Eigenvalues of differential equations by finite-difference methods

Abstract

The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue A(h) of the corresponding finite difference equations towards the eigenvalue A of the differential equation itself and it is shown that where h is the size of the interval of the grid covering the range of the independent variable; the constant v is usually a negative number and consequently A(h) may well be a lower bound to A. This convergence property is used in the numerical calculation of A by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

Improved formulas for complete and partial summation of certain series.

“…which is correct to around a unit in the 8th decimal. But use of the present tables for (1/j )-extrapolation in (6) or (7), for n = 20, yields the highly accurate Sw = 3.12868 93008 04617 359 correct to about 2 units in the 17th decimal, showing a gain of around 9 places. B.…”

“…For the lowest values of ra, there will be considerable deviation between the true value of the root z2m+\ and the function úm+i which is (ra + m)ir + an exact odd polynomial in l/(ra + m), making Sn+m an exact even polynomial in l/(ra + m). But at the inconvenience of having to compute S"+m for the initial values of ra, we may employ (6) or (7) to extrapolate for Sn+m for some larger ra to get 22m+i which will agree with the true value of the root z2Z+i to very high accuracy. Taking Suppose that the problem is to calculate the 14th zero of Jw>(z) or 25/2'.…”

“…For a similar example employing Table 4, and revealing the full accuracy of (6) or (7), we choose a modification of Sn+m , say S"+m , where (13) S"+m = (ra + m){z2Zli -(n + m)x], and where now z2m+\, instead of being the rath zero of J2m+i(x), is defined as a preassigned number of terms of the right member of (11) (for a = 0, v = 2m + $) which is the same for every ra. For the lowest values of ra, there will be considerable deviation between the true value of the root z2m+\ and the function úm+i which is (ra + m)ir + an exact odd polynomial in l/(ra + m), making Sn+m an exact even polynomial in l/(ra + m).…”

Approximate relations between series and integrals

“…However, it may be worth while to do this if the integral is required for many values of some parameter since it should be only necessary to estimate the error for two or three values of the parameter. It may also, in some cases, be possible to obtain a still better estimate by the method invented by L. F. Richardson and called by him the 'deferred approach to the limit' (for some interesting examples of this see Salzer [12] and Bolton and Scoins [2]). We calculate the sum S for a series of values of h (Ao, 2ho, • • • , nho say) and use simple polynomial extrapolation to obtain an estimate for h = 0.…”

“…If fix) tends to zero sufficiently rapidly at x = ± «> and is reasonably continuous we may expect (2) to give a good approximation to (1). In practice it is often found that (2) is a much closer approximation to (1) than would have been guessed simply by looking at the graph of fix).…”