VARIABLE-STEP VARIABLE-ORDER 3-STAGE HERMITE-BIRKHOFF-OBRECHKOFF ODE SOLVER OF ORDER 4 TO 14

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 4, Winter 6 VARIABLE-STEP VARIABLE-ORDER 3-STAGE HERMITE-BIRKHOFF-OBRECHKOFF ODE SOLVER OF ORDER 4 TO 4 TRUONG NGUYEN-BA, HEMZA YAGOUB, YU ZHANG AND RÉMI VAILLANCOURT ABSTRACT Variable-step variable-order 3-stage Hermite- Birkhoff-Obrechkoff methods of order 4 to 4, denoted by HBO(4-4)3, are constructed for solving non-stiff systems of first-order differential equations of the form y = f(x, y), y(x ) = y These methods use y and y as in Obrechkoff methods Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep-type and Runge-Kutta-type order conditions which are reorganized into linear Vandermonde-type systems Fast and stable algorithms are developed for solving these systems to obtain Hermite-Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions The order and stepsize of these methods are controlled by four local error estimators When programmed in Matlab, HBO(4-4)3 is superior to Matlab s ode3 in solving several problems often used to test higher order ODE solvers on the basis of the number of steps, CPU time, and maximum global error It is also superior to the variable step 3-stage HBO(4)3 of order 4 on some problems Programmed in C++, HBO(4-4)3 is superior to DP(8,7)3M in solving expensive equations over a long period of time Introduction In this paper, the 3-stage variable step (VS) HBO(4)3 of order 4 constructed in [9] is extended to variable-step variable-order (VSVO) 3-stage Hermite-Birkhoff-Obrechkoff methods of This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Centre de recherches mathématiques of the Université de Montréal AMS subject classification: Primary: 65L6; Secondary: 65D5, 65D3 Keywords: general linear method for nonstiff ODE s, Hermite-Birkhoff method, Obrechkoff method, Vandermonde-type systems, maximum global error, number of function evaluations, CPU time, Matlab s ode3, comparing ODE solvers Copyright c Applied Mathematics Institute, University of Alberta 43

2 44 TRUONG NGUYEN-BA, ET AL order 4 to 4, denoted by HBO(4-4)3 Both of these methods are designed for solving nonstiff systems of first-order initial value problems of the form () y = f(x, y), y(x ) = y, where = d dx, where the derivative f (x, y) can be obtained analytically or recursively There are many such problems, for instance in dynamical systems [], [3] The variable order HBO(4-4)3 has the remarkable property that the norm of the principal local truncation error coefficient decreases rapidly as the order increases The competitiveness of HBO(p)3 of order p comes from the surprisingly stable fast solution of ill-conditioned Vandermonde-type linear systems in O(p ) operations (see [, p 87]) by taking the special structure of these systems into account, in contrast with Gaussian elimination with pivoting in O(p 3 ) operations, which requires more storage, is slower and, at times, leads to unstable solutions when the stepsize is very small It is seen that HBO(4-4)3, when programmed in Matlab, requires fewer steps, uses less CPU time, and has higher accuracy than Matlab s ode3 on several problems often used to test higher order ODE solvers The VSVO HBO(4-4)3 is shown to perform better than the VS HBO(4)3 on some problems When programmed in C++, HBO(4-4)3 is superior to DP(8,7)3M [] in solving expensive equations over a long period of time HB methods of order 9, and have been studied in [8] Threestage HB(5-5)3 of order 5 to 5 and 4-stage HBO(5-4)4 of order 5 to 4 can be found in [] and [], respectively The contents of the paper is as follows Section defines VSVO HBO(4-4)3 of order 4 to 4 Order conditions are listed in Section 3 In Section 4, HBO(4-4)3 is represented in terms of Vandermonde-type systems In Section 5, the reader is referred to Section 5 and the Appendix of [9] for the symbolic construction of elementary matrices which decompose the coefficient matrices of these systems Section 6 deals with fast solutions of Vandermonde-type systems Section 7 considers the regions of absolute stability of constant step HBO(4-4)3 and its principal local truncation error coefficients Section 8 deals with step and order controls Numerical results are listed in Section 9 VSVO HBO(4-4)3 The defining formulae of HBO(p)3, of order p, depends on its variable backstep points and three off-step points,

