L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Comput., 18(1):1--22, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....f(x) 1. 40) The function f is given by f(x) cos #x ) and the exact solution is u(x, t) f(x exp( t) In this problem, no boundary condition is required. We solve (40) using the Chebyshev collocation method in space and a Runge Kutta method (the built in Matlab solver ode45, see [Sha Rei] for details) in time. We compare two di#erent methods. The first method uses the original matrix given by formulas (3) to (6) and the second method uses (3) and the NST. In Figure 23 we plot the errors at time t = 1 versus the number of collocation points. The NST method produces more accurate ....

Shampine, L. F. and Reichelt, M. W.: The Matlab ODE suite, SIAM J. Sci. Comput., 18, 1-22 (1997).

.... a specified accuracy, TOL, the numerical method, M, generates a piecewise polynomial, z TOL (x) defined for x 2 [a; b] and satisfying, kz TOL (x) Gamma y(x)k KM TOL: This assumption is satisfied by most state of the art ODE software and certainly by the built in numerical methods of MATLAB [8]. Note that KM can depend on the method and the problem and can be interpreted as the numerical condition number associated with method M. The restrictions we imposed on the verification tools we developed include: 1. The tool must be easy to use If possible the calling sequence should be ....

L.F. Shampine and M.W. Reichelt. The MATLAB ODE suite, SIAM J. Sci. Comp., 18, 1997, pp.1-22.

....has no critical points other than where 0. # Theorem 2 implies that the minimization of can be solved by local gradient descent, e.g. ## # for any initial condition when is of full rank, and almost all initial conditions otherwise. We have successfully used a stiff integrator [26], Powell s method [24] and stochastic descent [2] We find that simplex descent [22] and (branch and bound) global optimizations [12,14] do not work well, however, and may wander to a solution 0 that is not near the observed time series. Convergence results are discussed in detail in the ....

....figure with Fig. 2 which uses the same selection of states. Fig. 5. Evolution of deltas in a gradient descent minimization for the uppermost point of Fig. 4. The nine point trajectory segment gives a total of 18 deltas. The gradient descent was performed with the ode15s stiff integration routine [26] of Matlab. In panel (a) the deltas are plotted as a function of the integration control points, which are not uniformly spaced in time. The equivalent nominal integration time of these control points are shown in panel (b) note this is base 10 logarithm time. In this example the stable component ....

L.F. Shampine, M.W. Reichlet, The Matlab ODE suite, Technical Report, The MathWorks, Inc., Prime Park Way, Natick, MA, 1995.

....nonlinear equations which are commonly solved using some form of Newton iteration. For linear multistep methods we apply the theory of Dorsselaer and Spijker [11] to justify a new stopping criterion for the Newton iteration. The ideas are tested on the Matlab ODE suite of Shampine and Reichelt [10] and are shown to lead to a significant improvement in overall efficiency for some problems. Key words. stiff problems, IVPs, ODEs, BDFs, Newton method. AMS subject classifications. primary 65L05, 65L99. 1 Introduction We consider stiff initial value problems (IVPs) of the form = f(y(t) ....

....direct LU decomposition. We adopt an entirely different approach. We intend to reduce the overall number of linear systems that has to be solved. In order to illustrate the approach we conduct numerical experiments using the Matlab ODE suite code written by Shampine and Reichelt known as ode15s [10]. ode15s is a stiff ordinary differential equation solver that implements a modified form of the BDFs and the modified Newton method. The code can also be used to implement the standard BDF. The linear algebraic systems are solved by LU decomposition of the iteration matrix. This matrix is formed ....

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Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Stat. Comput., 18, No. 1:1--22, January 1997. 22

....formulas, NDFs. Optionally, it uses the backward differentiation formulas, BDFs (also known as Gear s method) that are usually less efficient. Like ode113, ode15s is a multistep solver. If you suspect that a problem is stiff or if ode45 has failed or was very inefficient, try ode15s. [7] ode23s is based on a modified Rosenbrock formula of order 2. Because it is a one step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective. 7] ode23t is an implementation of the trapezoidal rule using a ....

