Exponentially convergent parallel algorithm for nonlinear eigenvalue problems

Transcription

1 IMA Joural of Nuerical Aalysis (7) 7, doi:.93/iau/drl4 Advace Access publicatio o Jauary 7, 7 Expoetially coverget parallel algorith for oliear eigevalue probles I. P. GAVRILYUK Berufsakadeie Thürige, Staatliche Studieakadeie, A Warteberg, D-9987 Eiseach, Geray AND A. V. KLIMENKO, V. L. MAKAROV AND N. O. ROSSOKHATA Departet of Nuerical Matheatics, Istitute of Matheatics of Natioal Acadey of Scieces of Ukraie, 3 Tereshchekivs ka Street, 6 Kyiv-4, Ukraie [Received o Jue 6; revised o 4 Noveber 6] A ew algorith for oliear eigevalue probles is proposed. The uerical techique is based o a perturbatio of the coefficiets of differetial equatio cobied with the Adoia decopositio ethod for the oliear part. The approach provides a expoetial covergece rate with a base which is iversely proportioal to the idex of the eigevalue uder cosideratio. The eigepairs ca be coputed i parallel. Nuerical exaples are preseted to support the theory. They are i good agreeet with the spectral asyptotics obtaied by other authors. Keywords: oliear eigevalue proble; parallel algorith; expoetially coverget algorith.. Itroductio Aog the variety of uerical techiques for eigevalue probles for differetial equatios, two pricipally differet approaches ca be idetified: ethods based o a approxiatio of solutios of the differetial equatio (e.g. fiite-differece (Collatz, 96; Adrew, 3, ; Adrew et al., 995), fiite-eleet (Babuška & Osbor, 99) or spectral approxiatios (Trefethe, ; Babeko, 986)) ad ethods based o a approxiatio of the coefficiets of differetial equatio (e.g. the piecewise costat approxiatio (Kryloff & Bogolioubov, 96), the piecewise polyoial approxiatio (Gordo, 969; Pruess, 973; Pruess & Fulto, 993; Bailey et al., 99; Marletta & Pryce, 99; Greeberg, 99), etc.). Both of these approaches have their areas of applicability, ad hece, advatages ad drawbacks. The pricipal peculiarity of the first approach is the capability to calculate oly a restricted uber of the first few eigevalues ad the correspodig eigefuctios. The uber of coputable eigevalues ad eigefuctios is restricted by the size of the grid step. That is why uerical techiques based o the first approach, such as fiite eleet or fiite differece ethods, are effective for fidig low-idexed eigevalues, but ieffective for fidig high-idexed eigevalues. A iportat exaple of a proble where these ethods are ieffective is the coputatio of the spectral desity fuctio, where about eigevalues ad eigefuctios should be calculated (Pryce, 993, p. 73). Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, Eail: ipg@ba-eiseach.de Eail: akarov@iath.kiev.ua Correspodig author. Eail: ataross@gail.co c The author 7. Published by Oxford Uiversity Press o behalf of the Istitute of Matheatics ad its Applicatios. All rights reserved.