3 VSVO 3-STAGE HBO ODE SOLVER 45 which in this paper are taken to be () c =, c = 3, c 3 = This choice, which is based on extensive numerical experimentation, is simple and involves only one new point, thus reducing computation time An HBO(p)3 requires the following four formulae to perform the integration step from x n to x n+, where, for notational simplicity, c = is used in the summations and c 3 is used instead of its value to keep track of it and distinguish the values y n+c3 and f n+c3 from y n+ and f n+, respectively The floor of a real number q, denoted by q, is the largest integer smaller than or equal to q A Hermite-Birkhoff polynomial (HBP) of degree (p ) is used as predictor P to obtain y n+c to order (p ), (3) (p3)/ ) y n+c = y n + h n+ (a f n+c + β j f nj + h n+ ( (p4)/ j= j= γ j f nj ) An HBP of degree (p ) is used as predictor P 3 to obtain y n+c3 order (p ), to (4) ( (p3)/ ) y n+c3 = y n + h n+ a 3j f n+cj + β 3j f nj j= + h n+ ( (p4)/ j= j= γ 3j f nj ) An HBP of degree p is used as integration formula IF to obtain y n+ to order p, (5) y n+ = y n + h n+ ( 3 j= + h n+ (p3)/ ) b j f n+cj + β j f nj ( (p4)/ j= j= γ j f nj )

4 46 TRUONG NGUYEN-BA, ET AL An HBP of degree p is used as step control predictor P 4 to obtain ỹ n+ to order (p ), (6) ( (p3)/ ) ỹ n+ = y n + h n+ a 4j f n+cj + a 43 f n+ + β 4j f nj j= + h n+ ( (p4)/ j= γ 4j f nj ) j= We note that f n is computed only once at each step x n 3 Order conditions for VSVO HBO(4-4)3 As in similar search for ODE solvers, we impose the following Runge-Kutta-type simplifying assumptions [7], [8], [] on HBO(4-4)3: (7) i a ij c k j + k!b i(k + ) = k + ck+ i, j= where (8) B i (j) = (p3)/ l= [ β il η j l+ ] + (p4)/ (j )! { i =, 3, j =,,,, p, with η l+ = and η l+ = by notation, and (9) η j = h n+ (x n x n+j ) = h n+ l= { i =, 3, k =,,,, p 3, [ γ il ] η j l+, (j )! j h ni, j =, 3,, 6 Equation (9) will be frequently used in this paper without further reference There remain two sets of equations to be solved: () () i= 3 b i c k i + k!b (k + ) =, k =,,, p, k + 3 i c p b i j a ij (p )! + B i(p ) + B (p) = p!, i= i= j=

5 VSVO 3-STAGE HBO ODE SOLVER 47 where () B (j) = (p3)/ i= β i η j i+ (j )! + (p4)/ i= j =,,, p + γ i η j i+ (j )!, We note that equations (), for k =,,, p, are multistep-type order conditions On the other hand, equation () for k = p and equation () are Runge-Kutta-type order conditions The numbers B (k), B (k) and B 3 (k) are associated with IF, P and P 3, respectively 4 Vandermonde-type formulation of HBO(p)3 4 Integration formula IF The p-vector of the reordered coefficients of IF in (5), u = [b, γ, b 3, b, β, γ, β, γ, β 3, γ 3,, β, (p3)/, γ, (p4)/ ] T, is the solution of the Vandermonde-type system of order conditions (3) M u = r, where (4) M c 3 c η η (p3)/ + c = 3 c η η!!! η (p3)/ + c p 3 c p η p η p η p (p3)/ + (p)! (p)! (p)! (p)! (p)! and r = r ( : p) has components! η (p4)/ + η p (p4)/ + (p)! r (i) =, i =,,, p i!

6 48 TRUONG NGUYEN-BA, ET AL The leading error term of IF is [ c p 3 b 3 p! + b c p (p3)/ p! + η p j+ β j p! j= (p4)/ + j= γ j η p ] j+ (p )! h p+ n+ (p + )! y(p+) n The detailed structure of columns 5 to p of M R p p is as follows: M (i, j) = η i j/ /(i )!, M (i, j + ) = d dη j/ M (i, j), provided j + p in the second equation { i =,,, p, j = 5, 7,, (p )/ +, 4 Predictor P The (p )-vector of the reordered coefficients of P in (3), u = [a, γ, β, γ, β, γ, β 3, γ 3,, β, (p3)/, γ, (p4)/ ] T, is the solution of the system of order conditions (5) M u = r, where (6) M = η η (p3)/ + η! η η p3 (p3)! η p4 (p4)! η (p3)/ +! η (p4)/ + η p3 (p3)/ + (p3)! η p4 (p4)/ + (p4)! and r = r ( : p ) has components r (i) = ci, i =,,, p i!

