Methods tuned on the physical problem. A way to improve numerical codes

We consider the problem on how the numerical methods tuned on the physical problem can contribute to the enhancement of the performance of the codes. We illustrate this on two simple cases: solution of time independent one-dimensional Schroedinger equation, and the computation of integrals with oscillatory integrands. In both cases the tuned versions bring a massive gain in accuracy at negligible extra cost. We presented two simple problems where successive levels of tuning enhance significantly the accuracy at negligible extra cost. These problems should be seen as representing only some illustrations on how the codes can be improved but we must also mention that in many cases tuned versions still have to be developed. Just for a suggestion, quadrature formulae which involve the integrand and a number of successive derivatives of this exist, but no formula is available when some of these derivatives are missing, for example when we dispose of y and y'' but not of y'. A direct application will be on the case when the integrand involves the solution of the Schrodinger equation by the method of Numerov. (author)