A Continuous Formulation of A(α)-Stable Second Derivative Linear Multistep Methods for Stiff IVPs in ODEs

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1 Joural of Algorithms & Computatioal Techology Vol. 6 No. 79 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs R. I. Ouoghae ad M. N. O. Ihile Departmet of Mathematics, Uiversity of Bei, Bei City, P.M.B 54, Bei City, Nigeria. Received: 4/0/00; Accepted: 0/0/0 ABSTRACT This paper cosiders family of A(α)-stable secod derivative liear multistep methods of order p = + for step umber 5 for the solutio of stiff IVPs i ODEs. The methods are demostrated to be A(α)-stable for. At = 4, the method is stable but ot A(α)-stable. The istability of the ew methods sets i whe > 5. Numerical examples are give to demostrate the applicatio of the methods. Keywords: Collocatio method, liear multistep methods, iitial value problem, boudary locus AMS subject classificatio: 65L05, 65L06.. INTRODUCTION Recetly, [8] itroduced Secod derivative multistep methods for stiff ODEs y = f( x, y), x [ x0, X]. yx ( 0)= y0 () m m f : R R R. The geeral form of the secod derivative discrete LMM discussed i [8] is αjy j h β, jf j h β, jf j α = +, 0. () Correspodig author. ouoghae0@yahoo.co.u, moihilo@yahoo.com

2 80 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs The stiff stability of a specific methods i () was ivestigated i [8] by the meas of boudary locus method. The method was foud to be stiffly stable for 7 ad ustable for > 8. I the spirit of [8], we ivestigate a aother class of secod derivative liear multistep methods (SDLMM) for the umerical solutio of ordiary differetial equatios (ODEs) i (). Cotiuous liear multistep methods (CLMM) form a super class of LMM with properties that embed the characteristics of LMM ad hybrid methods. The motivatio lies i the fact that the ew formulatio offers the advatage of a cotiuous solutio of the iitial value problems (IVPs) ulie the discrete solutio geerated from the methods i [8]. The success of these methods is i their attaiable stiff stability characteristics useful for resolvig the problem posed by stiffess i the IVPs. Util ow, some classes of methods have bee developed ad used successfully i the umerical solutio of () such iclude that of [,,, 4], [7], [8, 9], [, ], [, 4],[8] [],[, 4] ad [] if () is stiff. Other methods for ostiff IVPs iclude that of [5], [6], [7], [5], [6], [8, 9, 0], [], [] ad for sigularity IVPs, see [9, 0] ad etc. The hybrid couter part of () is α, jy j + α, jy vj = h( β, jf j + β, jf vj h ( δ f + δ f ), j j, j vj ) + () α 0. For ay give, the parameter α, j, α, j, β, j, β, j, δ, j, δ, j may be chose i a variety of ways, but usually the objective is to mae the order as high as possible, subject to the coditio of stability, while also desirig small error costat ad miimum umber of fuctios evaluatio. Iterestigly, special cases of these methods i [8], [], [4], [] ad [] are ow to be stiffly stable for the umerical solutio of the IVP (). We see the solutio of this problem by secod derivative CLMM ad its hybrid couterpart obtaied by reformulatio of the discrete secod derivative method i () ito its cotiuous methods. The secod derivative CLMM form a super class of [8] ad the classical secod derivative methods () i geeral. Now cosider the secod derivative CLMM yx ( + ( t+ ) h)= y + h βj() t f j + hβv() t f v + h γ() t f. + (4)

