Smooth Numerical Solutions of Ordinary Differential Equations

  • C. W. Gear
  • Thu Vu
Part of the Progress in Scientific Computing book series (PSC, volume 2)


When the function evaluation in parameter estimation by error norm minimization or in a range of related optimization problems involves the numerical solution of differential equations, the smoothness of that solution with respect to parameter changes is critical to the behavior of the minimization code (which might be trying to estimate partial derivatives numerically). If fixed stepsize, fixed order methods are used for integration, it is easy to see that the integrands have the desired smoothness, but such methods are not always efficient or even possible. Modern automatic integrators have very poor smoothness properties-some-times not even continuous. If the solution of y′ = f(y, t, p), y(0) =y0(p) is denoted by y(t, p) where p is a parameter, and the numerical solution from a code using tolerance ε is y(t, p;ε), we hope that
$$||\;{\rm{y(t,p)}}\;{\rm{ - }}\;{\rm{y(t,p;\varepsilon )}}\;{\rm{||}}\;{\rm{ = }}\;{\rm{0(\varepsilon )}}$$
In a “smooth“ method we also want
$$||\;{{{\partial ^{\rm{S}}}{\rm{y}}\left( {{\rm{t,p}}} \right)} \over {\partial {{\rm{p}}^{\rm{S}}}}}\; - \;{{{\partial ^{\rm{S}}}{\rm{y}}\left( {{\rm{t,p;\varepsilon }}} \right)} \over {\partial {{\rm{p}}^{\rm{S}}}}}\;||\; \leqq \;0(\delta )$$
In practice, the second term is computed numerically. If the integrator is not smooth, we are forced to use ε= 0(δS(Δp)S) which can be very expensive. In this paper we examine methods for which the above inequality can hold with = 5. Preliminary experiments with Runge-Kutta like codes show some promise.


Integral Curve Euler Method Automatic Integrator Minimization Code Related Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Gear, C.W., Runge-Kutta starters for multistep methods, ACM TOMS 6(3), September 1980, 263–279.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Vu, Thu V., Modified Runge-Kutta methods for solving odes, Dept.Rpt. UIUCDCS-R-81-1069, Dept. Computer Sci., Univ. Illinois, 1981.Google Scholar

Copyright information

© Birkhäuser Boston 1983

Authors and Affiliations

  • C. W. Gear
  • Thu Vu

There are no affiliations available

Personalised recommendations