Applied Mathematics Letters

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1 Applied Mathematics Letters 5 (01) Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joural homepage: O ew computatioal local orders of covergece Miquel Grau-Sáchez a Miquel Noguera a Àgela Grau a José R. Herrero b a Techical Uiversity of Cataloia Departmet of Applied Mathematics II Jordi Giroa 1-3 Omega Barceloa Spai b Techical Uiversity of Cataloia Departmet of Computer Architecture Jordi Giroa 1-3 C Barceloa Spai a r t i c l e i f o a b s t r a c t Article history: Received 15 February 01 Received i revised form 18 April 01 Accepted 18 April 01 Keywords: Order of covergece Noliear equatios Iterative methods Four ew variats of the Computatioal Order of Covergece (COC) of a oe-poit iterative method with memory for solvig oliear equatios are preseted. Furthermore the way to approximate the ew variats to the local order of covergece is aalyzed. Three of the ew defiitios give here do ot ivolve the ukow root. Numerical experimets usig adaptive arithmetic with multiple precisio ad a stoppig criteria are implemeted without usig ay kow root. 01 Elsevier Ltd. All rights reserved. 1. Itroductio Oe-poit iterative methods with memory for solvig a oliear equatio f (x) = 0 where f : I R R ad I is a eighborhood of the root α usually cosider a sequece S = {x } N defied by x +1 = φ (x ; x... x j ) 0 where φ is the iteratio fuctio. A sequece S is said to coverge to α with local order of covergece R 1 if there exists the followig limit log e +1 = lim log e where e k = x k α is the error i the kth iterate (see [1]). This limit is also equal to R-order defied i [3]. For oe-poit method with memory (1) the error equatio is (1) () e +1 = C e ) (3) where C is a real umber 0 < < 1 ad we will cosider (1 + 5)/. The ozero costat C is called the asymptotic error costat. The local order of covergece of a iterative method i a eighborhood of a root is the order of the correspodig sequece. If it is the the method approximately multiplies by the umber of correct decimals after each iteratio. That is from () we get log 10 e +1 log 10 e for large eough. I the ext sectios the way to approximate four ew variats of the local order of covergece is aalyzed ad umerical experimets usig adaptive arithmetic with multiple precisio ad a stoppig criteria are implemeted without usig ay kow root for three of the four techiques. Correspodig author. addresses: miquel.grau@upc.edu (M. Grau-Sáchez) miquel.oguera@upc.edu (M. Noguera) agela.grau@upc.edu (À. Grau) josepr@ac.upc.edu (J.R. Herrero) /$ see frot matter 01 Elsevier Ltd. All rights reserved. doi: /j.aml

2 04 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) Defiitios ad first result Next we give the defiitios of Computatioal Local Order of Covergece (CLOC) that is a variat of COC [45] Approximated Local Computatioal Order of Covergece (ACLOC) Extrapolated Local Computatioal Order of Covergece (ECLOC) ad Petković Local Computatioal Order of Covergece (PCLOC). These three last cocepts are variats of ACOC ECOC [6] ad PCOC [7] respectively. After the work of Weerakoo ad Ferado [4] may other authors have cosidered the COC i their research (see [86] ad refereces therei). I all those papers the COC is used to test umerically the order of covergece of the methods preseted. Cosiderig () we provide a ew parameter with lower cost tha COC: Defiitio 1. The computatioal local order of covergece (CLOC) of a sequece S is defied by λ = log e log e (4) where x ad x are two cosecutive iteratios ear the root α ad e = x α. Notice that the last defiitio has lower cost because we use the logarithm fuctio applied to oly oe variable say e istead of a quotiet such as e /e which is used i [4]. The mai drawback of COC ad CLOC is that they ivolve the exact root α which i a real situatio it is ot kow a priori. To avoid this we itroduce three variats of CLOC that do ot use the exact root. Firstly we give a ew parameter cosiderig three cosecutive poits: Defiitio. The approximated computatioal local order of covergece (ACLOC) of a sequece S is defied by λ = log e log e (5) wheree = x x. Secodly i order to avoid the requiremet of the kowledge of the exact root α we cosider three cosecutive iterates x x x ad usig Aitke s extrapolatio we give the followig approximatio of α α = x (δ x ) δ x (6) where δ is the forward differece operator δx k = x k+1 x k ad (6) is the δ -Aitke procedure [9]. The we ca defie a ew approximatio for the errore = x α ad a ew computatioal order of covergece: Defiitio 3. The extrapolated computatioal local order of covergece (ECLOC) of a sequece S is defied by λ = log e log e (7) where ẽ = x α ad α is give by (6). Fially aother way to avoid formulas ivolvig the exact root α cosists i usig the values of two cosecutive iterates. That is from f (x ) ad f (x ) the ew computatioal order of covergece is: Defiitio 4. The Petković computatioal local order of covergece (PCLOC) of a sequece S is defied by λ = log f (x ) log f (x ). (8) This last parameter PCLOC is defied i hoor of Petković who i [710] cosider i aalogy of COC the followig value = log f (x +1)/f (x ) log f (x )/f (x ). As we show below for all sequece {x } covergig to α with startig poits x j... x 1 x 0 close eough to α the values of λ λ λ ad λ coverge to whe. There exist umerical problems where a huge umber of sigificat digits of the solutio is eeded. Such problems require the use of methods with a high order of covergece together with adequate arithmetics. We compute with a multiple precisio arithmetic or symbolic maipulators as Maple that allow us to work with a adaptive arithmetic that is to update the legth of the matissa i each iteratio by meas of the formula Digits := [ ( log e + )] (9)