3 Joural of Algorithms & Computatioal Techology Vol. 6 No. 8 where v=, t, ad { y + j } are the approximatios of [, ] order p to {( yx, where ow f f x y, x = + j )} j= ( j, j) x + th, ad α (t) =. { β j (), t () }, () t, ad () t, are cotiuous coefficiets i t β v γ presumed to be real ad satisfyig the ormalizatio coditio α (t) =, t [a,b] ad h = x x is a fixed mesh size. The parameter v is icorporated to provide of grid collocatio poit x v i the ope iterval (x, x ). By costructio, the order of a method is p which is the degree of the basis polyomial fuctio. Stiffly stable methods which are useful for resolvig the problem of stiffess ca be derived from (4). Also, a family of methods at the hybrid poit x v which is preset i the implicit secod derivative scheme is derived to give the umerical values of the solutio y v at the poit x v. The hybrid solutio y v at the poit x v is give by the cotiuous method yx ( + vh)= α ( t) y + hβ ( t) f + hβ ( t) f, v= t j j,, +, t =. (5) I this paper, cotiuous collocatio methods for the solutio of () are proposed such that the collocatio is doe at all grid poit {x j } j = 0 ad oe off grid poit x v with v=. This paper is arraged as follows, sectios ad will cotai a discussio of the derivatio of the methods. The way the methods are derived differs from Taylor s series expasio approach demostrated i [], [7], [8], [], ad [4]. Sectio 4 cotais the derivatio of the iterval of absolute stability of the schemes, see []. I sectio 5, we preset the results of some umerical experimets usig oe of the methods.. DERIVATION OF THE SECOND DERIVATIVE CONTINUOUS LINEAR MULTISTEP METHODS (CLMM) The CLMM which icorporate aalytic property of the secod derivative lie the secod derivative methods discussed i [8], we wish to derive is give i (4). Let the cotiuous solutio to the IVPs be i the form m yx ( )= ajφ j( x) (6)

4 8 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs m with m = +, where { φ are polyomial basis fuctio give by α j (x) = x j j ( x)}, ad the a j 's are the real parameter costats to be determied. From (6), we have Differetiatig (6) with respect to x give + y ( x)= f ( x, y)= j( j ) a φ ( x) j= + y ( x)= f( x, y)= ja φ ( x). j= j j j j (7) (8) Collocatig (7) ad (8) at x= x j, (), + ad iterpolatig (6) at x = x ad x = x, we obtai the liear system of equatios x v 0 φ0( x) φ( x)... ( + ) φ+ ( x) a0 f 0 φ0( x ) φ( x )... ( + ) φ+ ( x + ) a f =. 0 φ0( x ) φ( x )... ( + ) φ + ( x + ) a f 0 φ0( x v) φ( x v)... ( + ) φ+ ( x v) a + f v 0 0 φ 0( x )... ( + )( + ) φ+ ( x ) a+ f + φ0( x ) φ( x ) φ( x )... φ+ ( x ) a+ y (9) Solvig equatio (9) by Gaussia elimiatio methods the values of a j 's are determied ad substitutig the resultig values a j 's ito (6) with t = (x x )/h, α (t) = ad settig x = x t+ o the left had side of (6), a specific scheme will emerge for a fixed value of. For example, for case =, v= i (), the value of the cotiuous coefficiets {β j (t)}, β v (t), γ (t), the cotiuous method ad the cotiuous error costat C p+ (t) ad the order p of the CLMM i (4) are 4 4 α0()=, α()=, β0()= t t β 6, ( )= 8t t t t t + + t, 4 4 β()= t 7 γ 6 t t t t, ( )= t t t t + t +, 4 4 7t t t t t yx ( + ( t+ ) h)= y + h[ + t + + t f f 6 ) + ( 6 ) ( (0) () ()

5 Joural of Algorithms & Computatioal Techology Vol. 6 No t 4 t t t f ] + h + + t f + ( 4 8t t ) f ] + h ( t t + t + ) f +, + () (4) Fixig t = ito (0), (), (), () ad (4) gives the discrete coefficiets, the method ad error c p+ ad the order, p of the CLMM i (4) to be y = y C 5 h + f + f + f 6 ( 4 ), 5 (5) hy ( x) =, p = (5) (6) For =, the value v = agai yields β()= t t t t t t β 4 5, ( )= 8 t 6t 4t 6t + + t + 9 5, β 4 5 t t 5t t t t y( x ( t ) h)= y h[( f t ) ( t t t t t t t ( f 9 5 ) ( t 7t ) f ] h ( t t t t ) f, α()=, t α()=, t 5 β0()= t t t t t , ()= t 9 t 4t t 7t γ 8 6 0, ( )= t t t t + + t 4 8 5, 6 (6) t ( 0 + t(60 + t( 5+ 4 t( 9+ 5 t)))) h y ( x) C6()= t, p = Isertig t = ito (8), (9), (0), ad () gives α() =, α() =, β0() =, β()= 70 60, t f 5 ) 4 5 (7) (8) (9) (0) () ()