2 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 89 Sice the secod approach is based o the approxiatio of the coefficiets of the differetial equatio by piecewise polyoials istead of the approxiatio of eigefuctios o a fiite grid, there is o restrictio o the uber of coputable eigevalues ad correspodig eigefuctios. However, i order to fid the solutio of the approxiate differetial equatio explicitly (i ters of special fuctios o each subiterval), the degree of the polyoials i the piecewise approxiatio of the coefficiets of the differetial equatio is o higher tha two (Gordo, 969). Hece, this approach is effective for piecewise sooth coefficiets, but sees ieffective i ay cases where the coefficiets are varyig rapidly. I spite of the fact that both of the uerical approaches etioed above are well-established for liear eigevalue probles ad there are uerous publicatios o the uerical solutio of liear eigevalue probles as well as the fact that the aalytical properties of oliear spectral probles are relatively well studied (Appell et al., 4; Heiz, 986, 995, 996; Heid et al., ; Zhidkov, ), to the author s kowledge there are far fewer publicatios o oliear eigevalue probles (see, e.g. Appell et al., 4; Heiz, 995, 996; Heid et al., ; Shibata, 99, 995, 997, a,b; Adrew, 994, ad the refereces therei). The reaso for this is that uerical techiques for oliear eigevalue probles require special attetio to approxiate their oliear parts. Oe such approach is the Adoia decopositio ethod which is sufficietly well studied for boudary-value probles (see, e.g. Abbaoui et al.,, ad the literature cited therei). I Kao & Jiag (5), the classical power ethod for the liear part is cobied with the Adoia decopositio ethod for the oliear part to develop a Hailtoia iverse iteratio ethod for oliear eigevalue probles, but o theoretical justificatio is give. I this paper, we develop a ew uerical techique for oliear eigevalue probles. The techique is based o a approach siilar to the hootopy ethod ad cosists of ibeddig the origial proble ito a paraetric set of probles (with paraeter t), such that it becoes a liear eigevalue proble with costat coefficiets for t = ad the origial oliear proble for t =. The trasitio fro t = to t = usig the Taylor series results i a recurrece algorith. Such a ethod, called the fuctioal-discrete ethod (FD ethod), was proposed for Stur Liouville probles i Makarov (99, 997) ad the developed i Makarov & Ukhaev (997), Makarov et al. (4), Badyrskiĭ & Makarov () ad Badyrskiĭ et al. (999, a,b, 5). It provides a expoetial covergece rate which iproves with icreasig eigevalue idex. The idea of the FD ethod (i.e. the derivatio of the algorith) is close to the idea of the hootopy ethod (see e.g. Adas & Plu (a,b); Plu (99, 99, 997)), but the aalysis ad ipleetatio are origial. To approxiate the oliear ter, i the preset paper we use the Adoia decopositio ethod. Such a atural cobiatio of the techique fro Makarov (99, 997) with the Adoia decopositio ethod possesses a iportat advatage, i that our algorith for oliear eigevalue probles is parallelizable ad coverges with the sae (expoetial) characteristics as the FD algorith for liear probles. The drawback of this approach for oliear probles as copared with the FD ethod for liear probles is that we ca oly guaratee a priori its (expoetial) covergece startig fro soe idex defied by the covergece coditio rather tha for all idexes. The paper is orgaized as follows: I Sectio, we describe the uerical techique for oliear eigevalue probles icludig a additioal differetial oralizig coditio, that of a vaishig first derivative. We prove a covergece theore which shows the expoetial covergece rate. Nuerical exaples are give to support the theory. Sectio 3 is devoted to the oliear eigevalue proble with a additioal itegral oralizig coditio. We obtai a covergece result siilar to the result obtaied i Sectio, ad illustrate the algorith with uerical exaples which cofir the theoretical results. We also copare the uerical eigevalues obtaied by our algorith Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

3 8 I. P. GAVRILYUK ET AL. with the correspodig spectral asyptotics fro Shibata (a,b). They are foud to be i good agreeet.. Nuerical algorith for a differetial oralizig coditio We cosider the eigevalue proble Au u (x) + λu(x) N(u(x)) =, x (, ), (.) u() =, u() = with the oralizig coditio u () = M, (.) where N: R R is a aalytic fuctio such that N (k) u N() =, (u) = d k N(u) du k N (k) ( u ), u R, k =,,..., ad N( u ) is a sooth fuctio with oegative derivatives. Note that i cotrast with the liear case where the oralizig coditio iflueces a ultiplicative costat oly, the solutio of a oliear eigevalue proble depeds essetially o the oralizig coditio. The geeral schee of our uerical approach is the followig: we ibed proble (.), (.) ito the paraetrical proble set where t > is a paraeter. u(x, t) x + λ(t)u(x, t) t N(u(x, t)) =, x (, ), u(, t) =, u(, t) =, u(, t) x = M, REMARK. The existece of a solutio to (.4) follows fro the results of Zhidkov (). We obviously have that u(x, ) = u(x), λ() = λ, (.3) (.4) Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, which suggests the idea to look for the solutio of (.4) i the for λ (t) = λ ( j) t j, u (x, t) = u ( j) (x)t j. (.5) Thus, settig t = we obtai λ = j= j= λ ( j), u (x) = j= j= u ( j) (x), (.6)