3 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) where is the local order of covergece of the method ad [x] deotes the iteger part of x. Notice that the legth of the matissa is icreased approximately by the order of covergece. I our experiece i order to guaratee all the sigificat digits required we add uits to log e. Our first aim is to express e as a fuctio of e +1. I a first approximatio from (3) if we cosider e +1 = C e the we get e = C 1/ e 1/ +1. Substitutig this result i the secod term of the right side of (3) we obtai e +1 = C e / ) +1 ad e = C 1 e +1 / +1 ). Therefore expressig e i terms of e +1 we ca state the followig propositio: Propositio.1. Cosiderig true the hypothesis i (3) we have e = C 1/ e 1/ +1 / +1 ). (10) 3. Computatioal Local Order of Covergece (CLOC) A relatioship betwee λ ad is derived. I fact we prove that λ coverges to whe e 0. That is λ i the sese that lim λ = 1. Propositio 3.1. If λ is the CLOC defied i (4) ad is the order of covergece the log C λ = 1 + O log e where C is give i (3). Proof. To prove (11) we express λ i terms of e. Takig ito accout (3) we take log e = log C e ) = log e + log C + O(e ). (1) The λ = log e log e = log e + log C + O(e log e ad we obtai the assertio of the propositio. ) Notice that for the calculus of the CLOC (4) ad for updatig the adaptive arithmetic process (9) it is ecessary to kow the exact root α. I this case the followig stoppig criterio is applied: (11) e = x α < 10 η where η is the maximum umber of correct decimals ad 10 η is the required accuracy. (13) 4. Approximated Computatioal Local Order of Covergece (ACLOC) A relatioship betwee λ ad is obtaied. A ew techique to update the umber of sigificat digits i a adaptive multi-precisio arithmetic is give ad a ew stoppig criterio is suggested. Propositio 4.1. Let λ be the ACLOC defied i (5) ad the order of covergece the log C λ = 1 + O log e where C is give i (3). Proof. From the followig expressio: log e = log e e = log e + O( e /e ) (15) ad (5) we get λ = log e log e = log e + O( e /e ) log e + O( e /e ). (14)

4 06 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) Applyig (1) we obtai λ = log e + log C + O(e ν ) log e + O( e /e ) where ν = mi{( 1) } ad the proof is complete. Observe that from (11) ad (14) the expressios of λ ad λ +1 are idetical. That is if we approximate the theoretical value of the local order by the computatioal values λ adλ : = λ ± λ = λ +1 ± λ +1 the λ +1 λ. Our umerical experimets cofirm this relatio. Repeatig (3) twice we obtai e = C +1 e ) ad ow we writee /e i terms of e : e = e e = C +1 e e e e = C e O e τ + + O(e C e where τ = mi{ 1 }. Moreover we get ) C e + + O(e ) e + O(e + ) e = C (+1)/ e 1/ / ) sice from (10) e = C 1/ e 1/ / ) ad e = C 1/ e 1/ Substitutig (17) i (16) we have the followig propositio: / ). (16) (17) Propositio 4.. If we set e = x α ade = x x the e C 1/(1 ) e e /( 1) (18) where is the order of covergece ad C is give i (3). The result (18) allows us to substitute the error term i (9) by a expressio that does ot ivolve the exact root. Ideed we implemet the followig adaptive multi-precisio arithmetic scheme: 3 Digits := 1 e log +. (19) e Moreover from (18) we propose the followig stoppig criterio istead of (13): e < 10 η ( 1)/. e (0) 5. Extrapolated Computatioal Local Order of Covergece (ECLOC) We give a relatioship betwee λ ad a ew techique to update the umber of sigificat digits i a adaptive multi-precisio arithmetic ad a ew stoppig criterio. We start by derivig a expressio of ECLOC as a fuctio of the local order of covergece. Propositio 5.1. If λ is the ECLOC defied i (7) ad φ is the order of covergece the log C λ 1 + ( 1) log e where C is give i (3). (1)

