Gladwell, I., Shampine, L. F. and Brankin, R. W. "Automatic selection of the initial step size for an ODE solver." J. Comp. and Applied Maths 18 pp175-192 (1987).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of suitable orders and stepsizes is non trivial and has been the subject of much research. Given suitable methods to compute and advance the integration formulae, it can make the difference between a good and a bad code. A number of different approaches have been suggested for this problem [1, 2, 4, 5, 8, 9, 10, 11]. This work concentrates on and extends two of these: one implemented by Shampine in his code RDEAM [11] and a second, simpler, version based on the difference between a predictor and corrector formula. Communicated by Prof. C. T. H. Baker, this document originally appeared as an IWR preprint ....

....Numerical schemes should acknowledge this. We hope to present a more detailed examination of this subject in a later paper. Recent research suggests that startup strategies may be one of the key factors Consider as starting order sequence of k = 1; 2; 3; 14 in efficient code design [5]. Other workers [2, 9] are active also in this field. 12 Relation to other methods It should be noted that this work differs fundamentally from more conventional schemes using trial steps or Runge Kutta formulae to estimate appropriate stepsizes. Instead of making potentially expensive function ....

Gladwell, I., Shampine, L. F. and Brankin, R. W. "Automatic selection of the initial step size for an ODE solver." J. Comp. and Applied Maths 18 pp175-192 (1987).

....to emphasize the fact that the code internally works with the squares of norms most of the time. If there were a direct estimate e i for the error in y i (which it turns out there isn t) we would be interested in keeping jjejj somewhat less than 1. 4. GETTING THE STARTING STEPSIZE I recommend [Gladwell et al. 1987] and [Watts 1983] as starting points for information on what has been done by others. My biases have led to different choices than made in these papers. Most important, if one changes the problem by replacing y with ffy, t with fit, changes the i by a factor of ff, and changes all t output points ....

Gladwell, I., Shampine, L. F., and Brankin, R. W. 1987. Automatic selection of the initial step size for an ODE solver. Journal of Computational and Applied Mathematics 18, 175--192.

....(see section 7.6 in [24] Users usually don t wish to be involved in the choice of an initial stepsize, and often they are not able to provide a good value, so that an automatic selection of the initial stepsize is necessary. Some bibliography is available about this matter, to which we refer [16][24] 25] 26] 31] in which automatic selection of the initial stepsize are described. The algorithms used to compute the initial stepsize, if the user doesn t supply any, can be of two kind. In some codes, a fixed fraction of the length interval, like 50 or 1 , is used as initial stepsize. In ....

Gladwell I., Shampine L.F., Brankin R.W., Automatic selection of the initial stepsize of an ODE solver, J.Comp.Appl.Math. (1987) 18, 175--192.

....of suitable orders and stepsizes is non trivial and has been the subject of much research. Given suitable methods to compute and advance the integration formulae, it can make the difference between a good and a bad code. A number of different approaches have been suggested for this problem [1, 2, 4, 5, 8, 9, 10, 11]. This work concentrates on and extends two of these: one implemented by Shampine in his code RDEAM [11] and a second, simpler, version based on the difference between a predictor and corrector formula. Communicated by Prof. C. T. H. Baker, this document originally appeared as an IWR preprint ....

....fail. Numerical schemes should acknowledge this. We hope to present a more detailed examination of this subject in a later paper. Recent research suggests that startup strategies may be one of the key factors 8 Consider as starting order sequence of k = 1; 2; 3; in efficient code design [5]. Other workers [2, 9] are active also in this field. 12 Relation to other methods It should be noted that this work differs fundamentally from more conventional schemes using trial steps or Runge Kutta formulae to estimate appropriate stepsizes. Instead of making potentially expensive function ....

Gladwell, I., Shampine, L. F. and Brankin, R. W. "Automatic selection of the initial step size for an ODE solver." J. Comp. and Applied Maths 18 pp175-192 (1987).

....of the pair, and possibly the scale of the problem. Another scheme is to use f(x; y) to estimate the derivative terms in the local error estimate. This has the advantage over the first scheme of using more information about the problem. A more elaborate, three phase scheme is proposed in [32]. 9.1.2 After an accepted step A step is accepted if jj est jj TOL. In this case, we have the possibility of increasing h for the next step in an attempt to improve efficiency. The new stepsize is often calculated using the formula fi TOL est 1=p h old (9.3) where h old is the ....

I. Gladwell, L. Shampine, and R. Brankin, Automatic selection of the initial stepsize for an ODE solver, J. of Computational and Applied Maths., 18 (1987), pp. 175--192.

....first step. In the usual fashion this estimate is used to adjust the step size. However, rather than go on to the next step, the step is repeated until an on scale step size is found. Only then does the integration proper begin. This starting step size algorithm is essentially that proposed in [19]. It combines the best of corresponding procedures in RKF45 and the NAG RK Merson code, D02PAF. Providing a good guess for the initial step size when using a variable order Adams or BDF code is especially difficult for users because the codes start with a surprisingly low order, typically one. ....

I. Gladwell, L.F. Shampine and R.W. Brankin, Automatic selection of the initial step size for an ODE solver J. Comp. Appl. Math. 18 (1987) pp. 175-192.

....values of the derivatives of the dependent variables may be unknown. On request, the integrator will attempt to calculate these values before starting the integration proper, see [22] for details. ii) In all the integrators a special starting step size algorithm similar to that described in [2] is used. This is designed to permit the integrator to get on scale immediately. The earlier NAG stiff solvers used a primitive version of this algorithm [1] but many other stiff solvers do not insist on starting on scale . iii) When calling any of the integrators associated with the sparse ....

Gladwell, I., Shampine, L.F. and Brankin, R.W. (1987) Automatic Selection of the Initial Stepsize for an ODE solver, J. Comput. Appl. Math., 18, 175--192.

....is input, a starting step size is computed automatically. An initial stepsize should be supplied only if an on scale value is known from a previous integration of a similar problem. When the code computes an on scale step size internally, it uses a modified version of the procedure described in [7]. A user supplied step size is checked by this procedure too and modified if necessary; that is, the user cannot control the step size by specifying it through the setup routine. Finally, a work array WORK(1:LENWRK) must be supplied. The setup routine checks that this array is long enough for ....

I. Gladwell, L.F. Shampine and R.W. Brankin,"Automatic Selection of the Initial Step Size for an ODE Solver", J. Comp. Appl. Math., 18 (1987) 175-192.

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