4 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 8 provided that the series (.6) coverges. I order to fid the suads of these series, we itroduce additioally the ew coefficiets A ( j) (x) by N u ( j) (x)t j = A ( j) (x)t j, j= A ( j) (x) = j! j= j N ( j= u ( j) (x)t j) t j. t= (.7) Substitutig (.5) ad (.7) ito (.4) ad coparig the coefficiets, we obtai the followig recurrece relatios: where ad A ( j) (u () d j+) u( dx (x) + λ () j+) u( u ( j+) () =, u ( j+) () =, F ( j+) (x) = j),..., u( ) = A ( j) (x) = p= N (α ) u (u () α + +α j = j j) [u( (x)] α j, j > (α j )! (x) = F ( j+) (x), x (, ), du ( j+) () =, j =,,,..., dx λ ( j+ p) u (p) (x) + A ( j) A () (u () ) = N(u () ) (u () (x))[u() (x)] α α (α α )! are the Adoia polyoials (Abbaoui et al., ). Note that (.8) j),..., u( ) (.9) j ) [u( (x)] α j α j, (α j α j )! (.) Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, N(u (x)) = j= A ( j) (x). (.) The solvability coditio for (.8) yields λ ( j+) = + λ ( j+ p) u (p) (ξ)u () p= A ( j) (u () (ξ)dξ / j) (ξ),..., u( (ξ))u () (ξ)dξ u () (.)

5 8 I. P. GAVRILYUK ET AL. ad uder this coditio the solutio of (.8) ca be expressed as u ( j+) (x) = û ( j+) (x) x si [π(x ξ)] = π λ ( j+ p) u (p) p= (ξ) + A ( j) (u () (ξ),..., u( j) (ξ)) dξ, (.3) ad u () = [u() (x)] dx is the usual L -or. For λ () ad u () (x), we have the so-called base uperturbed proble with the solutio d dx u() (x) + λ() u() (x) =, x (, ), u () () =, u() () =, du () () dx u () We defie the approxiatios of rak by λ = λ ( j), si (π x) (x) = M, x (, ), π λ () = (π), =,,.... j= = M (.4) (.5) u(x) = u ( j) (x). (.6) The error of the algorith (.8) (.6) ca be estiated i the followig way: u u u ( j), λ λ j= j=+ λ ( j) j=+ where u = ax x [,] u(x). Forulas (.) ad (.3) together with (.3) ad (.) iply the estiates u ( j+) λ ( j+ p) u (p) π p= + N (α ) ( u () ) u() α α (α α + +α j = j α )! u( j ) α j α j j) u( α j (α j α j )! (α j )!, λ ( j+) λ ( j+ p) u (p) p= + α + +α j = j N (α ) ( u () ) u() α α, α j α j (α α )! u( j ) (α j α j )! (.7) j) u( α )/ j u () (α j )!. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

6 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 83 Itroducig the ew variables v j+ = leads to the followig syste of iequalities: j+) u( u () (π) j+, μ j+ = λ ( j+) (π) j (.8) where A j (N, v,..., v j ) = with v j+ μ j+ α + +α j = j μ j+ p v p + A j (N, v,..., v j ), p= μ j+ p v p + A j (N, v,..., v j ), p= u () α N (α ) ( u () ) The, we write the correspodig syste of equatios as follows: μ j+ = v j+ = v α α (α α )! v μ j+ p v p + A j (N, v,..., v j ), p= α j α j j α j j (α j α j )! v (α j )!. μ j+ p v p + A j (N, v,..., v j ) = v j+ μ j+ v, j =,,..., p= v = u() u () =. (.9) (.) Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, Obviously, there exists a uique solutio to syste (.) which ajorizes the solutio of the syste of iequalities (.9), i.e. v j+ v j+, μ j+ μ j+. For further coveiece, suppose u () = M, which is ot a pricipal liitatio of our π aalysis. The, the eliiatio of μ j+ fro syste (.) leads to v j+ = v j+ p v p + ( + v ) p= α + +α j = j α j α j j N (α ) ( u () v α α ) (α α )! v (α j α j )! v (α j )!. α j j (.)

