The timeintegration problem^{65} for ordinary differential equations (ODEs) is traditionally written as follows: We are given an integer (the number of ODEs to integrate), a “righthandside” function , and the value of a function satisfying the ODEs
We wish to know (or approximate) for some finite interval .This is a wellstudied problem in numerical analysis. See, for example, Forsythe, Malcolm, and Moler [71, Chapter 6] or Kahaner, Moler, and Nash [92, Chapter 8] for a general overview of ODE integration algorithms and codes, or Shampine and Gordon [140], Hindmarsh [84], or Brankin, Gladwell, and Shampine [38] for detailed technical accounts.
For our purposes, it suffices to note that highly accurate, efficient, and robust ODEintegration codes are widely available. In fact, there is a strong tradition in numerical analysis of free availability of such codes. Notably, Table 3 lists several freelyavailable ODE codes. As well as being of excellent numerical quality, these codes are also very easy to use, employing sophisticated adaptive algorithms to automatically adjust the step size and/or the precise integration scheme used^{66}. These codes can generally be relied upon to produce accurate results both more efficiently and more easily than a handcrafted integrator. I have used the LSODE solver in several research projects with excellent results.

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