....or if ode45 has failed or was very inefficient, try ode15s. 7] ode23s is based on a modified Rosenbrock formula of order 2. Because it is a one step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective. [7] ode23t is an implementation of the trapezoidal rule using a free interpolant. Use this solver if the problem is only moderately stiff and you need a solution without numerical damping. ode23tb is an implementation of TR BDF2, an implicit Runge Kutta formula with a first stage that is a ....

Shampine, L. F. and M. W. Reichelt, "The MATLAB ODE Suite," (to appear in SIAM Journal on Scientific Computing, Vol. 18-1, 1997).

....Control Strategies 2 STEP STEP (ODE, DE, INTRP) by Shampine Gordon [47] is an implementation of VC PECE AMs of orders p = 1, 12, with local extrapolation, i.e. the order of the corrector is one order higher. The code is also implemented in the MATLAB solver ODEll by Shampine Reichelt [48]. The stepsize control strategy in STEP can briefly be described as follows. In case of a successful step, one estimates a possible stepsize ratio according to Equation (6.3) i.e. 2All codes presented here are well established LMM codes available at www. netlib. org. where e = 0. ol, with 0 ....

....5 0.2693 0.3349 0.0949 6 0.2072 0.2791 0.0678 7 0.1594 0.2326 0.0484 8 0.1226 0.1938 0.0346 9 0.0943 0.1615 0.0247 10 0.0725 0.1346 0.0176 11 0.0558 0.1122 0.0126 12 0.0429 0.0935 Table 6.1: Stepsize control set points for different orders for LSODE. LSODE; cf. Shampine Reichelt [48]. Some Remarks on the Conventional Stepsize Selection Schemes All coefficients in the stepsize control algorithms above, except for the exponent in (6.3) are purely empirical. The reason for imposing lower and upper limits on the stepsize ratio, COmi n COn 1 Wmx, is based on a natural ....

L. E Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAMJ. Sci. Cornput., 18(1):1-22, 1997.

....differential equations (ode s) is solved by a general ode solver. This approach is called method of lines. The recent version of MATLAB R contains a set of sophisticated general purpose codes (the so called odesuite) for solving ode s and and differential algebraic equations (dae s) of index 1 [36, 37]. In the present context, the solvers of the odesuite can be applied successfully unless (i) the mass matrix is singular and statedependent and or (ii) the semi discretized pde constitutes an index 2 dae and or (iii) the system is of such a size such that the sparse direct solver of MATLAB R ....

....the solvers contained within the odesuite the BDF implementation ode15s seems to be of most interest. This code is able to solve dae s of index (0 and) 1 provided that the mass matrix M does not depend on u. Among the features of ode15s the following are worth being mentioned at this point (cf. [36, 37]) In order to use the MATLAB R capabilities as efficient as possible the BDF method for variable stepsizes is implemented in a fixed coefficient formulation. In addition to the BDF methods of orders 1 to 5 a modification of these formulae is implemented which aims at a smaller truncation ....

Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ode suite. SIAM J. Sci. Comput., 18(1):1--22, 1997.

....from the NAG numerical library. See [3] for more details and discussion of this approach. The second approach involves the development of low overhead, low order, new methods completely within the PSE. For example, consider the recent sequence of improvements to the ODE component of MATLAB: [16], 14] and a similar improvement in the ODE capability of MAPLE [15] There is a need for both approaches as they complement each other. The latter is ideal for rapid prototyping and numerical experiments while the former is often required for final production runs and large scale simulations. ....

L.F. Shampine and M.W. Reichelt. The MATLAB ODE suite, SIAM J. Sci. Comp., 18, 1997, pp.1-22.

....developed to address the needs in IVP problem area. We mention here Plod [1] Odexpert [5] and Godess [6] all without capabilities to solve very large systems of ODEs with sequential, parallel or distributed methods) Solving ODEs is also an actual issue for computer algebra systems developers [10]. 2. Software requirements. Non scientific software for small computers raises the standard by which scientific software is judged. Users expect a similar level in technical applications. 2.1. Human versus automated expertise. In addition to knowledge of the software options, the user of a ....