7 VSVO 3-STAGE HBO ODE SOLVER 49 The detailed structure of columns 3 to p of M R (p) (p) is as follows: M (i, j) = η i j/+ /(i )!, M (i, j + ) = d dη j/+ M (i, j), { i =,,, p, j = 3, 5,, (p 3)/ +, provided j + p in the second equation A truncated Taylor expansion of the right-hand side of (3) about x n gives p+ S (j)h j n+ y(j) n with coefficients j= (7) S (j) = M (j, : p )u = r (j) = cj, j =,,, p, j! S (j) = (p3)/ i= β i η j (p4)/ i+ (j )! + i= j = p, p, p + γ i η j i+ (j )!, We see by (7) that P is of order p The leading error term of P is [ S (p ) ] cp h p n+ (p )! y(p) n 43 Predictor P 3 The (p )-vector of the reordered coefficients of P 3 in (4), u 3 =[a 3, γ 3, a 3, β 3, γ 3, β 3, γ 3, β 33, γ 33,, β 3, (p3)/, γ 3, (p4)/ ] T, is the solution of the system of order conditions (8) M 3 u 3 = r 3,

8 4 TRUONG NGUYEN-BA, ET AL where (9) M 3 = c η η (p3)/ + c! η! η c p (p)! η p (p)! η p3 (p3)! η (p3)/ +! η (p4)/ + η p (p3)/ + (p)! η p3 (p4)/ + (p3)! and r 3 = r 3 ( : p ) has components () r 3 (i) = ci 3, i =,,, p, i! r 3 (p ) = b 3 [ ] p! b S (p ) B (p) We see by () that P 3 is of order p The last equation of (8) corresponds to order condition () The detailed structure of columns 4 to p of M 3 R (p) (p) is as follows: M 3 (i, j) = η i j/ /(i )!, M 3 d (i, j + ) = M 3 (i, j), dη j/ { i =,,, p, j = 4, 6,, (p )/, provided j + p in the second equation A truncated Taylor expansion of the right-hand side of (4) about x n gives with coefficients p+ j= S 3 (j)h j n+ y(j) n S 3 (j) = M 3 (j, : p )u 3 = r 3 (j) = cj 3, j =,,, p, j! (p3)/ S 3 (j) = a 3 S (j ) + (p4)/ + γ 3i i= i= η j i+ β 3i η j i+ (j )!, j = p, p, p + (j )!

9 VSVO 3-STAGE HBO ODE SOLVER 4 44 Step control predictor P 4 The p-vector of the reordered coefficients of P 4 in (6) is ũ 4 = [a 4, γ 4, β 4, γ 4, β 4, γ 4,, β 4, (p3)/, γ 4, (p4)/, a 43, a 4 ] T By setting a 43 = b 3 + ω 3 and a 4 = b + ω, ũ 4 reduces to the (p )- vector u 4 which is the solution of the system of order conditions () M 4 u 4 = r 4, where M 4 = M and r 4 = r 4 ( : p ) has components c i 3 c i r 4 (i) = i! (b 3 + ω 3 ) (i )! (b + ω ), i =,,, p (i )! For arbitrary nonzero ω 3 and ω, P 4 yields ỹ n+ to order (p ) A good experimental choice is ω 3 = 5 and ω = 9 The column ordering in matrices (4), (6), and (9) makes it easy to go from order p to order p + simply by adding a bottom row and a far-right column This ordering differs from the one used in [9] 5 Construction of elementary matrices A fast solution of Vandermonde-type systems (3), (5), (8) and (), with coefficient matrices () M l R m l m l, l =,, 3, 4, where (3) m = p, m = p, m 3 = p, m 4 = p can be achieved in O(m l ) operations by decomposing (M l ) into the product of elementary matrices For l =,, 3, the construction of lower bidiagonal matrices L l, initializing upper tridiagonal matrices U l, and upper bidiagonal matrices U l for decomposing (M l ) is described in Section 5 and in the Appendix of [9] and will not be repeated here This construction is most easily achieved by means of symbolic computation The case l = 4 was reduced to the case l = in Section 44 The elementary matrix functions L l k and U k l, l =,, 3, 4, are constructed only once as functions of η,, η 6 Then they are used by fast