7 84 I. P. GAVRILYUK ET AL. Itroducig the geeratig fuctio for the sequece {v j } by f (z) = z j v j, (.) j= we have N( f (z)) = = j= j= z j N( f (z)) ( j) z z= z j α + +α j = j α j α j j N (α ) ( u () v α α ) (α α )! v (α j α j )! v (α j )!. (.3) Multiplyig (.) by z j+, suig both sides over j fro to ad takig ito accout (.) ad (.3), we obtai the followig equatio: f (z) v = [ f (z) v ] + ( + v )zn( f (z)). Settig f (z) = f (z) v = f (z) f (), we arrive at the followig quadratic equatio with respect to f (z): or [ f (z)] f (z) + ( + v )zn( f (z) + f ()) = [ f (z)] [ ( + v )zn (v )] f (z) + q =, where q = ( + v )z[n( f (z) + v ) N (v ) f (z)]. Due to the assuptios o N, we have q >. Thus, there exists a positive z ax = R such that z ax < ( + v )N (v ) α j j (.4) ad for all z [, R] the series (.) coverges, i.e. there exists a positive geeratig fuctio. O the other had, this eas that R j v j c/j +ε Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, with soe positive ε ad a positive costat c. Returig to (.8), we obtai Aalogously, we arrive at the iequality u ( j+) c ( j + ) +ε (π R) j+. λ ( j+) c R( j + ) +ε (π R) j.

8 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 85 Thus, the series (.6) coverges ad (.7) iplies the estiates provided that u u j=+ λ λ u ( j) c ( + ) +ε ( ) c R( + ) +ε, π R ( ) +, π R (.5) r = <. (.6) π R The last coditio is fulfilled at least for large eough. Hece, we have obtaied the followig covergece result. THEOREM. Let coditio (.3) holds ad let the idex of the eigepair satisfies (.6) with R give by (.4). Uder these coditios, the uerical algorith (.8) (.6) coverges super-expoetially with the estiate (.5). COROLLARY. Oe ca see fro (.6) that the covergece iproves with the icrease of the idex of the eigepair uder cosideratio. That is why we call this behaviour as a super-expoetial covergece. COROLLARY.3 Usig the ajorat series (.6) by usual arguets, we ca show that the series (.5) coverges uiforly i t ad represets the solutio of proble (.4). Moreover, the assuptios of Theore. guaratee the aalytical depedece of the eigevalues ad the eigefuctios for correspodig o the paraeter t at least i the disc t, ad we ca write λ ( j) = j! u ( j) (x) = j! d j λ (t) dt j, t= j u (x, t) t j. t= To illustrate the algorith, we cosider the followig uerical exaples. (.7) EXAMPLE.4 Usig the stadard iitial-value proble solver of Maple, oe ca easily fid (e.g. by the shootig ethod) that the exact first eigevalue λ (e) of the proble u (x) + λu(x) sih [u(x)] =, x (, ), u() = u() =, u () = belogs to the iterval ( , ). The algorith described above provides the followig values: λ = λ () = π = , λ () = ad () λ = () λ + λ () = with the error r () = λ (e) () λ = EXAMPLE.5 Let us cosider the eigevalue proble u (x) + [λ u (x)]u(x) =, x (, ), u() =, u() =, u () =. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

9 86 I. P. GAVRILYUK ET AL. TABLE Approxiatios of λ = λ δ TABLE Approxiatios of λ 3 = λ We have calculated the exact eigevalues (withi the accuracy ε =. ) by the bisectio ethod usig the Maple procedure dver78 with the paraeters abserr =., relerr =. ad with digits = 4. The, we have foud the approxiate eigevalues accordig to our approach. The results are give i Tables 3. The error o a logarithic scale is depicted i Fig.. Fro Fig., we ote that the absolute value of the deviatio δ = λ exact λ of the approxiate eigevalues fro the exact oes obeys the fuctioal depedece log δ a + c ad is i good agreeet with Theore.. Oe ca also see that the slope icreases as the idex of the trial eigevalue icreases. This cofirs Corollary.. 3. A itegral oralizig coditio I ay applicatios, the itegral oralizig coditio δ Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, u (x)dx = M (3.) is used for proble (.). I this case, istead of (.4) we have the proble u(x, t) x + λ(t)u(x, t) t N(u(x, t)) =, x (, ), u(, t) =, u(, t) =, u (x, t)dx = M.