6 84 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs y = y β () = 8 47, β()= γ 45 40, ()= 0, h h + f + f + f f f + 70 (4 448 ) 6, 70 C 6 6 (6) hy x = ( ) p 4400, =5. () (4) (5) For =, v =, we have β β ()= β 5 ()= t 49 t 6t t t t , t t t t t t t , ()= t t t t 9t t , β α ()=, t α ()=, t ()= t t + t + t t + t , t 59 t 7t 55t 7t t , β ()= + + ()= t t 7t t 7t t , γ (6) (7) (8) (9) (0) () () t t t t t yx ( ( t ) h)= y h[( ) f t t t t t + ( + t ) f t 9t 9t t + ( + t ) f +

7 Joural of Algorithms & Computatioal Techology Vol. 6 No. 85 ( t 8t 8t 8t t + 45 ) f t 7t 55t 7t t + ( ) f ] ( t 7t t 7t t + h ) f, () 7 ( ( t t t t t t h y C (4) 7()= t + ) ( ( (089 + ( ( 5 + ))))) 7) ( x), p = Substitutig t = ito (6), (7), (8), (9), (0), (), () ad (4) gives y = y h h + f + f f + f f 5400 ( ) f +, 90 5 C 7 7 (7) hy x = ( ) p , =6. (5) (6) Followig the procedure described i (0)-(8), ad the examples for =,,, v=, ad t = above, the cotiuous coefficiets, the cotiuous methods, the cotiuous error costats, the discrete coefficiets, the discrete methods ad the discrete error costats of the SDCLMM i (4) for = 4,5,6,7,8,9,0,,,,4,5 are obtaied, see []. THE DERIVATION OF THE SECOND DERIVATIVE CONTINUOUS HYBRID PREDICTOR Similarly, let the cotiuous solutio of (5) be m yx ( )= bjφj( x), v= t +, t = / (7) with degree m = +, where variable b j 's are the real parameter costats to be determied. Differetiatig (7) with respect to x twice, gives + y ( x)= f( x, y)= jbφ ( x), j= + j y ( x)= f ( x, y)= j( j ) b ( x) j= j jφj (8)

8 86 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs Collocatig (8) at x = x j, j = 0() ad iterpolatig (7) at x = x ad x = x, we obtai the geeral matrix equatio to be φ0( x) φ( x) φ( x)... φ+ ( x) b0 y φ0( x ) φ( x ) φ( x )... φ+ ( x + ) b y = φ0( x ) φ( x ) φ( x )... φ + ( x + ) b y 0 φ0( x ) φ( x )... ( + ) φ+ ( x ) b + f 0 0 φ ( x )... ( + )( + ) φ ( x ) b f (9) Solvig equatio (4) gives the values of b j 's. Substitutig the resultig values b j 's ito (7) with t = (x x )/h, α v (t ) =, v= ad settig x o = x t + the left had side of (7), a specific scheme is obtaied for 5 with v= i (7). We give below examples of the derived cotiuous coefficiets, the cotiuous hybrid predictors, the cotiuous error costat c p+ (t ), the discrete predictors, the discrete error costats c p+ ad the order p of the hybrid formula i (5) for =,, respectively. α ( t )= t, α ( t )=, α ( t )= + t, 0 (40) β, t t ( t )= t t, β ( t )= +,, (4) yx ( + ( t+ ) h)=( t) y + ( + t ) y + h( t t ) f h t t + ( + ) f, (4) C 4 4 (4) t t h y x ()= t ( + ) ( ), p =. 4 (4)

9 Joural of Algorithms & Computatioal Techology Vol. 6 No. 87 Fixig t = t = ito (40), (4), (4), ad (4) gives the discrete coefficiets, the predictor ad error costat of the hybrid formula i (5) to be h h y = 7 y y f + f +, C 4 4 (4) hy ( x) =, p =. 84 (44) (45) For =, ad v = i (5) yields β, 4 4 α0( t t t t t )= + + α , ( t )= t + t t, α 4 ( t )=, α( t )= 7 t t t t , 4 4 ( t )= 5 t t 5t t β, , ( )= t t t t + + t , 4 t t t t yx ( ( t ) h)=( y t t t y ) ( ) + t + ( t t t ) [( 5t t 5t t y ) h f + h ( t t 4 t t + f ), + (46) (47) (48) (49) Settig t = t = i (46), (47), (48), (49), ad (50) gives h h y = 05 y + y + y f + f (5) t t t h y x C5()= t ( + ) ( + ) ( ), p =4. 0 C (5) hy x = ( ) p 80, =4., (50) (5) (5)