5 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) Proof. Now we writeλ i terms of e. To do that we express log e as a fuctio of e ad e : log e = log e e log e e + e = log e + O( e /e ) log e + O( e /e ) We obtai = log e /e λ = log e log e = log + O( e /e ). () e /e log e /e 3 From (3) we deduce e = C e 1+O(e Next we get log C e 1 λ ) log C 1/ e 1/ / ) + O( e /e ) + O( e /e 3 ). ( 1) log e + log C + O(e ) ( 1) log e + log C + O(e / ) ad the proof is complete. Propositio 5.. Give e = x α ade = x α the ) ad takig ito accout (10) we have e 3 = C 1/ e 1/ 1+O(e / ). e C β e /( 1) where β = 1 1. (3) Proof. Takig ito accout e = C 1/ e 1/ / ) we write ẽ i terms of e : ẽ = (e e ) e e + e = C e +1 / C e ) + e C e / ) e + C 1/ e 1/ = C 1/ e ( 1)/ where τ = mi{ 1 }. Now from (4) ad e = C 1/ e 1/ ẽ = C 1/ C 1/ e 1/ / = C (1 )/ e ( 1)/ / ) τ/ ) (4) / ) we get ) ( 1)/ C 1/ 1 + O e 1/ / ) τ/. (5) τ/ ) From (5) we have e ( 1)/ C ( 1)/ ẽ from which the proof immediately follows. Notice that (3) allows us to implemet a iterative method (1) with a multi-precisio adaptive arithmetic. Istead of (9) we ow cosider the expressio: 3 Digits := 1 log ẽ +. (6) I additio as a alterative to (13) (3) provides the followig stoppig criterio ẽ < 10 η ( 1)/. (7) 6. Petković Computatioal Local Order of Covergece (PCLOC) I this sectio we provide a relatioship betwee λ ad. I additio we derive a ew techique to update the umber of sigificat digits i a adaptive multi-precisio arithmetic ad a ew stoppig criterio.

6 08 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) Propositio 6.1. If λ is the PCLOC defied i (8) ad is the order of covergece the log C Γ λ O log e where C is give i (3) ad Γ = f (α). Proof. Settig (8) f (x k ) = Γ e k + O(e k ) = Γ e k ( k )) k = 1 (9) ad from (3) e = Ce ) we have ) λ = log f (x ) log f (x ) = log e + log Γ C + O( e log e + log Γ + O( e ) Dividig the umerator ad deomiator i the right side of the precedig equatio by log e the proof is complete.. Propositio 6.. If we deote Q = f (x ) f (x ) the e C Q. Proof. Takig ito accout (9) ad (10) we have Q = C 1 1 e / ) ad the proof immediately follows. (30) The result (30) allows us to substitute the error i (9) by a expressio that does ot ivolve the exact root. Ideed we implemet the followig adaptive multi-precisio arithmetic scheme: Digits := 1 f (x log ) f (x ) +. (31) Moreover from (30) we propose the followig stoppig criterio istead of (13): f (x ) f (x ) < 10 η ( 1)/. (3) 7. Iterative methods ad umerical results We cosider i this sectio six iterative methods φ k k = 1 6 with local covergece order equal to ad respectively. The first three methods are oe poit iterative methods without memory kow as Newto s method Chebyshev s method [11] ad Schröder s method [1]. The other three schemes are iterative methods with memory amely the Secat method ad two variats (see [13]). They are defied as where φ 1 (x ) = x u(x ) φ (x ) = φ 1 (x ) 1 L(x ) u(x ) (34) 1 φ 3 (x ) = φ (x ) L(x ) M(x ) u(x ) (35) φ 4 (x ) = x [x x ] 1 f f (x ) φ 5 (x ) = φ 4 (x ) [x φ 4 (x )] 1 f f (φ 4 (x )) (37) φ 6 (x ) = φ 4 (x ) [x φ 4 (x ) x ] 1 f f (φ 4 (x )) (38) u(x) = f (x) f (x) L(x) = f (x) f (x) u(x) M(x) = f (x) 3! f (x) u(x) ad [x y] 1 y x f = f (y) f (x). We poit out that CLOC ad PCLOC ca be computed if 1 ACLOC if ad ECLOC if 3. If we have a method of higher order of covergece the multi-precisio arithmetic is required ad will be used where ecessary to obtai may correct figures. I geeral from guess poits x j... x 1 x 0 we obtai the admissible poits x 1... x I. Notice that if we use a arithmetic with at most η decimal digits with stoppig criterio e I+1 < 10 η the x I+1 will ot be cosidered sice it would eed a matissa with higher legth to hold all the correct decimals. Hece as x I is the best admissible poit we will take = I i the defiitios of CLOC ACLOC ECLOC ad PCLOC. (33) (36)

