E. Hairer, S. P. Nrsett, and G. Wanner, Solving Ordinary Differential Equations I, Springer Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the vectorial problem are integrated according to the same rule. Solving for z and i 1 and using (3.7) wehave (z# i 1 ) 1 h i (1b Q) 1 h i b # Omega D ( i # f) 3.9) From this expression, we conclude that (3. 2) can be written as a s stage RK method whose tableau [10]isgiven by c 1b b : 3.10) Finally,we test the Gauss Legendre, Lobatto and Radau quadrature rules and compute the corresponding tableaux (3.10) The results are summarized in Table (3.1) Correspondencebetween quadrature rules for BD finite elements and RK methods. Quadrature Rule RK ....

....for the quadraturepoint values. However, these two sets of values are simply related by a linear transformation. Furthermore, note that all RK schemes appearing in the table are of the collocation type, where polynomials interpolate the twointerface values i , i 1 and the internal stages z [10]. This link between finite elements and RK processes allows a somewhat unique wayoflooking at the discretization procedure: from the finite element point of view we see jumps in the field variables, since wehave discontinuous interpolations of the internal fields, glued together from one element ....

E. HAIRER, S.P. NRSETT, AND G. WANNER, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, 1987.

....[13] which switches automatically between a non stiff and a stiff integration algorithm along the solution. LSODAR also provides a root finder. RK45 78 Runge Kutta Fehlberg solvers of orders 5 and 8 with variable stepsize of Kraft Fuhrer using the PrinceDormand coefficients according to [11]. DASSL RT Multistep solvers of Petzold [19] for DAEs (DASSL) and for DAEs with root finder (DASSLRT) ODASSL RT Multistep solvers of Fuhrer [5] based on DASSL DASSLRT of Petzold for ODAEs (ODASSL) and for ODAEs with root finder (ODASSLRT) MEXX Extrapolation solver of Lubich [14] for a ....

Hairer, E.; Norsett, S.P.; Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. SpringerVerlag, Berlin, 1987.

....or other methods relying on second derivatives of # could be spoiled by these second order discontinuities, while the Gauss Newton method that requires only first derivatives is robust against them. The trajectory is computed in the data segment by integrating Eq. 7b) using the code RETARD [20]. The sensitivities of the trajectory with respect to the parameters, needed in the optimisation process, are obtained by solving the variational equations along with Eq. 7b) 17,20] The cost function is computed by Eq. 4) Then x(t) is projected onto the spline manifold h of the ....

....is robust against them. The trajectory is computed in the data segment by integrating Eq. 7b) using the code RETARD [20] The sensitivities of the trajectory with respect to the parameters, needed in the optimisation process, are obtained by solving the variational equations along with Eq. 7b) [17,20]. The cost function is computed by Eq. 4) Then x(t) is projected onto the spline manifold h of the following subinterval by simply reading out the function values and derivatives, i.e. 10a) r jk 0, K, r jk =x(T 0,K. Now the continuity constraints read r j (r j0 ....

E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I, Springer, Berlin, 1987.

....at a number of spline knots and its first derivatives at the two end points W W [37] By choosing an appropriate density of the spline knots, the number of degrees of freedom of the initial curve can be adapted to the problem. The integration of the DDE is done by the code RETARD [38]. Details of the method and aspects of implementation are described in [39] Applications to ordinary differential equations can be found in [36, 40] APPLICATION TO THE ELECTRONIC CIRCUIT Nonparametric model fit To recover the driving system s nonlinearity and the damping factor from the ....

Hairer, E., Nrsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I, Springer, Berlin, 1987.

....appear in the literature, it is becoming evident that the choice of proper test systems can greatly ease code comparisons and algorithm development. Compilations of test problems have become well established in other fields, for example in optimization [9 11] and differential equations [12,13]. The CASP project [14] Critical Assessment of techniques for protein Structure Prediction) provides a testbed for objective testing and comparison of methods for identifying protein structure from sequence. To our knowledge, no similar collection exists for design and evaluation of molecular ....

E. Hairer and G. Wanner, Solving ordinary differential equations. Volume II, Springer Series in Comput. Mathematics, Vol. 14, Springer-Verlag 1996, http://www.zib.de/uwe.poehle/ode.html

....asymptotic A1 expansion on N. Lets say v L L. 1 1 (x)dx ao al . 2The next monography from Hairer et al. [8] is expected to contain a full review of the uses of trees in numerical analysis, including the Hopf algebra formulation. Meanwhile, see [1] For an introduction to numerics, the book [7] is a handy tool Of course the first term, a0, will be the value of the integral if the integral exists. On other hand, we are more interested in the integral of a vector field, 3 The unit in Butcher group is the trivial method x, x, e. The inverse of a method x, y, e is the method y, x, e that ....