10 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 87 TABLE 3 Approxiatios of λ 6 = λ δ6 FIG.. Depedece of the absolute error o the discretizatio paraeter of algorith (.6) o a sei-logarithic scale. Usig the approach described i Sectio, we arrive at the followig algorith for the uerical solutio of proble (.), (3.). Algorith EVIN(,) The algorith coputes the approxiate eigepair correspodig to the idex give by (.6).. Give, ad the solutio u () (x) = M si (π x), x (, ), λ () = (π) (3.) Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, of the base proble d dx u() u () (x) + λ() u() (x) =, x (, ), () =, u() () =, u() = M, (3.3) set λ = λ (), u (x) = u () (x). (3.4)

11 88 I. P. GAVRILYUK ET AL.. For j =,,...,. Copute λ ( j+) = + λ ( j+ p) u (p) (ξ)u () p= A ( j) (u () j) (ξ),..., u( (ξ)dξ / (ξ))u () (ξ)dξ M. (3.5). Solve with.3 Copute d j+) u( dx (x) + λ () u () j+) u( (x) = F ( j+) (x), x (, ), u ( j+) () =, u ( j+) () =, j+) (x)u( (x)dx = F ( j+) (x) = p= j+ λ = λ j + λ ( j+), p= u (p) λ ( j+ p) u (p) (x) + A ( j) (x)u ( j+ p) (x)dx, (x, u () (3.6),..., u( j) ). (3.7) j+ u (x) = j u (x) + u ( j+) (x). (3.8) I order to ivestigate the covergece of algorith EVIN, we look for the solutio u ( j+) (x) of the ohoogeeous boudary-value proble (3.6) i the for u ( j+) (x) = B ( j+) si (π x) + û ( j+) (x), (3.9) where û ( j+) (x) is give by (.3). Substitutig u () (x) = M si (π x) ito the last equatio i (3.6) for the ukow costat B ( j+), we obtai which yields M j+) B( = M j+) B( = u () + p= u () j+) (x)û( (x)dx j+) (x)û( (x)dx B (p) p= si (π x)û ( j+ p) (x)dx + p= u (p) B (p) B ( j+ p) p= û (p) (x)u ( j+ p) (x)dx, (x)û ( j+ p) (x)dx. (3.) Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

12 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 89 Further, relatios (3.5), (.3), (3.9) ad (3.) iply λ () M u() A() = B () π A(), u() N( u () ), M π A() Let us set b = M ad v = M, ad prove by iductio that for all j =,,..., the estiates λ ( j) l j (π) j,. B ( j) b j (π) j, (3.) u ( j) v j (π) j hold with soe positive ubers l j, b j ad v j. Actually, supposig the validity of these iequalities for j =,...,, we obtai fro (3.5) for j = +, λ (+) M(π) + α + +α = l + p v p p= N (α ) ( u () ) (v ) α α (α α )! (v ) α α (α α )! (v ) α (α )! }. (3.) Relatios (.3) ad (3.) yield B (+) M M (π) + l + p v p + M A (N, v,..., v ) p= + b p b + p + ( p ) b p l + p k v k + A p (N, v,..., v p ) + p= p= k= p l p k v k + A p (N, v,..., v p ) p= k= p l + p k v k + A p (N, v,..., v p ). (3.3) k= Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, Takig ito accout (3.) ad (3.3), we obtai fro (3.9) u (+) b (π) l + p v p + A (N, v,..., v ). (3.4) p=