L.F. Shampine, and M.W. Reichelt, The Matlab ODE Suite, SIAM Journal on Scientific Computing, Vol. 18, No. 1 (1997), pp. 1-22.

....and it is expected to be nalized at the end of this year. The facilities of ODEtools from Maple, and the similar tools from other CAS are far to cover all the user needs (for example, the sti IVP solving case) Recent reports demonstrate the e ort to improve these tools. For example, the paper [14] describes mathematical and software developments for a suite of programs for solving ODEs in Matlab. 5 D NODE package has similar facilities with EpODE (ExPert system for ODEs) recently presented in [9] and [11] and available at http: www.info.uvt.ro petcu epode: a large collection of ....

....on processes and parallel stages for the above mentioned methods. Like EpODE, D NODE will include a function to detect such process stages distributions. In order to show the performance of the methods on semi discrete PDEs we include in our tests the linear IVP obtained from the following PDE [14]: u(x; t) t = e t 2 u(x; t) x 2 (4) with initial and boundary conditions u(x; 0) sin(x) u(t; 0) u(t; 0. As the second test problem we take the nonlinear IVP obtained by the semidiscretization of the following nonlinear convection di usion problem [1] particular case for ....

Shampine, L.F., Reichelt, M.W., The Matlab ODE Suite. SIAM J. Sci. Comput. 18, No. 1 (1997), 1-22.

....method of lines approach, with the spatial dependence modelled using surface spherical harmonics. The resulting system of ordinary di#erential equations is solved by a sti# solver in this case ODE15s from the MATLAB ODE Suite, which is based on numerical di#erentiation formulae of order 1 to 5 [49]. The exponential (in space) convergence of spherical harmonic approximation means that we have to solve only relatively small systems of ordinary di#erential equations (typically of order 100 in size) and simulations of the special case (1.2) took times of the order of minutes on an up todate ....

L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comput. 18:1-22, 1997.

....method of lines approach, with the spatial dependence modelled using surface spherical harmonics. The resulting system of ordinary di erential equations is solved by a sti solver in this case ODE15s from the MATLAB ODE Suite, which is based on numerical di erentiation formulae of order 1 to 5 [46]. The exponential convergence of spherical harmonic approximation means that we have to solve only relatively small systems of ordinary di erential equations (typically of order 100 in size) and simulations of the special case (1.2) took times of the order of minutes on an up to date platform ....

L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comput. 18:1-22, 1997.

....nonlinear equations which are commonly solved using some form of Newton iteration. For linear multistep methods we apply the theory of Dorsselaer and Spijker [11] to justify a new stopping criterion for the Newton iteration. The ideas are tested on the Matlab ODE suite of Shampine and Reichelt [10] and are shown to lead to a significant improvement in overall efficiency for some problems. Key words. stiff problems, IVPs, ODEs, BDFs, Newton method. AMS subject classifications. primary 65L05, 65L99. 1 Introduction We consider stiff initial value problems (IVPs) of the form y 0 = f(y(t) ....

....direct LU decomposition. We adopt an entirely different approach. We intend to reduce the overall number of linear systems that has to be solved. In order to illustrate the approach we conduct numerical experiments using the Matlab ODE suite code written by Shampine and Reichelt known as ode15s [10]. ode15s is a stiff ordinary differential equation solver that implements a modified form of the BDFs and the modified Newton method. The code can also be used to implement the standard BDF. The linear algebraic systems are solved by LU decomposition of the iteration matrix. This matrix is formed ....

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Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Stat. Comput., 18, No. 1:1--22, January 1997. 22

....explicit terms to account for possible changes in the value of the dollar. Equation (5.2) is solved numerically through Matlab 5. 3 using a variable order stiff differential equations solver based on the Klopfenstein Shampine family of numerical differentiation formulas of orders one through five [80]. 5.4. Stage 1: The Professor Model Results. In this section we present numerical solutions of Equation (5.2) We examine three cases: 1. Maintaining the same levels (relative to inflation) of government funding over the years. 12 2. Decreasing government funding by 5 per year (relative to ....