10 88 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs For =, v = 5, we have 4 5 α( t )=4 t t 5 t t t + α5, ( t )=, 4 5 ( t )= t + t + t t + t , α α β, 4 5 ( t )= 4 t 0t t 7t t , α ( t )= t 4t t t t , 4 5 ( t )= 4t t t + t t , β, t 4 5 t t t t t ( )= + + +, 4 (5) (54) (55) (56) (57) (58) 4 5 4t 0t t 7t t yx ( + ( t+ ) h)=( + + y ) t t ( t t t ) y 4 5 5t t + (4t t t + ) y C t 0t 47t 0t 85t + ( y ) t t t t h( t ) f 4 5 t t t t t + h ( ) f, 6 (6) t t t h y x ()= t ( + ) ( + ) ( ), p =5. 70 (59) (60)

11 Joural of Algorithms & Computatioal Techology Vol. 6 No. 89 Substitutig t = t = ito (5), (54), (55), (56), (57), (58), (59), ad (60) gives h h y 5 = y y + y + y f + f +, C 6 6 (6) hy x = ( ), p =5. 07 (6) (6) Followig the approach above for =,,, v= ad t = t = i (5), we obtaied the cotiuous coefficiets, the cotiuous hybrid predictor, the cotiuous error costats, the discrete coefficiets, the discrete hybrid solutio formula ad the discrete error costats of the hybrid predictor i (5) for = 4()5. 4. STABILITY OF THE METHODS BY PLOTTING THE BOUNDARY LOCUS I this sectio, we ivestigate the stability properties of the secod derivative cotiuous liear multistep method (SDCLMM) defied i (4) for a give value of 4. I examie the stability property of (), we give the followig defiitios: Defiitio : A umerical itegrator is said to be zero stable if the roots of are ρ(r) = 0 iside the uit circle or simple o the uit circle, where j ρ()= r α r is the first characteristics polyomial for the umerical j itegrator. Defiitio : A umerical itegrator is said to be A-stable if absolute value of the root(s) of the stability polyomial of the umerical itegrator lies i the ope left half of the z-plae of the stability regio. Defiitio : A umerical algorithm is said to be A(α) stable for some π α if the wedge Sα ={ z: Arg( z) < α, z 0} is cotaied i its regio 0, of absolute stability. The largest α (α max ) is regarded as the agle of absolute stability o the argumet of stability. The defiitio of stiffly stability give i the spirit of Gear

12 90 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs [, 4] show that stiff stability implies A(α)-stability. Applyig methods (4) for a give to the scalar test problem y' = λy, Re(λ) <0, ad substitutig the hybrid solutio (5), y v at poit x v for a correspodig at the hybrid poit we obtai the cotiuous stability polyomials to be j π(,,, rztt)= r r z β () tr zβ ()( t α ( t) r + zβ j v, ( t ) r + z β, ( t ) r ) z γ ( t) r, z= λh. j j (6) for which the boudary locus are plotted to reveal the stability iterval. Methods (4) is said to be stable, if all roots of the polyomial π (r, z, t, t ) lie i or o the uit ad with those lyig o the uit circle are simple roots. For example, settig =, t = 0, v = ad t = i (6) yield the stability polyomial of the method i (4) to be π(,,0,. For =, )= 4 z rz rz rz rz r t =, v = ad t = i (6) give the stability polyomial to be π(,,,. I similar maer, )= z rz rz 7r z 7r z rz r r we obtai the stability polyomials for the scheme i (4) for =,4,5, 6,7,8,9,0,,,,4. Plottig the roots of the methods for 5, we observed that the algorithm i (4) are A(α)-stable for, at =4, the iterval of absolute stability is (.,0), ad ustable whe 5. The ew scheme is foud to be A-stable for step umber =, ad 4. Followig the idea of [C.W. Gear, Numerical Iitial Value Problems i Ordiary Differetial Equatios. Pretice-Hall. Eglewood Cliffs. New Jersey, (97), pp. 8-9], the agle (α) of θ absolute stability of the ew scheme i (4) is computed usig α = ta. D Agai, see Fatula []. Table () below give the values of α ad D respectively for the method i (4) for. The boudary locus curve of algorithm (4) is foud by settig r = e iθ i (6), where, 0 θ π. The curve is obtaied by plottig the root(s) of the stability polyomial i (6) ad is observed to be symmetric about the real axis. The upper half is obtaied for 0 θ π, ad a mirror image of this through the real lie completes the regio of absolute stability. See appedix for the Matlab code for plottig the stability regio of

13 Joural of Algorithms & Computatioal Techology Vol. 6 No = 6 = 9 = = 5 Im(z), z=λ h = = Re(z), z=λ h = = 4 = 5 = 7 = 8 = 0 = Figure. Boudary locus for the A(α)-stable secod derivative LMM i (4) for Im(z), z= λ h Re(z), z= λ h Figure. Boudary locus for the secod derivative LMM i (4) for = 4 shows that the method i (4) has small iterval of absolute stability ad such methods are ot suitable for solvig stiff IVPs. At = 5, the algorithm i (4) ad (5) are observed to be ustable.

