# §30.16 Methods of Computation

## §30.16(i) Eigenvalues

For small we can use the power-series expansion (30.3.8). Schäfke and Groh (1962) gives corresponding error bounds. If is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93).

Another method is as follows. Let be even. For sufficiently large, construct the tridiagonal matrix with nonzero elements

and real eigenvalues , , , , arranged in ascending order of magnitude. Then

30.16.2

and

The eigenvalues of can be computed by methods indicated in §§3.2(vi), 3.2(vii). The error satisfies

### ¶ Example

For , , ,

30.16.5

which yields . If is odd, then (30.16.1) is replaced by

## §30.16(ii) Spheroidal Wave Functions of the First Kind

If is large, then we can use the asymptotic expansions referred to in §30.9 to approximate .

If is known, then we can compute (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions , if is even, or , if is odd.

If is known, then can be found by summing (30.8.1). The coefficients are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).

A fourth method, based on the expansion (30.8.1), is as follows. Let be the matrix given by (30.16.1) if is even, or by (30.16.6) if is odd. Form the eigenvector of associated with the eigenvalue , , normalized according to

Then

For error estimates see Volkmer (2004a).

## §30.16(iii) Radial Spheroidal Wave Functions

The coefficients calculated in §30.16(ii) can be used to compute , from (30.11.3) as well as the connection coefficients from (30.11.10) and (30.11.11).

For other methods see Van Buren and Boisvert (2002, 2004).