Approximate evaluation of functional integrals generated by the relativistic Hamiltonian

An approximate evaluation of matrix-valued functional integrals generated by the relativistic Hamiltonian is considered. The method of evaluation of functional integrals is based on the expansion in the eigenfunctions of Hamiltonian generating the functional integral. To find the eigenfunctions and the eigenvalues the initial Hamiltonian is considered as a sum of the unperturbed operator and a small correction to it, and the perturbation theory is used. The eigenvalues and the eigenfunctions of the unperturbed operator are found using the Sturm sequence method and the reverse iteration method. This approach allows one to significantly reduce the computation time and the used computer memory compared to the other known methods.

A Modification of the Fast Inverse Square Root Algorithm

We present a new algorithm for the approximate evaluation of the inverse square root for single-precision floating-point numbers. This is a modification of the famous fast inverse square root code. We use the same “magic constant” to compute the seed solution, but then, we apply Newton–Raphson corrections with modified coefficients. As compared to the original fast inverse square root code, the new algorithm is two-times more accurate in the case of one Newton–Raphson correction and almost seven-times more accurate in the case of two corrections. We discuss relative errors within our analytical approach and perform numerical tests of our algorithm for all numbers of the type float.

Approximate evaluation of functional integrals with centrifugal potential

Approximate evaluation of functional integrals containing a centrifugal potential is considered. By a centrifugal potential is understood a potential arising from a centrifugal force. A combination of the method based on expanding into a series of the eigenfunctions of a Hamiltonian generating a functional integral and the Sturm sequence method for the eigenvalue problem is used for approximate evaluation of functional integrals. This combination allows one to significantly reduce a computation time and a used computer memory volume in comparison to other known methods.