2 δ 1, + [n 1 nδ 2, δ1, 2 Q Z λ, Z λ] 1/2 13 for I n and λ = 1, 2. Snce we can apply two types of nversons n the calculaton of the sums 9, by combnng the nversons 1 and Ic n 13 we are n the possblty to construct four ncluson methods. 4. Convergence of the mproved methods Before consderng convergence propertes of the smultaneous nterval method 13 and ntal condtons for ts convergence, we wll gve some necessary estmates. It s easy to show that where z Z j + C λ z j = z ζ j + ξ λ j ε λ+1 j ; r j, ξ 1 j = Σ 1,j and ξ 2 j 1 + ε j Σ 1,j For brevty, let us set for λ = 1, 2: Σ 2 1,j = + Σ 2,j 2 + 2ε j Σ 1,j + ε 2 j Σ2 1,j + Σ 2,j. h λ j = md z Z j + C λ z j = z ζ j + ξ λ j ε λ+1 j, w λ j = 1 d λ j = h λ j r j h λ j q λ = ns λ 2, n n 1, r j s λ k, = = n 1 q qλ n/ε δ 1, 2, η = 25 2 v λ n j h λ j 1 z z j + C λ k k = 1, 2, j λ 2, λ s 1, f = nδ 2, δ1, 2 q λ, Lemma 2. Let the nequalty 12 hold. Then d λ j 5r 3ρ 2 ; w λ j 12 11ρ ; 41nn 1 nn 1r ε, γ = ρ3 5ρ 3., Laguerre-lke methods wth correctons f λ γr 1 ε 2 n 5 0. 2 v v n 1 f λ 1 + v λ ; γr 1; ε 5ρ ; n 1f λ ; η ; The proofs of the assertons v are smlar wth those gven n [13] and wll be omtted to save a space. Let IM be an teratve numercal method whch generates k sequences z m 1,..., z m k for the approxmaton of the solutons z1,..., zk. To estmate the order of convergence of the teratve method IM we usually ntroduce the error-sequences ε m = z m z = 1,..., k. The convergence analyss of ncluson methods wth correctons needs the followng asserton, whch s a specal case of Theorem 3 gven n [5]: Theorem 1. Gven the error-recurson 14 ε m+1 α k m tj ε, Ik ; m = 0, 1, 2,..., j where t j 0, α 0, 1, j k. Denote the matrx of exponents appearng n 14 wth T k, that s T k = [t j ] k k. If the non-negatve matrx T k has the spectral radus ρt k 1 and a correspondng egenvector x ρ 0, then the R-order of all sequences ε m I k s at least ρt k. In the sequel the matrx T k = [t j ] wll be called the R-matrx because of ts connecton wth the R-order of convergence. Let O R IM denote the R-order of convergence of an teraton method IM. For the total-step methods 13 we can state Theorem 2. Assume that ntal dsks Z 0 1,..., Z0 n I n and the nequalty Z 0 are chosen so that ζ 15 ρ 0 3n 1r 0 holds. Then the ncluson methods 13 are convergent and the followng s true for each I n and m = 1, 2,... : 1 ρ m 3n 1r m ; 142 M. S. Petkovć, D. M. Mloševć 2 ζ Z m for each I n and m = 1, 2,... ; 3 the lower bound of the R-order of convergence of the nterval methods 13 s λ + 4 λ = 1, 2, f INV1 = I c, O R = 4.646, f INV 1 = 1. Proof. Let us note that the condton 15 provdes that ntal dsks Z 0 1,..., Z0 n be dsjont. Indeed, for arbtrary par, j I n j we have z 0 z 0 j ρ 0 3n 1r 0 2r 0 r 0 + r 0 j, whch means that Z 0 Z 0 j = accordng to 5. The assertons of Theorem 2 wll be proved by mathematcal nducton. In the sequel we wll often use the nequalty 12 n the form 16 r ρ 1 3n 1 1 6, often wthout explct ctaton. Frst, let m = 0 and let us take nto consderaton the ntal condton 15. Then, accordng to Lemma 1, we mmedately obtan the mplcatons ζ Z ζ Z λ := Z C λ z I n ; λ = 1, 2. We should also prove that the ncluson dsks Z λ 1,..., Z n λ λ = 1, 2 are also dsjont. It s not dffcult to estmate so that we have md Z λ Nz 2r, Hz 2r, md Z λ j = z C λ z z j + C λ z j z z j C λ z C λ z j ρ 4r 3n 1r 4r r + r j. Thus, Z λ Z λ j = j because of 5. The above facts are