E. Hairer, S.P. Norserr et G. Wanner, Solving Ordinary Differential Equa- tions, I

....of #. It can be seen that the right most zero minimizes the local error in this example. Figure 3 illustrates our theoretical results experimentally on a specific ODE. It plots the local error of several global Hermite filters (GHF) as a function of the evaluation time for the Lorenz system (e.g. [HNW87]) It is assumed that # ### # # # is constant (# # # # ## # #) In addition, we assume that, in each mean value filter composing the GHF, the distance between the evaluation time and the rightmost interpolation point is constant. In the graphs, ## # ## # ##### ##### and # # # # # # # #####. The ....

....(OREG) a stiff problem, model famous chemical reactions, the Lorenz system (LOR) is an example of the so called strange attractors , the Two Body problem (2BP) comes from mechanics, and the van der Pol (VDP) equation describes an electrical circuit. All these problems are described in detail in [HNW87]. We also consider a problem from molecular biology (BIO) the Stiff DETEST problem D1 [Enr75] and another stiff problem (GRI) from [Gri72] Finally, we consider four problems (LIEN, P1, 12 0 0.5 1 1.5 2 2.5 3 3.5 4 GHF(3,3) GHF(4,4) 1.5 2 2.5 3 3.5 4 4.5 5 5.5 GHF(3,3) GHF(4,4) ....

E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

....behaves well as long as it is used away from the maximum possible precision [AL97] This quantity depends both on the equation which is to be solved and on the precision arithmetic. To give an idea of this phenomenon, the Figure 1 shows the behavior of the global error and R i using DOPRI5(4) [HNW87] when the tolerance varies on the real ODE: 10 (y Gamma x y(0) 0:02; on [0; 2] Its solution is y(x) 0:02 0:2 x x . This is a very unstable problem. It can be seen that from tolerance 10 to 10 , the global error decreases linearly in logarithmic scale. In this zone, the ....

E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, 1987.

....uses finite elements and constraints without a reference frame. While it naturally leads to problems in the higher frequencies and even to instabilities, see also Shabana [10] stabilization techniques exist which add an artificial numerical damping, like the HHT method, see e.g. Hairer et al. [6, 7]. Alternatively, Gonzalez and Simo [5] derived stable energy momentum methods for Hamiltonian systems and it has been extended to contact and impact problems by Demkowicz and Bajer [3] The finite element method has been used for the modelling of multibody dynamics problems with contact, see e.g. ....

E. Hairer, S. P. Nrsett, and G. Wanner, Solving ordinary differential equations I, nonstiff problems, Springer--Verlag Berlin Heidelberg (1987).

....km of the highest possible order in the stepsize h, i.e. for any fixed m such that 1 m s, we seek r = r 1 ; r m ) such that k i = O(h ) 29) for q as large as possible. For this purpose we need to use some theory of order conditions for RK methods. It is well known (see e.g. [6]) that y 1 as well as each of the stages k i has a B series. This is an expansion which we write in the form B(a; y) t2T ae(t) ae(t) a(t)F (t) y) 30) 17 Here T is the set of rooted trees, ae(t) is the number of nodes in the rooted tree t, a assigns to each t 2 T a real number, ....

....and F (t) depends on the derivatives of the function f and is called an elementary differential. The theory now tells us that there are sequences u i , k i , and y 1 such that (formally) u i = B(u i ; y 0 ) k i = B( k i ; y 0 ) y 1 = B(y 1 ; y 0 ) 31) for i = 1; s. Hairer et al. [6] provide the following recursion formulas for k i and u i u i ( 1; k i ( 1; y 1 ( 1; u i (t) k j (t) k i (t) ae(t)u i (t 1 ) Delta Delta Delta u i (t m ) y 1 (t) b j k j (t) 32) where the trees t 1 ; t m are such that t = t 1 ; t m ] ....

E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I, Nonstiff Problems. Springer-Verlag, Second revised edition, 1993.