10 4 TRUONG NGUYEN-BA, ET AL Algorithm 4 in [9] to solve the systems M l u l = r l, for l =,, 3, 4, in (3), (5), (8) and () at each integration step as described in Section 6 Since the Vandermonde-type matrices M l can be decomposed into the product of a diagonal matrix containing reciprocals of factorials and a confluent Vandermonde matrix, the factorizations used in this paper hold following the approach of Björck and Pereyra [4], Krogh [5], Galimberti and Pereyra [], and Björck and Elfving [3] Thus pivoting is not needed in this stable O(p ) decomposition because of the special structure of Vandermonde-type matrices This contrasts with the standard O(p 3 ) Gaussian decomposition with pivoting which uses more storage, is slower and may become unstable in solving Vandermondetype equations at very small stepsize 6 Fast solution of Vandermonde-type systems for HBO(p)3 In this section we describe O(p ) solutions to the Vandermonde-type systems for HBO(p)3 Firstly, the elimination procedure of Section 5 in [9] is applied to M l to construct m l m l lower bidiagonal matrices L l k, k = 3,, m l, of the form (4) L k = defined by the multipliers I k τ k+ τ m (5) τ i = i + k µ l (k) = L l k(i, i), i = k +, k +,, m l, where (6) µ l (k) = { M l (, k), if M l (, k) =, M l (3, k), if M l (, k) =, k = 3, 4,, m l Left multiplying M l triangular matrix by L l k, k = 3,, m l, produces the upper (7) L l M l = L l m l Ll 4 Ll 3 M l

11 VSVO 3-STAGE HBO ODE SOLVER 43 Secondly, we construct the m l m l upper tridiagonal matrix U l which transforms M l into the matrix of first order divided differences M l U l of the columns of M l whose first component is and where the divisors are taken from the second row of M l We note that U 4 = U since M 4 = M For given p and l, Table lists the diagonal elements of U l for i =,,, m l The nonzero nondiagonal elements of U l are given in the i U (i, i) U (i, i) = U 4(i, i) U 3 (i, i) 3 /c 3 /η /c 4 /(c c 3 ) /(η c ) 5 /(η c ) /(η 3 η ) 6 /(η 3 η ) 7 /(η 3 η ) /(η 4 η 3 ) 8 /(η 4 η 3 ) 9 /(η 4 η 3 ) /(η 5 η 4 ) /(η 5 η 4 ) /(η 5 η 4 ) /(η 6 η 5 ) /(η 6 η 5 ) 3 /(η 6 η 5 ) 4 TABLE : The diagonal entries of U l following three equations: (8) U (i, i) = U (i, i), i = 3, 7, 9,, m, U (3, 4) = U (4, 4), U (4, 5) = U (5, 5), (9) U (i, i) = U (i, i), i = 3, 5, 7,, m, (3) U 3 (i, i) = U 3 (i, i), i = 3, 6, 8,, m 3, U 3 (3, 4) = U 3 (4, 4)

12 44 TRUONG NGUYEN-BA, ET AL The remaining elements of U l are zero The first two rows of M l U l are [ ] (3) L l M l U( l :, : m l ) = Thirdly, the elimination procedure of Section 53 in [9] is used to construct m l m l upper bidiagonal matrices Uk l, k =,, m l, of the form (3) Uk l = defined by the divisors (33) σ i = I k σ k+ σ k+ σ k+ σ ml σ ml σ ml σ ml σ ml k µ l (i) µ l (i k) = U l k(i, i), i = k +, k +,, m l, where µ l (i) and µ(i k) are defined in (6) Right multiplying L l M l by U l k, k =,, m l, produces the diagonal matrix where and D l (i, i) = D l = L l m l Ll m l Ll 3 M l U l U l U l m l, D l (i, i) =, i =,, 3, (i )! [µ l (3)] [µ l (4)] [µ l (i )], i = 4, 5,, m l Lastly, M l is decomposed into the product of elementary matrices: M l = ( L l m l Ll m l 3) Ll D l ( U l U l U m l ) l and the solution of M l u l = r l is (34) u l = U l U l U l m l (D l ) L l m l L l m l L l 3 r l, where fast computation goes from right to left