14 9 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs the algorithm i (4). Figure, below shows the boudary of the stability regio of each method for. Table. The step-umber, scaled variable t ad agle of absolute stability for the method i (4). t Agle- D p(4) p (5) (α) of c p+ (5) c p+ (5) Absolute Stability for (4)+(5)

15 Joural of Algorithms & Computatioal Techology Vol. 6 No (.,0) Not 7 6 A(α)- stable Ustable Ustable Table. The step-umber, methods, order ad the agle (α) of absolute stability BDF [5] p α zero ustable D ustable ustable SDLMM [8] p α ustable D ustable The agle of iterval of absolute stability of method i (4) for give value of is give i Table (). The agle of iterval of absolute stability of Gear s method (Bacward differetiatio formula) ad Eright s method i for 8 is give i Table (). Comparig Table () ad Table (), observe that the ew scheme i (4) step umber termiate at = 4, while that of BDF ad SDLMM step umber termiate respectively at = 6 ad = 7. Also, the order of the algorithm i (4) are far higher tha the existig stiffly stable BDF ad SDLMM which are also A(α)-stable as discussed i [E. Hairer, ad G. Waer, Solvig Ordiary Differetial Equatios II. Stiff ad Differetial-Algebraic Problems, Spriger-

16 94 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs Verlag, Berli, (996), pp. 5-6] ad the error costat of method i (4) are smaller tha that of the BDF [5] ad the SDLMM i [8]. Ifact, these properties show that the ew scheme i ((4) ad (5)) have sigificats advatage over the BDF i [5] ad the SDLMM i [8] respectively. 5 NUMERICAL EXPERIMENTS Our aim i this sectio is to see how well our methods compare with Ode5s of MATLAB code i [6] whe solvig the followig iitial value problems: Problem []: Liear problem discussed i [8] ad [6] e e y = y, y(0)=, yx ( )= e e Problem []: Noliear chemical problem solved i [8] ad [6] 4 y = 0.04y + 0 yy, y(0) =, 4 7 y = 0.04y + 0 yy 0 y, y(0) = 0, 7 y = 0 y, y(0) = 0, xε[0,]. Problem []: Oscillatory problem i [0] 0 x 0 α e ( cosα( x) + siα( x)) α x e ( cosα( x) siα( x)) 4 x e y = yy, (0)=, yx ( )= x. e x e 0.x e For problem [], x 0(0.000)5, ad for problem [], we have x 0(0.000). I problem, eigevalues are ( 0., 0.5,, 4, 0, +iα), α = ad x 0(0.000). I solvig the iitial value problems above, the implicitess i the methods has bee resolved usig the Newto Raphso iterative scheme as suggested by [8], [] ad [], while the iverse Euler method 0.x 0 x 00 x 000 x

17 Joural of Algorithms & Computatioal Techology Vol. 6 No. 95 [ s] hyy y = y +, s =0,,,... y hy (64) i [, ] is used to geerate the startig values for the iterative schemes. The results of the discrete versio of the A(α)-stable LMM of order 8 o the stiff problems above are show below i Tables (), (4) ad (5) respectively. Table. The Numerical Solutio of Problem []. x Exact Solutio A(α)-Stable Ode5s SDLMM [8] LMM ((4)+(5)) Table 4. Errors i A(α)-Stable LMM Calculatio Versus Ode5s ad SDLMM [8] for Compariso of Results i Table (). x A(α)-Stable Ode5s, Error SDLMM [8], Error LMM ((4)+(5)), Error e e e e e e e e e e e e e e-007