....covers variable step, varying smoothness and the possibility of a vanishing statedependent lag. The paper provides a systematic, self contained and rigorous analysis valid under these conditions. We assume familiarity with the concept of a continuous explicit Runge Kutta (CERK) method (see [5] p.176 et seq. for initial value problems in ordinary differential equations (ODEs) t) f(t; y(t) for t t 0 and y(t 0 ) y 0 : 1) The CERK methods outlined in this paper are suitable for solving DDEs of the form (t) F (t; u(t) u(fl(t; u(t) for t 0 t t N T; 2) Mathematics ....

E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations -- I, SpringerVerlag, Berlin, 1987.

....methods for which a ij = 0 when i j and i j, respectively. In the explicit case, each stage, k i , depends only on previously computed stages k j , j i. With diagonal implicit methods we need to solve a system of equations in each stage. Runge Kutta methods are extensively studied in e.g. [2, 5]. 2.2 Crouch Grossman Methods The Crouch Grossman methodswere introduced in [3] The methodsare basedupon the existence of a frame on a differentiable manifold, M, with tangent space TM p , p 2 M, and tangent bundle TM defined as TM = S p2M TM p . A vector field on M is a section of TM, F : M ....

....defined as follows. Algorithm 2. 2 (Munthe Kaas Methods) y 0 = p for i = 1; 2; s u i = h j=1 a ij k i = f (c i h; u i ; y 0 ) k i = dexpinv(u i ; k i ; q) end y 1 = v; y 0 ) where the coefficients are given by the qth order classical Runge Kutta method s Butcher tableau [5], and the dexpinv function is defined by dexpinv(u; v; q) v Gamma 2 [u; v] q Gamma1 k=2 B k k k z [u; u; u; v] where [ is the matrix commutator and B k is the kth Bernoulli number. The matrix commutator is defined by [A; B] AB Gamma BA when A and B ....

[Article contains additional citation context not shown here]

E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Springer Verlag, 1993.

....of such changes. Typically these changes are combined with interpolation procedures for obtaining newly required back information, y n Gamma2 ; y n Gamma3 ; y n Gammak ; the just computed value y n Gamma1 at t n Gamma1 is unchanged. Other representations of the BDF are summarised in [3]. 1.2) is a formula for y n which in this paper we regard as an approximation to Y (t n ) the value of the local solution of (1.1) at t n = t n Gamma1 h where Y = f(t; Y ) t t n Gamma1 ; Y (t n Gamma1 ) y n Gamma1 : 1.3) We assume throughout that unique local solutions exist. This ....

....this quantity can be expanded in powers of h about t n Gamma1 and it is the principal term which is estimated in most codes in order to control overall accuracy. An alternative definition of local solution and truncation error for linear multistep methods is given by Hairer, Norsett and Wanner [3], pp. 315 317. 2 At a typical step (1.2) requires the solution of the algebraic system ff k;i y n Gammai Gamma hfi k f(t n ; y) 0; F : D ae R ; 1.6) and in section 1 sufficient conditions are given for the existence and uniqueness of a solution y j y n . The conditions are ....

E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations 1. Berlin: Springer Verlag 1987.

.... In particular, we compare the efficiency and absolute stability of a fifth order parallel ICE with the fifth order Dormand Prince method [4] with a fifth order Hermite approximant [7] 2 A Brief Review of ICEs We assume that the reader is familiar with the theory for a stage CERK method (see [5] p.176 et seq. as defined by the continuous Runge Kutta (RK) tableau c A (1) in which c 1 = 0 and c = 1. The key to constructing parallel ICEs is the calculation of an appropriate order continuous quadrature extension (CQE) Definition 2.1 A CERK method (1) has continuous quadrature order q ....

E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations - I, Springer-Velag, Berlin, 1987.

....= y 0 and a set of parameters 2 IR n . The (numerical) solution can be noted by the concept of flux [3] by y( y 0 ; In most cases the given differential equation system may be nonlinear. Therefore the solution noted by the flux incorporates a procedure of numerical integration [6,7]. 6 R. Seppelt M. M. Temme In the notation of the Petri net, a differential equation system is applied to a transition t j by the following steps. y 0 : w Gamma (p i ; t j ) i=1; jI M (t j )j and e w Gamma (p i ; t j ) 1 for (p i ; t j ) 2 AM : w Gamma (p i ; t j ) ....