13 VSVO 3-STAGE HBO ODE SOLVER 45 Process Procedure (34) is implemented by the following two steps: Step Algorithm 4 in [9] overwrites r l = r l ( : m l ) with U l U l m l (Dl ) L l m l Ll m l Ll 3 rl in O(m l ) operations The input is M = M l ; m = m l ; r = r l ; L k = L l k, k = 3, 4,, m l ; U k = U l k, k =,, m l ; and D = D l Step For each value of l, one of the following three cases is computed: Case (l = ) The following iteration overwrites r = r ( : m ) with U r : r (3) = r (3)U (3, 3), r (4) = r (4)U (4, 4), r (i) = r (i)u (i, i), i = 5, 7,, (m + )/, r () = r () r (3), r (3) = r (3) r (4), r (4) = r (4) r (5), if m 5, r (i) = r (i) r (i + ), i = 5, 7,, (m + )/ 3 Case (l = ) The following iteration overwrites r = r ( : m ) with U r : r (i) = r (i)u (i, i), i = 3, 5, 7,, (m + )/, r (i) = r (i) r (i + ), i =, 3, 5,, (m + )/ 3 Case 3 (l = 3) The following iteration overwrites r 3 = r 3 ( : m 3 ) with U 3 r3 : r 3 (3) = r 3 (3)U 3 (3, 3), r 3 (i) = r 3 (i)u 3 (i, i), i r 3 () = r 3 () r 3 (3), r 3 (3) = r 3 (3) r 3 (4), if m 3 4, = 4, 6, 8,, m 3/, r 3 (i) = r 3 (i) r 3 (i + ), i = 4, 6, 8,, m 3 /

14 46 TRUONG NGUYEN-BA, ET AL Order α/4 α/ p HBO(p)3 ABM(p, p ) TABLE : For order p, the table lists the scaled abscissae of absolute stability for HBO(p)3 and ABM(p, p ) 7 Regions of absolute stability and principal error term The unscaled regions of absolute stability, R, of HBO(p)3, p = 4, 5, 4, shown in Figure are obtained by applying the root condition (see [3, pp 56 57]) to the characteristic equation (35) (p)/ j= γ j r j =, which comes from an application of HBO(p)3 with constant h to the linear test equation y = λy, y = Let ABM(4-3) denote the family of ABM methods, ABM(p, p ), with predictor of order p and corrector of order p in PECE mode Table lists the scaled abscissae of absolute stability, α/4 and α/, of HBO(4-4)3 and ABM(4-3) [4, p 35 4], respectively It is seen that HBO(p)3 has a larger scaled interval of absolute stability than ABM(p, p ) for p > 7 The principal error term of HBO(p)3 is of the form (36) [δ {f p } + δ (p){{f p }f} + δ 3 { f p } + δ 4 { 3 f p } 3 ] h p+,

15 VSVO 3-STAGE HBO ODE SOLVER HBO(5) HBO(4) -3α α HBO(6) HBO(7) α - α HBO(9) HBO(8) 5-5 α HBO() 5 - α α - -5 HBO() 5-8 α HBO() HBO(3) 4 5 α α HBO(4) α FIGURE : Unscaled regions of absolute stability of HBO(4-4)3

16 48 TRUONG NGUYEN-BA, ET AL PLTC of HBO(p)3 4 PLTC PLTC p δ δ δ 3 δ 4 HBO(p)3 ABM(p, p ) e 86e 38e 44e 98e 3 8e 46e 3 e 96e 3 83e e 3 75e 444e 4 65e 7e 4 57e 4e 4 5e 58e 5 45e 46e 5 TABLE 3: For given order p, the table lists the principal local truncation error coefficients (PLTC) of HBO(p)3 and the scaled norms 4 PLTC and PLTC for HBO(p)3 and ABM(p, p ), respectively where {f p }, {{f p }f}, { f p }, { 3 f p } 3 are elementary differentials defined in [6], [6] and [] and the principal local truncation error coefficients (PLTC) are [δ, δ, δ 3, δ 4 ] One may attempt to reduce the norm PLTC by varying the parameters c and c 3 in HBO(4-4)3 The PLTC of ABM(p, p ) are [β k C p, C p+ ] [7, p 7] Table 3 lists the principal local truncation error coefficients (PLTC) of HBO(p)3 Also listed are the scaled norms 4 PLTC and PLTC for HBO(p)3 and ABM(p, p ), respectively, the former being rapidly decreasing and being smaller that the latter 8 Controlling stepsize and order A variant of the procedure described in [4] is used to control the stepsize and order, p, of VSVO HBO(4-4)3 For simplicity, in this section the order of the step control predictor P 4 will be denoted by q = p The program computes the maximum norm E q = y n ỹ n,q, where ỹ n,q := ỹ n is the value obtained by P 4 The stepsize h n+ is obtained by the formula (see [4]): { ( ) } /κ tolerance (37) h n+ = min h max, β h n, 4 h n, E q