18 96 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs Table 5. The Numerical Solutio of Problem []. x A(α)-Stable LMM ((4)+(5)) Ode5s SDLMM [8] e e e e e e e e e-005 Note that absolute error ( Error ) is the modulus of the exact solutio mius the computed solutio. Our aim is ot to optimize the code with respect to the computig time but to rather cosider the performace of the methods with respect to stability for a fixed step size h ad t respectively. 6. CONCLUSION I this paper, we described the costructio of a class of A(α)-stable LMM for step umber 5 of order p = + which are appropriate for the umerical solutio of stiff differetial systems. The scheme was foud to be A(α)-stable for, but have a small regio of absolute stability at =4 ad ustable for 5 as revealed by the boudary locus i figures (,, ad ) respectively. The umerical results are from codes writte i MATLAB usig a Dell Laptop of processor of 70MHz, RAM of 50MB ad hard dis space of 40 Gb. The umerical solutios i Table (), Table (4) ad Table (5) show that the methods are promisig ad idicate that the A(α)-stable LMM i (4) are competitive with the state-of the-art of MATLAB ode5s code i [6] ad [8], while Tables (6) ad (7) show that the A(α)-stable LMM outperform the MATLAB ode5s code i [6] for stiff differetial equatios. ACKNOWLEDGEMENTS This wor has beefited from Prof. J. C. Butcher of the departmet of Mathematics, Uiversity of Auclad, New Zealad through persoal commuicatio with the use of sype, especially i plottig the boudary locus of the ew scheme discussed i this paper. The authors also expresses their gratitude to the uow referees for their suggestios ad very istructive commets. APPENDIX: EXAMPLE OF THE MATLAB CODE FOR PLOTTING THE STABILITY REGION OF THE SCHEME IN (4). For = i (6), the stability polyomial is π rz,,, =

19 Joural of Algorithms & Computatioal Techology Vol. 6 No. 97 Table 6. The Numerical Solutio of Problem []. x Exact Solutio A(α)-Stable LMM((4)+(5)) Ode5s SDLMM [8] e e e e e e e e-009 Table 7. Errors i A(α)-Stable LMM Calculatio with Ode5s ad SDLMM [8] for Compariso of Results i Table (6). x A(α)-Stable LMM((4)+(5)), Error Ode5s, Error SDLMM [8], Error e e e e e e-0

20 98 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs z rz rz 7r z 7r z r r+ +. The matlab code for plottig the stability regio is give below: = 00; rr = exp(iispace(0,8 pi, + )); z = [0] for = : r = rr(); roo = roots([( 7r /40), ( r/60 + 7r /80), (/60 r/60), (r r)]); [what, which] = mi(abs(z(ed) roo)); z = [z, roo(which)]; ed z = z(: ed); plot(z) xlabel('re(z), z =λh') ylabel('im(z), z =λh') ad i this same maer, we plot the stability regios of the scheme i (4) for 5. REFERENCES. J.C. Butcher, A modified multistep method for the umerical itegratio of ODEs. J. Assoc. Comput. Mach. Vol., (965), J.C. Butcher, A geeralizatio of sigly-implicit methods. BIT. Vol., (98), pp J.C. Butcher, The Numerical Aalysis of Ordiary Differetial Equatio: Ruge Kutta ad Geeral Liear Methods,Wiley, Chichester, (987). 4. J.C. Butcher, Some ew hybrid methods for IVPs. I. J. R. Cash ad Glad Well (eds), Computatioal ODEs. Claredo press, Oxford (99), J.P. Colema, ad S.C. Duxbury, Mixed collocatio methods for y = f (x, y ), Uiversity of Durham, Dept. of Mathematical Scieces. Research Report NA-99/0, (999). J. Comput. Appl., (000), pp W.B. Gragg, ad H. J. Stetter, Geeralized multistep predictor corrector methods, J. Assoc. Comput. Mach., Vol. (964), pp G. Dahlquist, O stability ad error aalysis for stiff oliear problems. Part,