13 83 I. P. GAVRILYUK ET AL. Thus, (3.) is satisfied for j = + provided that the ubers l +, b + ad v + are defied by the recursive syste of equatios l + = M l + p v p + p= α + +α = b + = M l M + p v p + M A (N, v,..., v ) p= p= + ( p ) b p l + p k v k + A p (N, v,..., v p ) + p= k= N (α ) ( u () ) (v ) α α (α α )! (v ) α α (α α )! (v ) α (α )!, b p b + p p p l p k v k + A p (N, v,..., v p ) l + p k v k + A p (N, v,..., v p ), p= v + = b + + k= l + p v p + A (N, v,..., v ). p= I order to solve this syste, we use the ethod of geeratig fuctios. The geeratig fuctios are itroduced by f (z) = v j z j, g(z) = l j+ z j, h(z) = b j z j (3.5) j= with f () = M. The, (3.), (3.3) ad (3.4) iply the followig syste of equatios: j= k= j= f (z) v = h(z) b + z[g(z) f (z) + N( f (z))], g(z) = M [g(z)[ f (z) v ] + N( f (z))], { Mz[g(z) (3.6) h(z) b = f (z) + N( f (z))] + M 4 [h(z) b ] } + z[h(z) b ][g(z) f (z) + N( f (z))] + z [g(z) f (z) + N( f (z))]. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, Eliiatig g(z) ad h(z), we arrive at the followig equatio for f (z): 4 [ f (z)]4 3 + [( M[ f (z)] ) M + ( ) ] MzN( f (z)) [ f (z)] [(3 + 4 )M M + (5 )MzN( f (z))] f (z) + ( + )M + 6M MzN( f (z)) + (3 )Mz [N( f (z))] =. (3.7)

14 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 83 I order to show that this equatio possesses solutios, we rewrite the secod equatio i syste (3.6) i the for g(z)[ M f (z)] = N( f (z)). (3.8) For z =, the fourth-order equatio (3.7) has the followig solutios: f () = { M, M, M, ( + ) M}. Sice f () = M = v, the solvig (3.7) with respect to f (z), we should choose that brach of the ultivalued fuctio f (z) for which f () = M. The fuctios g, f ad N are positive, ootoe ad ted to as z, which together with (3.8) ad a cotiuity arguet provide the existece of a fiite z ax = R such that (3.7) is solvable for z [, R] ad usolvable for all z > R. The value R is obviously the covergece radius for the series (3.5), ad therefore R j v j < with soe positive ε ad a positive costat c. Fro (3.4), we obtai Aalogously, estiate (3.) iplies c j +ε, R j l j < c j +ε, R j b j < c j +ε u ( j+) c ( j + ) +ε (π R) j+. λ ( j+) c R( j + ) +ε (π R) j. Thus, the series (.6) coverges ad (.7) iplies the estiates provided that u u j=+ λ λ ( ) c + ( + ) +ε, π R ( ) c R( + ) +ε, π R u ( j) (3.9) r = <. (3.) π R The last coditio is fulfilled at least for large eough. Thus, we have proved the followig stateet. THEOREM 3. Let coditios (.3) ad (3.) be satisfied; the the approxiate eigepair of the idex coputed by algorith (3.) (3.8) coverges super-expoetially to the correspodig eigepair of proble (.), (3.) ad the estiates (3.9) hold. COROLLARY 3. Oe ca easily see fro (3.) that the covergece rate of algorith (3.) (3.8) iproves with the icrease of the eigevalue idex. COROLLARY 3.3 By usig algorith (3.) (3.8), we ca calculate the eigevalues λ ad the correspodig eigefuctio u (x) for = N, N +,..., begiig with soe N. I order to fid the eigepairs (λ, u (x)), =,..., N, aother ethod should be used, e.g. the fiite-differece ethod or a ethod siilar to Babeko (986). Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

15 83 I. P. GAVRILYUK ET AL. FIG.. Fuctio f (z) for N( f ) = f 3 ad f () =. EXAMPLE 3.4 Let us cosider the eigevalue proble with the itegral oralizig coditio u (x) + λu(x) u 3 (x) =, x (, ), u() = u() =, u =. (3.) The brach of the ultivalued fuctio f (z) with f () = is preseted i Fig.. Oe ca easily see that.5874 < R < Theore 3. asserts that our ethod coverges if /( π.) <, i.e. for 3. However, this is a sufficiet coditio ad for a give proble, the ethod ca coverge also for which is uch saller tha this value. For exaple, let us copute the first eigevalue of proble (3.). Sice the exact solutio of this proble is ot kow, i order to cotrol the error we aalyse the L -or of the residual r = ( ) d u (x) dx + ( u ) 3 λ u (x) (x) dx /. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, The solutio of the base proble is λ () = (π) ad u () = si (π x). This solutio represets the approxiatio of the rage zero, i.e. λ λ = λ () = π, u (x) u (x) = u () (x) = si (π x).