17 VSVO 3-STAGE HBO ODE SOLVER 49 with κ = p and safety factor β = 8 The coefficients of IF, P, P 3 and P 4 are obtained successively as fast solutions of the linear systems (3), (5), (8) and () The step to x n+ is accepted if E q tolerance, else it is rejected and the program returns to the previous step with smaller step 7 h n+ If the step to x n+ is successful, besides P 4, three other step control predictors of orders ρ = q ± and ρ = q, respectively, similar to P 4, [ (ρ)/ ] ỹ n+,ρ = y n + h n+ a 4j f n+cj + a 43 f n+ + β 4j f nj j= + h n+ [ (ρ)/ j= γ 4j f nj are used to control the order and stepsize by means of the following three maximum norms, ], j= E q± = y n+ ỹ n+,q±, E q = y n+ ỹ n+,q, which estimate the local error at x n+ had the step to x n+ been taken at orders q ± and q, respectively To choose the lowest satisfactory order the following rules are used (a) The order is lowered if E q min{e q, E q+ } or E q max{e q, E q } (b) The order is raised only if the following stronger conditions, are satisfied E q+ < E q < max{e q, E q }, (c) When the order q of P 4 is, E q+ is not available; thus, the order is lowered if E q max{e q, E q } (d) When q =, the order is raised only if E q+ < E q

18 43 TRUONG NGUYEN-BA, ET AL Then κ and E q are reassigned according to the selected order κ For example, if the order is to be lowered in the next step, κ new = κ old and E q = E q The stepsize h n+ is then controlled by formula (37) We summarize the procedure to advance integration from x n to x n+ (a) The order p is obtained by the above procedure Then, the stepsize, h n+, is obtained by formula (37) with κ = p (b) The numbers η,, η (p3)/ +, defined in (9), are calculated (c) The coefficients of IF, P, P 3 and P 4 are obtained successively as solutions of systems (3), (5), (8) and (), respectively (d) The values y n+c, y n+c3, y n+, and ỹ n+ are obtained by formulae (3) (6) (e) The step is accepted if y n+ ỹ n+ is smaller than the chosen tolerance in which case the program goes to (a) with n replaced by n + Otherwise the program returns to (a) with the same order p and smaller step 7h n+ 9 Numerical results 9 Comparing HBO(4-4)3 against ode3 The numerical performance of HBO(4-4)3 programmed in Matlab and Matlab s ode3 is compared on the following set of problems: Arenstorf s orbits [], the Brusselator and the Pleiades [, pp 44 49], Euler s equation and the restricted three-body problem [4, pp 3 59], and the following nonstiff DETEST problems: growth problem B of two conflicting populations, two-body problems D D5, and Van der Pol s equation E with ɛ = [4] HBO(4-4)3 is selfstarting with HBO(4)3 The initial step size, h, is chosen by a method similar to steps (a) and (b) of [, p 69] Computations were performed on a Mac with a dual 5 GHz PowerPC G5 and 4 GB DDR SSRAM running under Mac OS X Version 46 and Matlab Version 7435 (R4) Service Pack 9 CPU against maximum global error The maximum global error is taken in the uniform norm, MGE = max n { y n z n }, where y n is the numerical value obtained by HBO(4-4)3 and z n is the exact solution obtained by Matlab s ode3 with stringent tolerance 5 4

19 VSVO 3-STAGE HBO ODE SOLVER Brusselator Arenstorf Euler Pleiades E D FIGURE : CPU (horizontal axis) versus log ( MGE ) (vertical axis) HBO(4-4)3 and ode3