21 Joural of Algorithms & Computatioal Techology Vol. 6 No. 99 Report No TRITA-NA-7508, Dept. of Iformatio processig, Computer Sciece, Royal Ist. of Techology, Stocholm, (975). 8. W.H. Eright, Secod derivative multistep methods for stiff ODEs. SIAM. J. (974), pp.. 9. W.H. Eright, Cotiuous umerical methods for ODEs with defect cotrol. J. Comput. Appl. Math., Vol. 5 (000), pp W.H. Eright,T.E Hull ad B. Liberg, Comparig umerical methods for stiff of ODEs systems. BIT, Vol. 5 (975), pp S.O. Fatula, Numerical Methods for Iitial Value Problems i ODEs. Academic Press, New Yor, (978).. S.O. Fatula, Oe-leg multistep method for secod order ODEs. Comp. Math.Applic. Vol. 0, No., (984) pp. 4.. C.W. Gear, The automatic itegratio of stiff ODEs. pp i A.J.H. Morrell (ed). Iformatio processig 68: Proc. IFIP Cogress, Ediurgh (968), Nor-Hollad, Amsterdam. 4. C.W. Gear, Algorithm 407, DIFSUB for solutio of ODEs. Comm. ACM, Vol. 4 (97) pp C.W. Gear, Numerical Iitial Value Problems i ODEs. Pretice-Hall. Eglewood Cliffs. N.J., (97). 6. J.D. Higham ad J.N. Higham, Matlab Guide. SIAM. Philadelphia (000). 7. E. Hairer, S.P. Norset, ad G. Waer., Solvig Ordiary Differetial Equatios I. Nostiff Problems, Spriger-Verlag, Berli, (99). 8. E. Hairer, ad G. Waer., Solvig Ordiary Differetial Equatios II. stiff ad Differetial-Algebraic Problems, Spriger-Verlag, Berli, (996). 9. M.N.O. Ihile, Coefficiets for studyig oe-step ratioal schemes for IVPs i ODEs: III. Extrapolatio methods. Iteratioal J Comput. ad Maths with Applic. Vol. 47 (004), pp M.N.O. Ihile, The root ad bell s iteratio methods are of the same error propagatio characteristics i the simultaeous determiatio of the zeros of a polyomial, part I: Correctio methods Iteratioal J. Comput. ad Maths with Applic. Vol. 56 (008), pp M.N.O. Ihile, ad R.I. Ouoghae, Stiffly stable cotiuous extesio of secod derivative LMM with a off-step poit for IVPs i ODEs. J. Nig. Assoc. Math. Physics. Vol. (007), pp J.J. Kohfeld, ad G.T. Thompso, Multistep methods with modified predictors ad correctors. J. Assoc. Comput. March., Vol. 4 (967), J.D. Lambert, Numerical Methods for Ordiary Differetial Systems. The Iitial Value

22 00 A Cotiuous Formulatio of A(α)-Stable Secod Derivative Liear Multistep Methods for Stiff IVPs i ODEs Problems. Wiley, Chichester, (99). 4. J.D. Lambert, Computatioal Methods for Ordiary Differetial Systems. The Iitial Value Problems. Wiley, Chichester, (97). 5. A. Marthise, Cotiuous Extesios to Nystrom Methods for the Explicit Solutio of Secod Order Iitial Value Problems, Uiversity of Trodheim, Departmet of Mathematical Scieces, Preprit Numerics No. 4 (994), p.. 6. A. Marthise, Cotiuous Extesios to Nystrom Methods for Secod Order Iitial Value Problems, BIT, No. 6 (996), pp O. Nevalia, O the umerical itegratio of oliear IVPs by liear multistep methods. BIT, Vol. 7, (977). pp B. Owre, ad M. Zearo, Order Barriers for Cotiuous Explicit Ruge Kutta Methods, Uiversity of Trodheim, The Norwegia Istitute of Techology, Divisio of Mathematical Scieces, Preprit Mathematics ad Computatio, No. /89, (Preprit for) Math. Comput., 56 (99), pp , B. Owre, ad M. Zearo, Cotiuous explicit Ruge Kutta methods, pp i Computatioal ODEs., Proc. Cof. Lodo/UK 989. Ist. Math. Appl. Cof. Ser., New Ser. 9,(99). 0. B. Owre, ad M. Zearo, Derivatio of Efficiet Cotiuous Explicit Ruge Kutta Methods, Uiversity of Toroto, Computer Sciece Dept; Techical Report 40/90, 990 (Preprit for) SIAM J. Sci. Stat Comp., (99), pp R.I. Ouoghae, Stiffly Stable Secod Derivative Cotiuous LMM for IVPs i ODEs. Ph.D Thesis. Dept. of Maths. Uiversity of Bei, BeiCity. Nigeria. (008).. M. Selva, C. Arevalo, ad C. Fuherer, A collocatio formulatio of multistep methods for variable step size extesios. Appl. Numer. Math, Vol. 4 (00). pp U.W. Sirisea, P. Oumayi, ad J.P. Chollo, Cotiuous hybrid through multistep collocatio. ABACUS, Vol. 8; No. ; (00), pp