16 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 833 The or of this approxiatio is exactly oe ad the L -or of the residual r (x) coputed with Maple 9.5 is The first correctios are u =, r = λ () = 3, u () (x) = 3 si(π x) (si(π x)) 3 6π 4π. Correspodigly, the first approxiatio (the approxiatio of rak oe) is λ = λ + λ () = π + 3, u (x) = u (x) + u () (x) = si(π x) + 3 si(π x) (si(π x)) 3 6π 4π. The ors of the first approxiatio of the eigefuctio ad the correspodig residual are u =.55, r = Proceedig further i accordace with our algorith, we obtai the followig secod approxiatio: λ = λ + λ () = π π, u (x) = u (x) + u () (x) = si(π x) + 3 si(π x) 6π + (si(π x)) 3 4π 9 si(π x) 5π 4 si(π x)( 3(cos(π x)) + (cos(π x)) 4 ) 3π 4. The ors of this approxiatio of the eigefuctio ad the correspodig residual are u = , r =.468. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, The results are also give i Tables 4 6. The L -or of the residual o a logarithic scale, for differet eigevalues, is depicted i Fig. 3. Fro this figure, we see that the L -or of the residual obeys log r α + c. We ca also see that the slope icreases as the idex of the eigevalue icreases (copare with Corollary 3.).

17 834 I. P. GAVRILYUK ET AL. TABLE 4 Approxiatios of λ for M = λ r TABLE 5 Approxiatios of λ 3 for M = λ 3 r TABLE 6 Approxiatios of λ 6 for M = λ 6 r Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, FIG. 3. L -or of residual versus the rak of the approxiatio o a sei-logarithic scale.

18 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 835 TABLE 7 Approxiatios of λ 3 for M = λ 3 r TABLE 8 Approxiatios of λ 3 for M = λ 3 r TABLE 9 Approxiatios of λ 3 for M = 5 λ 3 r Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, FIG. 4. L -or of residual for λ 3 versus the rak of the approxiatio o a sei-logarithic scale.

19 836 I. P. GAVRILYUK ET AL. TABLE Nuerical eigevalues λ 3 (M) coputed by eas of the algorith EVIN ad the correspodig asyptotic eigevalues accordig to Shibata (a) M λ 3 (M) λ asyp 3 (M) δ [484.7, ] EXAMPLE 3.5 Let us cosider proble (3.) with u = M. Approxiatios of λ for differet M are show i Tables 5 ad 7 9. Fro these results, we ca coclude that the eigevalue depeds o M, aely, it icreases alog with the icrease of M. The L -or of the residuals o a logarithic scale, for differet values of M, is depicted i Fig. 4. Fro Fig. 4, we see that the slope decreases as M icreases, which eas that covergece becoes worse with the icrease of M. We also copare our uerical results with the asyptotic forula fro Shibata (a): λ asyp 3 (M) = M + 6 ( M M + O M M The results preseted i Table show good agreeet betwee the uerical eigevalues obtaied accordig to algorith EVIN ad the asyptotics fro Shibata (a). 4. Coclusios Based o the techique of perturbig the coefficiets of the differetial equatio ad the Adoia decopositio ethod, we proposed a recursive algorith for the uerical solutio of oliear eigevalue probles. By usig the ethod of geeratig fuctios, we proved a geoetric covergece rate with a deoiator iversely proportioal to the idex of the trial eigevalue. The latter allows us to coclude that covergece iproves alog with the icrease of the idex of the trial eigevalue. Nuerical results cofir the theoretical fidigs. As it follows fro exaples, the uerical results obtaied are i good agreeet with the asyptotics fro Shibata (a,b). The results preseted are also valid for a oautooous oliearity N(x, u) with N: (, ) R R beig a aalytic fuctio with respect to u [, R], such that ). Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5, k N(x, u) u k N(x, ) =, = d k N(u) du k N (k) ( u ), u R, k =,,..., (4.) ad N( u ) is a sooth fuctio with oegative derivatives.