20 43 TRUONG NGUYEN-BA, ET AL Problem CPU PEG Problem CPU PEG Arenstorf 4% D % Brusselator 34% D 3% Euler 3% D3 37% Pleiades 45% D4 5% Restricted 3-body 43% D5 49% B 6% E 3% TABLE 4: CPU percentage efficiency gain, CPU PEG, of HBO(4-4)3 over ode3 for the listed problems In Figure, CPU (horizontal axis) is plotted versus log ( MGE ) (vertical axis) for six problems in hand 9 CPU percentage efficiency gain of HBO(4-4)3 against ode3 The CPU percentage efficiency gain (CPU PEG) is defined by the formula (cf Sharp [5]), ( j (38) (CPU PEG) i = CPU ),ij j CPU,,ij where CPU,ij and CPU,ij are the CPU of methods and, respectively, associated with problem i, and j = log ( MGE ) The CPU time was obtained from the curves which fit the data (log ( MGE ), log (CPU)) in a least-squares sense using, say, Matlab s polyfit The CPU PEG for the set of problems in hand is listed in Table 4 It is seen from Figure and Table 4 that HBO(4-4)3 compares favorably with Matlab s ode3 on the basis of CPU versus log ( MGE ) for the set of problems in hand 93 Maximum global error against tolerance For six typical problems, time interval [, t f ] and given tolerance (TOL), Table 5 lists several numerical results related to the step control, namely: CPU time (CPU), number of function evaluations (NFE), number of failed attempts (REJ) and maximum global error (MGE) of HBO(4-4)3 and ode3 For HBO(4-4)3, NFE splits into 75% and 5% between f and f, respectively Similar results hold for the other problems in hand For equivalent MGE, it is seen from Table 5 that HBO(4-4)3 uses less CPU time and fewer function evaluations than ode3 It is also

21 VSVO 3-STAGE HBO ODE SOLVER 433 HBO(4-4)3 (left col) and ode3 (right col) Problem TOL CPU NFE REJ MGE BRUS e 6 64e 4 t f = e 9 74e e 3 e EULER e 4 89e 3 t f e 8 399e e 89e AREN e e t f e 6 8e e 9 39e 6 PLEI e 3 499e t f = e 6 7e e 9 9e 6 E e 5 6e 4 t f = e 9 5e e 7e 9 D e 4 8e t f = 6π e 6 8e e 9 39e 6 TABLE 5: For given problem, time interval [, t f ] and tolerance (TOL), the table lists CPU time (CPU), number of functions evaluations (NFE), number of failed attempts (REJ) and maximum global error (MGE), in left column for HBO(4-4)3 and right column for ode3, respectively seen that, for a given tolerance, the step control of both methods is problem dependent There is a compromise between a sharper step control and the number of failed attempts Similar results hold for the other problems considered in this paper 9 Comparing HBO(4-4)3 against HBO(4)3 In this section, VSVO HBO(4-4)3 is shown to perform better than VS HBO(4)3 [9] on some problems The starting values for HBO(4)3 are obtained with DP(5,4)7FM [8] In Figure 3, CPU is plotted versus log ( MGE ) for Van der Pol s equations for increasing ɛ = 3, 4, 5, 6 [, pp 5], with increasing CPU PEG = 6%, 7%, 6%, 7%, respectively In Figure 4, CPU is plotted versus log ( MGE ) for DETEST s B, C, E, and E5 with CPU PEG = 37%, 4%, %, and 8%, respectively 93 Comparing HBO(4-4)3 in C++ against DP(8,7)3M In Figure 5, CPU (horizontal axis) is plotted versus log MGE (vertical axis) for the Brusselator and the cubic wave equation [5] as obtained by HBO(4-4)3 and DP(8,7)3M The CPU PEG s, defined by formula (38), are % and 6% for

22 434 TRUONG NGUYEN-BA, ET AL ε = 3 ε = ε = 5 ε = FIGURE 3: CPU (horizontal axis) versus log ( MGE ) (vertical axis) for Van der Pol s equation HBO(4-4)3 and HBO(4)3 the Brusselator and the cubic wave over the intervals [e-4,e-] and [e-,e-], respectively Similar to the test results in [9], it is seen from Figure 5 and the CPU PEG that for the cubic wave problem whose derivative evaluations are relatively expensive, HBO(4-4)3 wins over DP(8,7)3M Conclusion A variable-step variable-order 3-stage Hermite- Birkhoff-Obrechkoff method, HBO(4-4)3, of order 4 to 4 was constructed by solving Vandermonde-type systems satisfying multistep-type and Runge-Kutta-type order conditions The stepsize and order are controlled by four local error estimators This method, in its vectorized Lagrange form, was tested on the Brusselator, Euler s equation, Arenstorf s orbits, the restricted three-body problem, the Pleiades, and the following nonstiff DETEST problems: two-body problems of class D, the growth problem of two conflicting populations of class B and Van der Pol s equation of class E Programmed in Matlab, HBO(4-4)3 was