....student population of our university to be S T = 3000; 5.8) and we assume that we initially have S(0) 500 (5.9) successful students in that population. The system of equations (5.5) and (5.7) is solved numerically through Matlab 5. 3 using a variableorder stiff differential equations solver [80]. 5.7. Stage 2: The Professor Student Model Results. In this section we present numerical solutions to the system of nonlinear equations describing the interaction between the professor and student populations, given by Equations (5.5) and (5.7) We again consider three scenarios: maintaining ....

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L. F. Shampine and M. W. Reichelt, "The MATLAB ODE Suite," SIAM J. Sci. Comp., vol. 18, no. 1, 1997. 29

....involve the method of nite di erences and the Fourier method; see ( Fornberg 1996] p. 130) Having selected an appropriate di erentiation matrix D, it remains to solve the nonlinear system of ordinary di erential equations (56) We shall use Matlab s built in function ode45.m for this purpose [Shampine and Reichelt 1997]. It is based on a Runge Kutta fourth and fth order pair, combined with Fehlberg s time step selection scheme. Table 14. Function for computing the right hand side of the sine Gordon system (56) function dw = sgrhs(t,w,flag,D) Function for computing the right hand side of the SG equation N = ....

....methods the Runge Kutta Fehlberg algorithm executed between 610 and 620 function evaluations as part of about 100 successful steps and 3 or 4 failed attempts on the interval 0 t 6 . If sti ness becomes a problem for larger N , the use of ode15s.m instead of ode45.m should be considered; see [Shampine and Reichelt 1997]. The code of Table 15 may be used to solve more challenging problems, perhaps involving moving or interacting solitons, or it may be modi ed to solve other nonlinear evolution equations. It may also serve as basis for the comparison of various numerical methods for solving PDEs such as these. ....

Shampine, L.F. and Reichelt, M.W. 1997. The matlab ode suite. SIAM J. Sci. Comput. 18, 1-22.

....done by differencing for the discrete problem, since analytic gradients are available by computation of the discrete adjoint state. This is not the case if higher order methods are used [8] If one uses variable step and variable order codes that control the local truncation error [13] 2] 1] [14] the error in will depend on the errors that come from the numerical integration of (1.3) and (1.5) Moreover, after (1.3) has been solved, the values of 1 obtained will have to be used in an interpolation during the integration of (1.3) That interpolation error will also affect the ....

L. F. SHAMPINE AND M. W. REICHELT, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1--22.

....done by differencing for the discrete problem, since analytic gradients are available by computation of the discrete adjoint state. This is not the case if higher order methods are used [8] If one uses variable step and variable order codes that control the local truncation error [13] 2] 1] [14] the error in rf will depend on the errors that come from the numerical integration of (1.3) and (1.5) Moreover, after (1.3) has been solved, the values of y obtained will have to be used in an interpolation during the integration of (1.3) That interpolation error will also affect the accuracy ....

L. F. SHAMPINE AND M. W. REICHELT, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1--22.

....BDFs are not appropriate here since they are A stable only up to order two. For higher orders, the imaginary axis is not part of their stability region. If the maximum order is not restricted by the user, numerical instabilities may occur, as shown on the right with the matlab BDF code ode15s [9]. The oscillations are artificial and completely wrong. Note that this behavior may occur with any BDF code, depending on the physical stiffness and the chosen tolerances. The reference solution on the left was computed by another stiff solver, RADAU5 [3] which has better stability properties for ....

Shampine, L.F., Reichelt, M.: The MATLAB ODE Suite, Rept. 94-6, Math. Dept., SMU Dallas, TX, 1994

....: y 0 = F (t; y) the Jacobian matrix J = F y has a wide range of eigenvalues. This introduces a huge constraint on the step size for explicit methods, and hence implicit methods such as Backward differentiation formulas (BDF) or Rosenbrock formulas are used in practice. The MATLAB ODE suite [10] implements BDF and Rosenbrock methods respectively in ode15s and ode23s. However, these methods require forming the Jacobian of the right hand side repeatedly. If the user doesn t provide a routine to compute the Jacobian matrix analytically, then these methods employ sparse finite differences to ....

L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, Users Documentation, (1995).

