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Some Analytical and Numerical Consequences of Sturm Theorems

The Sturm comparison theorems for second order ODEs are classical results from which information on the properties of the zeros of special functions can be obtained. Sturm separation and comparison theorems are also available for difference-differential systems under oscillatory conditions. The separation theorem provides interlacing information for zeros of some special functions and the comparison theorem gives bounds on the distance between these interlacing zeros. For monotonic systems Sturm theorems for the zeros do not exist because there is one zero at most. Instead, bounds on certain function ratios can be obtained using information on the coefficients of the system, and particularly monotonicity properties. Similar ideas that can be used to prove Sturm theorems can be considered for obtaining this type of bounds; the qualitative analysis of associated Riccati equations is a key ingredient in both cases. We review some applications for modified Bessel, parabolic cylinder and Laguerre functions and we also present related results for incomplete gamma functions. Sturm theorems, both for second order ODEs and first order DDEs can be applied for the computation of the real zeros of special functions. Recently a fourth order method based on the Sturm comparison theorem for computing the real zeros of solutions of second order ODEs was developed. We discuss the connection of this method with the Sturm theorem and we explain how this has been extended to the computation of complex zeros.