§30.16 Methods of Computation

For small $|{\gamma }^2|$ we can use the power-series expansion (30.3.8). Schäfke and Groh (1962) gives corresponding error bounds. If $|{\gamma }^2|$ 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 $n-m$ be even. For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $\mathbf{A}=[A_{j,k}]$ with nonzero elements

30.16.1		$A_{j,j}$	$=(m+2j-2)(m+2j-1)-2{\gamma }^2{\displaystyle \frac{(m+2j-2)(m+2j-1)-1+m^2}{(2m+4j-5)(2m+4j-1)}},$
		$A_{j,j+1}$	$=-{\gamma }^2{\displaystyle \frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)}},$
		$A_{j,j-1}$	$=-{\gamma }^2{\displaystyle \frac{(2j-3)(2j-2)}{(2m+4j-7)(2m+4j-5)}},$
ⓘ Symbols: $m$: nonnegative integer, $A_{j,k}$: tridiagonal matrix elements and ${\gamma }^2$: real parameter Referenced by: §30.16(i), §30.16(ii) Permalink: http://dlmf.nist.gov/30.16.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §30.16(i), §30.16 and Ch.30

and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. Then

30.16.2		$${\alpha }_{j,d+1}\le {\alpha }_{j,d},$$
ⓘ Symbols: $d$: dimension and ${\alpha }_{j,k}$: real eigenvalues Permalink: http://dlmf.nist.gov/30.16.E2 Encodings: TeX, pMML, png See also: Annotations for §30.16(i), §30.16 and Ch.30

and

30.16.3		$${\lambda }_n^m\left({\gamma }^2\right)=\underset{d\to \mathrm{\infty }}{lim}{\alpha }_{p,d},$$
$p=\lfloor \frac{1}{2}(n-m)\rfloor +1$.
ⓘ Symbols: $\lfloor x\rfloor$: floor of $x$, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.16.E3 Encodings: TeX, pMML, png See also: Annotations for §30.16(i), §30.16 and Ch.30

The eigenvalues of $\mathbf{A}$ can be computed by methods indicated in §§3.2(vi), 3.2(vii). The error satisfies

30.16.4		$${\alpha }_{p,d}-{\lambda }_n^m\left({\gamma }^2\right)=O\left(\frac{{\gamma }^{4d}}{4^{2d+1}{((m+2d-1)!(m+2d+1)!)}^2}\right),$$
$d\to \mathrm{\infty }$.
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.16.E4 Encodings: TeX, pMML, png See also: Annotations for §30.16(i), §30.16 and Ch.30

If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$.

If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd.

If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$ can be found by summing (30.8.1). The coefficients $a_{n,r}^m({\gamma }^2)$ 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 $\mathbf{A}$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $\mathbf{A}$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to

30.16.7		$$\sum _{j=1}^d{e}_{j,d}^2\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}=\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$$
ⓘ Symbols: $!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree and $d$: dimension Permalink: http://dlmf.nist.gov/30.16.E7 Encodings: TeX, pMML, png See also: Annotations for §30.16(ii), §30.16 and Ch.30

Then

30.16.8		$$a_{n,k}^m({\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}{e}_{k+p,d},$$
ⓘ Symbols: $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients Permalink: http://dlmf.nist.gov/30.16.E8 Encodings: TeX, pMML, png See also: Annotations for §30.16(ii), §30.16 and Ch.30

30.16.9		$${\mathsf{Ps}}_n^m(x,{\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}\sum _{j=1}^d{(-1)}^{j-p}{e}_{j,d}{\mathsf{P}}_{n+2(j-p)}^m\left(x\right).$$
ⓘ Symbols: : Ferrers function of the first kind, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.16.E9 Encodings: TeX, pMML, png See also: Annotations for §30.16(ii), §30.16 and Ch.30

For error estimates see Volkmer (2004a).

The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11).

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