....we discretize the time interval (0; T ) by introducing the points 0 = t 0 t 1 : t N t = T , and replace the time derivative by means of suitable difference quotients. We tested several multi step and Runge Kutta methods and we found the modified NDF method of Shampine and Reichelt [13] to be most efficient. We modified this method in order to take advantage of the linearity of (3.12) 3.13) The sequence U (l) n = U (l) t n ) n = 0; 1; N t , constructed in this fashion, gives the approximation of the exact solution u(x; t n ) as u h (x; t n ) N h X i=1 U ....

L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Comp., 18:1--22, 1997.

....# = exp( Hm ) Hm )e 1 0 1 # : This appears favorable when the dimension m is not too small. 7. Numerical experiments. We have implemented the method (5.8) with (and without) Krylov approximations in a Matlab code exp4. The program is written in the format used in the Matlab ODE suite [28], which is available via anonymous ftp on ftp.mathworks.com in the pub mathworks toolbox matlab funfun directory. The code exp4 can be obtained from na.uni tuebingen.de in the pub codes exp4 directory. A C version of exp4 is also available from this ftp site. 7.1. A reaction diffusion equation ....

....a nonstiff to a stiff problem. The solution of the problem for ff = 2 Delta 10 Gamma3 is shown in the movie on pp. 250ff in [11] In Figs. 7.1 7. 3 we show work precision diagrams for our exponential integrator exp4, and for the explicit Runge Kutta integrator ode45 from the Matlab ODE suite [28], which is based on a fifth order method of Dormand and Prince [3] The vertical axis shows the error at the end point t = 1, the horizontal axis gives the 10 7 10 8 10 9 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 ode45 exp4 error flops Fig. 7.1. Brusselator for ....

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L.F. Shampine and M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comp. 18 (1997), 1-22.

....with 16 basis elements. 3.3 Implementation The computer code was written in MATLAB version 5.1 (The MathWorks, Inc. Natick, MA) and computations were carried out on a Sun Sparc Ultra II workstation and a personal computer with a 180 MHz Intel Pentium Pro processor. The MATLAB routine ode15s [18] was used for time stepping, which is a variable order, variable step method based on numerical differentiation formulas. The relative and absolute error tolerances were set to 1 Theta 10 Gamma6 . Since this is a variable step method, at time t the solution over the previous delay interval had ....

L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18(1):1--22, 1997.

....to get portable and high performing linear algebra, all linear algebra operations are performed using BLAS or LAPACK and module interfaces for the FORTRAN 77 libraries have been included. The implementation has been inspired from the RADAU5 code of [HW91] as well as from the MATLAB ODE suite, [SR95]. 5.1.1. The Implemented Methods. The implemented methods are the stiffly accurate MIRKs with high stage order of Table 4.6 and 4.7. Even though the B stable methods of Section 4.2 seem to work well, Ben96c] the problems about parallel error estimation has not been solved and a parallel ....

L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. Technical report, The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760, 1995.

....a phantom diffusion coefficient a y after consulting the collaborating engineers who also provided realistic Dirchlet data f and g for (1. 1) The differential equation was solved up to time T = 6 Delta 10 4 [s] with the subroutine ode15s from Matlab s ode suite, cf. Shampine and Reichelt [17]; to this end we used a semidiscrete approach with 121 piecewise linear finite elements for the discretization of the spatial domain. To study the effect of measurement errors, the data have been perturbed by additive noise of up to 1 . A crucial part of the algorithm is the numerical ....

L. F. Shampine and M. W. Reichelt, The Matlab ode suite, SIAM J. Sci. Comput., 18 (1997), pp. 1-22.

....be the fastest in many cases, we comment that this method was devised for the problems with a negative spectrum and it is not well suitable if the Jacobian has complex eigenvalues. Apart from the schemes described above, we used the MR PC approach in the framework of the Matlab stiff code ode15s [23]. This is basically an enhanced version of LSODE, so that the modifications are similar to those made for LSODE. Although for all the problems ode15s ( MRPC) was faster than LSODE ( MRPC) in terms of number of function evaluations, we cannot yet present the CPU time speedup because ode15s is a ....