necessary for the ncluson method 13 to be well defned. As mentoned above, we can combne two types of nversons n the teratve formulas 13. In what follows supersubscrpt ndces e and c wll be used to mark the type of the used nverson n The case INV 1 = Ic Laguerre-lke methods wth correctons Let us consder frst the case INV 1, INV 2 = Ic. Applyng the centered nverson 2 and usng crcular arthmetc operatons, we get S 1, Z λ, Z λ n 1 n = z Z j + C λ z j = 1 wherefrom S 2 1, Z λ, Z λ = j n j s λ 1, u λ j ; dλ j 2; 2 s λ 1, Applyng and of Lemma 2, we get S 2, Z λ, Z λ n = j λ h j ; r j s λ 5n 1r 1, ; 3ρ 2, 5n 1r 5n 1r 2 3ρ 2 + 3ρ 2 s λ 2; 41n 12 1, γ1 r, γ 1 = 10ρ 3. j 1 z Z j + C λ z j s λ 2, ; γ 2r, γ 2 = 2 = n j 41n 1 10ρ 3. u λ j ; 2 dλ j Usng the above nclusons of the sums S1, 2 and S 2,, we obtan Q Z λ, Z λ = ns 2, Z λ, Z λ n n 1 S2 1, Z λ, Z λ ns λ 2, n λ 2; s 1, γr = q λ ; γr. n 1 Snce f λ = nδ 2, δ1, 2 qλ, accordng to the asserton of Lemma 2 we conclude that 0 f λ ; γr, so that we can calculate square root of a dsk n 1 nδ 2, δ1, 2 Q Z λ, Z λ = n 1 f λ ; γr. Further, puttng u λ = δ 1, + [ n 1f λ, and usng the asserton v of Lemma 2, from the teratve formula 13 we obtan u λ 17 Ẑ z ninv 2 ; η. Usng the dentty 7 we fnd ] 1/2 f λ = nδ 2, δ 2 1, q + q q λ = 1 = 1 n 1 n/ε δ 1, v λ. n 1 n/ε δ 1, 2 + q q λ 144 M. S. Petkovć, D. M. Mloševć Accordng to ths and v of Lemma 2 we obtan u λ = δ 1, + [ n 1f λ ] 1/2 = δ 1, + n/ε δ 1, 1 + v λ δ 1, + n/ε δ 1, 1; ε 5ρ = n/ε ; n ε δ 1, =: U. 5ρ Here we have taken nto account the crteron for the selecton of the proper value of the square root whch yelds n/ε δ 1, 2 = +n/ε δ 1,. Usng 6 and the nequalty we fnd 18 n ε δ 1, n 1 + ε u λ = 1 r n j 1 1 z ζ j n 1 + r ρ md U rad U = n ε n ε δ 1, n 5ρ r n. By 12 and 18 we obtan u λ 7n 1r 1 30ρ r n n 1, 6 7n 1 6 η 90n 7 25 r2 nn 1 90r 2 ρ 3 1 n 7 r nn n 1 = 1 n 7 r 90 25n 54n 1 2 1 n 11 0. r 20 Accordng to the last nequalty we conclude that 0 / u λ ; η so that the teratve processes 13 are well defned and Ẑ s a closed dsk. Then from 17 we obtan Hence Ẑ ˆD c ˆr = rad Ẑ 1 := z n u λ 16n 1 ε 3 r ρ 3, nη u λ ; u λ u λ η = η u λ η. 25n 2 n 1 ε 3 r 2ρ 3 n 7 n ρ Laguerre-lke methods wth correctons because of 2 n n 2 n From the above relaton we conclude that 19 and also, by 12, ˆr = O ɛ 3 r 16, for all n ˆr r 6. Snce ξ λ j wherefrom = O M 1, we have n Σ 1, s λ 1, = j Σ 2 1, s λ 1, 1 w λ j z ζ j = 2 = Σ 1, s λ 1, n j ξ λ j ε λ+1 j = O z ζ j h λ M ɛ λ+1, j Σ 1, + s λ 1, = O M ɛ λ+1 and Σ 2, s λ 2, = = n j n j 1 z ζ j 2 w λ j 1 w λ j z ζ j w λ j = O M ɛ λ+1. z ζ j Furthermore, usng the relaton q q λ = n we conclude that and Therefore, the quantty v λ Σ 2, s λ 2, n n 1 q q λ = O M ɛ λ+1 v = O M ɛ λ+3. Σ 2 1, s λ 1, 2, s very small so that we can use the approxmaton [ 1 + v λ ] 1/2 = 1 + vλ 2. 146 M. S. Petkovć, D. M. Mloševć Accordng to ths we have u λ n = δ 1, + δ 1, 1 + v λ n = δ 1, + δ 1, 1 + O ɛ ε ε λ+3 = n ε + O M ɛ λ+2. For the center ẑ of Ẑ we obtan from 13 and 17 whence 21 ˆε = ẑ ζ = ẑ = md Ẑ = z n u λ nε ε n + O O M ɛ λ+4 M ɛ λ+3 = n + O M ɛ λ+3 = O ɛ λ+4 snce n + O M ɛ λ+3 s bounded. Usng a geometrc constructon and the fact that the dsks Z m, and Z m+1 must have at least one jont pont the zero ζ, the followng relaton can be derved see [4] ρ m+1 ρ m r m 3r m+1. Usng 4.7 and the last nequalty for m = 0, we fnd ρ 1 ρ 0 r 0 3r 1 3n 1r 0 r 0 r0 2 6r1 3n 1 1 1, 2 wherefrom t follows 