23 VSVO 3-STAGE HBO ODE SOLVER B C E E FIGURE 4: CPU (horizontal axis) versus log ( MGE ) (vertical axis) HBO(4-4)3 and HBO(4)3 4 6 Brusselator 4 Cubic wave FIGURE 5: CPU (horizontal axis) versus log ( MGE ) (vertical axis) HBO(4-4)3 and DP(8,7)

24 436 TRUONG NGUYEN-BA, ET AL found to have lower global error, use less CPU time and require fewer function evaluations than the Matlab s ode3 HBO(4-4)3 was also found to perform better than HBO(4)3 of order 4 on Van der Pol s equations with ε = 3, 4, 5, 6, and DETEST s problems B, C, E and E5 Programmed in C++, HBO(4-4)3 is superior to DP(8,7)3M in solving expensive equations over a long period of time Acknowledgment The anonymous referee is deeply thanked for pointing out important modifications to this paper Philip W Sharp is heartily thanked for his suggestions which considerably improved the content and format of a preliminary version of this paper and for pointing out the cubic wave equation to the authors REFERENCES R F Arenstorf, Periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian elliptic motions, Amer J Math LXXXV (963), 7 35 R Barrio, F Blesa and M Lara, VSVO formulation of the Taylor method for the numerical solution of ODEs, Comput & Math with Applics 5 (5), 93 3 A Björck and T Elfving, Algorithms for confluent Vandermonde systems, Numer Math (973), A Björck and V Pereyra, Solution of Vandermonde systems of equations, Math Comp 4 (97), P J Bryant, Nonlinear wave groups in deep water, manuscript 6 J C Butcher, Coefficients for the study of Runge-Kutta integration processes, J Aust Math Soc 3 (963), 85 7 J C Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J Assoc Comput Mach (965), J R Dormand and P J Prince, A reconsideration of some embedded Runge- Kutta formulae, J Comput Appl Math 5 (986), 3 9 W H Enright and T E Hull, The test results on initial value methods for non-stiff ordinary differential equations, SIAM J Numer Anal 3 (976), G Galimberti and V Pereyra, Solving confluent Vandermonde systems of Hermite type, Numer Math 8 (97), 44 6 G H Golub and C F Van Loan, Matrix Computations, 3rd edition, The Johns Hopkins University Press, Baltimore MD, 996 E Hairer, S P Nørsett and G Wanner, Solving Ordinary Differential Equations I Nonstiff Problems, Springer-Verlag, Berlin, E Hairer and G Wanner, Solving Ordinary Differential Equations II Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 99

25 VSVO 3-STAGE HBO ODE SOLVER T E Hull, W H Enright, B M Fellen and A E Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J Numer Anal 9 (97), F T Krogh, Changing stepsize in the integration of differential equations using modified divided differences, in: Proc Conf on the Numerical Solution of Ordinary Differential Equations, University of Texas at Austin 97 (D G Bettis, ed), Lecture Notes in Mathematics No 36, Springer-Verlag, Berlin, 7, J D Lambert, Computational Methods in Ordinary Differential Equations, Ch 5, Wiley, London, J D Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, Chichester UK, 99 8 T Nguyen-Ba and R Vaillancourt, Hermite-Birkhoff differential equation solvers, Scientific Proceedings of Riga Technical University, in series Computer Science (46) (4), T Nguyen-Ba and R Vaillancourt, Hermite-Birkhoff-Obrechkoff 3-stage 6-step ODE Solver of order 4, Can Appl Math Quarterly 3() (5), 5 8 T Nguyen-Ba, H Yagoub, Y Li and R Vaillancourt, Variable-step variableorder 3-stage Hermite-Birkhoff ODE Solver of order 5 to 5, Can Appl Math Quarterly 4() (6), T Nguyen-Ba, H Yagoub, S J Desjardins and R Vaillancourt, Variable-step variable-order 4-stage Hermite-Birkhoff-Obrechkoff ODE Solver of order 5 to 4, Scientific Proceedings of Riga Technical University, in series Computer Science 3(48) (6), in press P J Prince and J R Dormand, High order embedded Runge-Kutta formulae, J Comput Appl Math 7() (98), E Rabe, Determination and survey of periodic Trojan orbits in the restricted problem of three bodies, Astronomical J 66(9), L F Shampine and M K Gordon, Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman, San Francisco, CA, P W Sharp, Numerical comparison of explicit Runge-Kutta pairs of orders four through eight, Trans on Mathematical Software 7 (99), Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada KN 6N5 address: tnguyen@mathstatuottawaca address: hy s@gmailcom address: zhangyujerry5@hotmailcom address: remi@uottawaca

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