Shampine L.F., Reichelt M.W., 1994, The Matlab ODE suite. Rept. 946, Math. Dept., Southern Methodist University, Dallas, TX, available from http://www.mathworks.com.

....is halted. Since we are only concerned with convergence results in real arithmetic we shall not consider this further. The crucial difference, as far as convergence properties are concerned, is that the routine uses a different embedded Runge Kutta pair, namely the Bogacki Shampine (2,3) pair [10, 9, 2]. This is designed to be operated in extrapolation mode and is a FSAL method (First Same As Last) so that while the higher order method has 3 stages, the lower order method uses the first stage of the next step and thus in reality has 4 stages. This means that the convergence results proved here ....

L.F. Shampine and M.W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Comp., 18:1--22, 1997.

....steps too long for the natural output , the values computed at each step, to result in a smooth graph formed by joining output points with straight lines, as is typical of plotting packages. Polking [35] makes this observation about the Matlab (4,5) pair. The new generation of Matlab ODE solvers [47] deals with this difficulty by evaluating a continuous extension at a number of points in the span of a step. This raises questions of how many points are needed for a smooth graph and where they should be placed. Another issue that becomes relevant in such a computing environment is the shape of ....

....we mention a few capabilities that seem to us worth adding to RKSUITE. One already mentioned is a continuous extension for the (7,8) pair. Recently one of us had to consider the needs of the dynamic system simulation package Simulink in the course of developing the codes of the Matlab ODE Suite [47]. Continuous simulation is an important application of ODE solvers that calls for capabilities not found in the popular general purpose codes. Simulation packages often provide a solver based on an explicit RK formula as the default, and they must provide the capabilities somehow. For instance, ....

L.F. Shampine and M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput. to appear.

.... software such as the computing environments MATLAB [40] and Maple [11] the CSEP project [1] and teaching packages such as that of Polking [33] and ODE Architect [31] We are still actively developing software with, e.g. the first author leading the development of the MATLAB ODE Suite [38] and the second involved in the ODE software used by the TI 85 [29] and TI92 calculators. In addition to developing software, recent publications include a monograph [35] on the practical numerical solution of IVPs and a paper [36] on the historical development of IVP software based on explicit ....

....expense of efficiency in execution time and storage. Because MATLAB is oriented towards numerical computation, it was relatively easy to modify RKF45 for the environment. The resulting code ode45 has been the main code for ODEs through the current release. Subsequently the MATLAB ODE Suite [38] of solvers was written specifically for the environment. Another tack is to use gateway software from MATLAB to standard numerical libraries, including in particular most of the ODE solvers of the NAG library. Making the popular codes for general scientific computation available in packages ....

L.F. Shampine and M.W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comp., to appear.

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L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Comput., 18(1):1--22, 1997.

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L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18(1997), 1-22.

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L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, preprint, 1995.

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L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, preprint, 1995.

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Shampine, L. F. & Reichelt, M. W. (1995). The MATLAB ODE suite, preprint, (available from ftp.mathworks.com:/pub/mathworks/toolbox/matlab/funfun.)

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Shampine, L. and M. Reichelt (1997). The MATLAB ODE Suite. SIAM Journal on Scientific Computing 18(1), 1--22.

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L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18(1997), 1-22.

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L. F. Shampine, M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18, (1997), no. 1, pp. 1--22.

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L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18(1):1--22, 1997.

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L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM J. Sci. Comput., 18(1):1-- 22, 1997.

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L. F. Shampine and M. W. Reichelt. The Matlab ODE Suite. SIAM Journal on Scientific Computing, Volume 18, Number 1 (1997), pp. 1-22. 16

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Shampine, L. F., & Reichelt, M. W., (1997). The MATLAB ODE suite. SIAM Journal on Scientific Computing, 18, 1-22.

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L. F. Shampine and M. W. Reichelt. The Matlab ODE Suite. SIAM Journal on Scientific Computing, Volume 18, Number 1 (1997), pp. 1-22.

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Shampine, L. F. and Reichelt, M. W. The MATLAB ODE Suite. Technical report, 1996.

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