22 ρ 1 4n 1r 1. Ths s the condton 14 for the ndex m = 1, whch means that all assertons of Lemmas 1 and 2 are vald for m = 1. Especally, the nequalty 20 of the form r 1 r 0 /6 ponts to the contracton of the new crcular approxmatons Z 1 1,..., Z1 n. Usng the defnton of ρ and 28, for arbtrary par of ndces, j I n j we have z 1 z 1 j ρ 1 3n 1r 1 2r 1 r 1 + r 1 j. Therefore, n regard to 5, the dsks Z 1 1,..., Z1 n produced by 13 are dsjont. Repeatng the above procedure and the argumentaton for an arbtrary ndex m 0 we can derve all above relatons for the ndex m+1. Snce these relatons have already been proved for m = 0, by mathematcal nducton t follows that, f the condton 12 holds, they are vald for all m 1. In partcular, we have 23 ρ m 3n 1r m Laguerre-lke methods wth correctons the asserton 1 and 24 r m+1 rm 6. Wth regard to the nequalty 24 we conclude that the sequence r m tends to 0; consequently, the ncluson methods 13 are convergent. Furthermore, takng nto account that 23 holds, the assertons of Lemmas 1 and 2 are vald for arbtrary m, whch means that the Laguerre-lke methods 13 are well defned at each teratve step. Suppose that ζ Z m for each I n. Then from 8 and 13 t follows that ζ Z m+1 accordng to the ncluson sotoncty. Snce ζ Z 0 the assumpton of the theorem, by mathematcal nducton we prove that ζ Z m for each I n and m = 0, 1,... the asserton 2. Fnally, we wll determne the lower bound for the R-order of convergence of the methods 13 the asserton 3. The sequences z m and r m of the centers and rad of the dsks Z m produced by the algorthms 13 are mutually dependent. For smplcty, we adopt 1 ɛ 0 = r 0 0, whch means that we deal wth the worst case model. We note that such assumpton s usual n ths type of analyss and has no nfluence on the fnal result of the lmt process whch we apply n order to obtan the lower bound for the R-order of convergence. By vrtue of 19 and 21 we notce that these sequences behave as follows ɛ m+1 ɛ m λ+4, r m+1 ɛ m 3 r m. [ ] From these relatons we form the R-matrx T c λ =. Its spectral radus s ρt c 2 = λ+4 and the correspondng egenvector xc ρ = λ+3/3, 1 Hence, accordng to Theorem 1, we get O R 13 c ρ T c 2 = λ + 4 λ = 1, 2. It remans to dscuss the case when the exact nverson 1 s appled n the fnal step, that s, INV 2 = 1. Startng from the ncluson 17 we obtan 25 and Ẑ ˆD e := z n u λ ū λ 2 η ; 2 u λ η 2 η 2 26 ˆr = rad Ẑ u λ nη 2 η 14n 1 ε 3 r 2 ρ 3. 148 M. S. Petkovć, D. M. Mloševć From 25 we fnd 27 ẑ = md Ẑ = md ˆD e = z n = z u λ 1 η 2 / u λ 2. u λ nū λ 2 η 2 Accordng to the estmatons derved above, we have η = Orɛ and u λ = O M 1/ɛ, whch gves η 2 / u λ 2 = Or 2 ɛ 4. Usng the development nto geometrc seres, from 27 we obtan ẑ = z n 1 + η2 u λ 2 + = z n + O u λ M r 2 ɛ 5, wherefrom ˆε = ε n u λ u λ = ε 3 O M αɛ λ+1 + βr 2 ɛ 2, + O M r 2 ɛ 5 = ε 3 O M ɛ λ+1 + O M r 2 ɛ 5 where α and β are some complex quanttes such that α = O1 and β = O1. Hence 28 ˆε = ε 3 O M ɛ λ+1 λ = 1, 2, and we see that the relatons 26 and 28 concde wth 19 and 21. Consequently, the lower bound of the R-order of convergence of the ncluson methods 13 when INV 1 = I c, INV 2 = 1 s the same as n the case when INV 1, INV 2 = I c. 2 The case INV 1 = 1 Havng n mnd that ξ λ j Σ 1, s λ 1, = = Accordng to ths we get n j n j = O M 1, n ths case we get 1 z ζ j ξ λ j h λ j h λ j z ζ j h λ j h λ j
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