7 M. Grau-Sáchez et al. / Applied Mathematics Letters 5 (01) Table 1 Test fuctios their roots ad the iitial poits cosidered. f (x) α x 0 {x 1 x 0 } f 1 (x) = x 3 3x +x {.5.60} f (x) = x 3 + cos x { } f 3 (x) = si x + 1 x { } f 4 (x) = (x+1) e x { } f 5 (x) = e x +7x { } f 6 (x) = e x + cos x { } f 7 (x) = x 3 l x { } Table Mi max iterval for error bouds. f 1 f f 3 f 4 f 5 f 6 f 7 I ( λ) I (λ) I (λ) I ( λ) φ [.8e 5 1.1e 3] φ [8.9e 5 3.3e 3] φ [8.8e 6 1.3e ] φ [8.1e 6 5.8e 4] φ [5.5e 5 3.0e 3] φ [3.6e 5 3.7e 3] [3.7e 5 1.5e 3] [1.0e 4 6.0e 3] [.0e 5 3.0e ] [7.9e 6 6.8e 4] [1.0e 4 4.4e 3] [1.4e 4 1.6e ] [5.6e 5.e 3] [1.4e 4 9.9e 3] [3.5e 5 5.1e ] [1.e 5 9.4e 4] [1.3e 4 7.e 3] [1.6e 4 1.7e ] [6.0e 5 1.e 3] [.1e 4 4.5e 3] [1.6e 3 1.e ] [3.e 5 5.5e 3] [1.9e 3 3.3e 3] [.8e 4 4.5e 3] Defiitio 5. The computatioal values cosidered i all umerical experimets are λ = log e I log e I 1 λ = log e I log e I 1 λ = log e I log e I 1 ad λ = log f (x I) log f (x I 1 ). (39) We have tested the precedig methods o seve fuctios usig the Maple computer algebra system. We have computed the root of each fuctio startig from the same iitial approximatio x 0 i (33) (35) ad {x 1 x 0 } i (36) (38). Depedig o the computatioal order of covergece used CLOC (4) ACLOC (5) ECLOC (7) or PCLOC (8) we stop the iterative method whe coditio (13) (0) (7) or (3) is fulfilled. Note that i all cases η = 00. The set of test fuctios preseted here were previously cosidered i [14]. Table 1 shows the expressio of each of these fuctios their root with twety five sigificat digits ad their iitial approximatio. The latter is the same for all the iterative methods cosidered cosiderig oe or two guess poits depedig o whether the algorithm works without or with memory. For each method ad fuctio we have applied the four techiques with adaptive multi-precisio arithmetic (9) (19) (6) ad (31). The umber of ecessary iteratios to get the desired precisio ad the values of iterated poits x 1... x I are the same. Table shows the umber of iteratios eeded to compute the root. I additio the last four colums show the iterval with miimum ad maximum error produced i the computatio of the correspodig Computatioal Local Orders of Covergece (CLOC ECLOC ACLOC or PCLOC) for the seve test fuctios. For istace cosiderig the CLOC ad Newto s method φ 1 let us deote I ( λ) the resultig error iterval obtaied i the computatio of the CLOC: [mi λ k max λ k ] for each fuctio f k k = From these umerical tests we ca coclude that the CLOC produces the best approximatios of the theoretical order of covergece of a iterative method. However the kowledge of the root is required. Coversely as we ca see i the defiitios of ACLOC (5) ECLOC (7) ad PCLOC (8) these parameters have the advatage that they do ot ivolve the expressio of the root α. Actually i real problems we wat to approximate the root which is ot kow i advace. For practical purposes (see Table ad Propositios ad 6.1) we recommed ECLOC sice it presets the best approximatio of the local order. Nevertheless PCLOC is a good practical parameter i may cases because it requires less operatios. Ackowledgmet The research was partially supported by the projects MTM C0-01 ad TIN of the Spaish Miistry of Sciece ad Iovatio. Refereces [1] D.D. Wall The order of a iteratio formula Math. Tables Aids Comput. 10 (1956) [] L. Torheim Covergece of multipoit iterative methods J. ACM 11 (1964) 10 0.

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