E. Hairer and G. Wanner. Solving Ordinary Differential Equations, volume 2. Springer--Verlag, 1980.

....= y 0 and a set of parameters 2 IR n . The (numerical) solution can be noted by the concept of flux [3] by y( y 0 ; In most cases the given differential equation system may be nonlinear. Therefore the solution noted by the flux incorporates a procedure of numerical integration [6,7]. 6 R. Seppelt M. M. Temme In the notation of the Petri net, a differential equation system is applied to a transition t j by the following steps. y 0 : w Gamma (p i ; t j ) i=1; jI M (t j )j and e w Gamma (p i ; t j ) 1 for (p i ; t j ) 2 AM : w Gamma (p i ; t j ) ....

E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations, volume 1. Springer--Verlag, 1980.

....allow the code to exploit the parallelism associated with the problem. To evaluate the behaviour of a number of proposed acceleration techniques, the test problem considered is a reactiondiffusion equation known as the diffusion Brusselator equation, defined on the unit square. It is described in [8] and has the form u t =B u 2 v Gamma (A 1)u ff 2 u x 2 2 u y 2 ; v t =Au Gamma u 2 v ff 2 v x 2 2 v y 2 : 15) The choice of initial values in this problem has a marked effect on the rate of convergence of the iterates. ....

E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Verlag, New York, (1987).

....can be used. For the test problem chosen in section 4.4 the banded option is the appropriate choice. 4. 4 The test problem In order to test the performance of our code on large problems, we chose as our test equation a reaction diffusion equation known as the diffusion Brusselator equation [16], defined on the unit square. It takes the form u t = B u 2 v Gamma (A 1)u ff i 2 u x 2 2 u y 2 j v t = Au Gamma u 2 v ff i 2 v x 2 2 v y 2 j 9 = 13) 20 with initial conditions u(0; x; y) 2:d0 1 y; v(0; x; y) 1 2 x ....

.... u n = 0; v n = 0: Here u and v denote chemical concentrations of reaction products, A and B are concentrations of input reagents which are taken to be constant and ff = d L 2 where d is a diffusion coefficient and L a reactor length. The solution of this problem is plotted in [16]. This problem is converted into a system of ordinary differential equations by the method of lines. That is, the second order spatial derivatives are replaced by central finite differences on a uniform array of N Theta N points. If the grid discretization parameter in both the x and y ....

Hairer, E., Norsett, S.P., Wanner G. (1987). Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Verlag, New York.

....of . It can be seen that the right most zero minimizes the local error in this example. Figure 3 illustrates our theoretical results experimentally on a specific ODE. It plots the local error of several global Hermite filters (GHF) as a function of the evaluation time for the Lorenz system (e.g. [HNW87]) It is assumed that t i 1 t i is constant (0 i 2k 2) In addition, we assume that, in each mean value filter composing the GHF, the distance between the evaluation time and the rightmost interpolation point is constant. In the graphs, t 0 ; t k ] 0; 0:01] and h = t k t 0 = 0:01. The ....

....(OREG) a stiff problem, model famous chemical reactions, the Lorenz system (LOR) is an example of the so called strange attractors , the Two Body problem (2BP) comes from mechanics, and the van der Pol (VDP) equation describes an electrical circuit. All these problems are described in detail in [HNW87]. We also consider a problem from molecular biology (BIO) the Stiff DETEST problem D1 [Enr75] and another stiff problem (GRI) from [Gri72] Finally, we consider four problems (LIEN, P1, 12 0 0.5 1 1.5 2 2.5 3 3.5 4 10 15 10 10 10 5 10 0 Time IHO(3,3) GHF(3,3) GHF(4,4) 1.5 2 2.5 ....

E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

..... It can be seen that the right most zeros minimizes the local error in this example. Figure 8 illustrates our theoretical results experimentally on a specific ODE. It plots the local error of several global Hermite filters (GHF) as a function of the evaluation time for the Lorenz system (e.g. [HNW87]) It is assumed that t i 1 t i is constant (0 i 2k 2) In addition, we assume that, in each mean value filter composing the GHF, the distance between the evaluation time and the rightmost interpolation point is constant. In the graphs, t 0 ; t k ] 0; 0:01] and h = t k t 0 = 0:01. The ....

....chemical reactions. Both OREG and HIRES are stiff pronlems. The Lorenz system (LOR) examplifies the so called strange attractors . the Two Body problem (2BP) comes from mechanics and the van der Pol (VDP) equation describes an electrical circuit. All these problems are described in detail in [HNW87]. We also consider a problem from molecular biology (BIO) and the Stiff DETEST problem D1 [Enr75] Finally, we consider four dynamical systems (LIEN, P1, P2, P3) where the function f contains more operations. LIEN, P2 and P3 are taken from [Per00] Overview of the Experiments The experimental ....