20 EXPONENTIALLY CONVERGENT PARALLEL ALGORITHM 837 It is easy to exted our algorith to probles (.), (.) ad (.), (3.) cosidered i a Hilbert space H with u: [a, b] H ad N(x, u): [a, b] H H beig a oliear aalytic fuctio i a eighbourhood of u, possessig all derivatives i the sese Frechet or Gâteaux satisfyig coditios (4.) with the correspodig or istead of absolute value o the left-had side. Existece ad uiqueess theores for such probles were studied i Heid et al. (). Ackowledgeets The authors would like to ackowledge the support provided by the Deutsche Forschugsgeeischaft. REFERENCES ABBAOUI, K., PUJOL, M. J., CHERRUAULT, Y., HIMOUN, N. & GRIMALT, P. () A ew forulatio of Adoia ethod: covergece result. Kyberetes, 3, ADAMS, E. & PLUM, M. (a) Free ad forced vibratios of trusses by Fourier decopositio, ad hootopy ethods for oliear atrix eigevalue probles (I) ethods. J. Math. Aal. Appl., 75, ADAMS, E. & PLUM, M. (b) Free ad forced vibratios of trusses by Fourier decopositio, ad hootopy ethods for oliear atrix eigevalue probles (II) properties ad suppleets. J. Math. Aal. Appl., 76, ANDREW, A. L. (994) Asyptotic correctio of coputed eigevalues of differetial equatios. A. Nuer. Math.,, 4 5. ANDREW, A. L. () Quadrature errors i fiite eleet eigevalue coputatios. Aust. N. Z. Id. Appl. Math. J., 4, ANDREW, A. L. (3) Asyptotic correctio of ore Stur-Liouville eigevalue estiates. BIT, 43, ANDREW, A. L., CHU, K.-W. E. & LANCASTER, P. (995) O the uerical solutio of oliear eigevalue probles. Coputig, 55, 9. APPELL, J., DE PASCALE, E. & VIGNOLI, A. (4) Noliear Spectral Theory. Berli, Geray: De Gruyter. BABENKO, K. I. (986) Foudatios of the Nuerical Aalysis (i Russia). Moscow, Russia: Nauka. BABUŠKA, I. & OSBORN, J. E. (99) Eigevalue probles. Hadbook of Nuerical Aalysis (P. G. Ciarlet & J. L. Lios eds). Asterda: North Hollad, pp BAILEY, P. B., EVERITT, W. N. & ZETTL, A. (99) Coputig eigevalues of sigular Stur-Liouville probles. Results Math.,, BANDYRSKIĬ, B. Ĭ., GAVRILYUK, I. P., LAZURCHAK, I. I. & MAKAROV, V. L. (5) Fuctioal-discrete ethod (FD-ethod) for atrix Stur-Liouville probles. Coput. Methods Appl. Math., 5, 5. BANDYRSKIĬ, B. Ĭ., GAVRILYUK, I. P., MAKAROV, V. L. & MAKAROV, I. L. (a) No-classical asyptotic expasios for eigevalue probles with a paraeter i the boudary coditios. Appl. Math. If., 7, BANDYRSKIĬ, B. Ĭ., LAZURCHAK, I. I. & MAKAROV, V. L. (b) A fuctioal-discrete ethod for solvig leftdefiite Stur-Liouville probles with a eigevalue paraeter i the boudary coditios. Coput. Math. Math. Phys., 4, BANDYRSKIĬ, B. Ĭ. & MAKAROV, V. L. () Sufficiet coditios for the eigevalues of the operator d /dx + q(x) with Ioki-Saarskiĭ coditios to be real-valued. Coput. Math. Math. Phys., 4, BANDYRSKIĬ, B. Ĭ., MAKAROV, V. L. & UKHANEV, O. L. (999) Sufficiet coditios for the covergece of o-classical asyptotic expasios for Stur-Liouville probles with periodic coditios. Differ. Equ., 35, COLLATZ, L. (96) The Nuerical Treatet of Differetial Equatios. Berli, Geray: Spriger. GORDON, R. G. (969) New ethod for costructig wavefuctios for boud states ad scatterig. J. Che. Phys., 5, 4 5. Dowloaded fro iaja.oxfordjourals.org at South Chia Agricultural Uiversity o Septeber 5,

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