E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

.... of the elementary differentials of order O(ffl i ) i.e. DeltaX i (x) 1 i X j d i;j f i;j (x) 29) Now, each Lie derivative (L Y ) i Gamma1 Y is a linear combination (with weights equal or greater than one) of the elementary differentials f i;j of order O(ffl i ) as well [17], i.e. 1 i (L Y ) i Gamma1 Y (x) 1 i X j a i;j f i;j (x) a i;j 1. By Lemma 2, we know that 1 i jj (L Y ) i Gamma1 Y jj R=2 ffl M 2 ffl M R i Gamma1 and, therefore, jj f i;j jj R=2 i ffl M 2 ffl M R i Gamma1 for all elementary differentials f i;j of order ....

Hairer, E., Norsett, S.P., and Wanner, G., Solving ordinary differential equations I. Nonstiff problems. second revised edition, Springer Verlag, 1993.

....1: you can make large timesteps except in the region of the two jumps) Thus, in order to be efficient, smoothed MD requires stepsize control schemes. Up to now, it has not become clear how to solve this problem most efficiently: in the scope of explicit, symmetric extrapolation schemes (cf. [4] or [2] or by use of symplectic discretizations [5] 6] 3 Statistical formulation of Molecular Dynamics This Section is concerned with the question of how to give an ensemble formulation of (1.2) and of the additional heat bath embedding of the molecular system. 3.1 Probability Density and ....

E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I, Nonstiff Problems. Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 2nd edition, 1993.

.... (or regridding) and order adjustment (see e.g. GB86] for finite elements, Hoh93] for collocation) On the other hand, h p methods are used to solve initial value problems by successively choosing appropriate local orders and stepsizes for the next step of the integration process (see e.g. [HNW87], Deu83] We shall deal with the latter which may be called evolutional h p methods to distinguish them from the former. Let us consider an initial value problem x 0 = f(x; t) x(t 0 ) x 0 of a first order ordinary differential equation. By Phi t;s we denote the associated flow from s ....

E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I, Nonstiff Problems. Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1987.

....valid for Runge Kutta solutions obtained with step size sequences fhng satisfying N X n=0 jh n 1=hn Gamma 1j C (5:1) ch hn h ; 0 n N ; 5:2) with a positive constant c. 27 Remark. Condition (5. 1) is familiar from the convergence analysis of linear multistep methods for ODEs, see [HaNW], Thm.III.5.7. Condition (5.2) may appear rather restrictive. However, if there is a finite subdivision of the integration interval into subintervals on which step sizes of different scales are used, then one can apply Theorem 5.1 separately on each of the subintervals. We do not give a proof of ....

E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, 2nd ed., 1993.

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E. Hairer, S. P. Nrsett, and G. Wanner, Solving Ordinary Differential Equations I, Springer Verlag, 1993.

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E. HAIRER,S.P.NRSETT, AND G. WANNER, Solving ordinary differential equations. I, Springer-Verlag, Berlin, second ed., 1993. Nonstiff problems.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, 2nd revised edition, 1993.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, 2nd revised edition, 1993.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: (2nd ed.), Springer-Verlag (Berlin), 1987.

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E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, 1987.

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E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I, Springer Verlag, 1993.

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E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. Second Revised Edition, Springer, Berlin, 1993.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I:, Springer-Verlag (Berlin), 1987.

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E.Hairer, S.P. Nrsett, G.Wanner, Solving Ordinary Differential Equations I, New York:

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Springer, 1987.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations (Springer-Verlag, Berlin, 2nd ed., 1993).

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E. Hairer, S. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, second edition, 1993.

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E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I, Non-stiff problems. Second edition, Springer, 1993.

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Hairer, E., Norsett, S., and Wanner, G. (1993). Solving ordinary differential equations. 2nd rev. ed. Series in Computational Mathematics 8. Springer-Verlag.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, 2nd revised edition, 1993.

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E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

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Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations, vol. II, Springer-Verlag, Berlin Heidelberg, Germany, 1996, 2 rev. ed.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving ordinary differential equations. I. Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I (Springer, Berlin, 1987).

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E. Hairer, S.P. N0rsett and G. Wannet, Solving Ordinary Differential Equations 1. Nonstiff Problems, Computational Mathematics 8 (Springer-Verlag, Berlin, 2nd rev. ed., 1993).

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E. Hairer, S.P. Nrsett, G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving ordinary differential equations I, Springer Ser. Comput. Math. 8, Springer-Verlag, 1987.

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