30 Spheroidal Wave Functions Computation 30.15 Signal Analysis 30.17 Tables

§30.16 Methods of Computation

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Permalink:: http://dlmf.nist.gov/30.16
See also:: Annotations for Ch.30

§30.16(i) Eigenvalues
§30.16(ii) Spheroidal Wave Functions of the First Kind
§30.16(iii) Radial Spheroidal Wave Functions

§30.16(i) Eigenvalues

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Keywords:: computation, eigenvalues, spheroidal differential equation
Notes:: See Meixner and Schäfke (1954, §3.93), Volkmer (2004a).
Referenced by:: §3.10(iii)
Permalink:: http://dlmf.nist.gov/30.16.i
See also:: Annotations for §30.16 and Ch.30

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 be even. For sufficiently large, construct the $d\times d$ tridiagonal matrix $\mathbf{A}=[A_{j,k}]$ with nonzero elements

30.16.1

$A_{j,j}=(m+2j-2)(m+2j-1)-2\gamma^{2}\frac{(m+2j-2)(m+2j-1)-1+m^{2}}{(2m+4j-5)(% 2m+4j-1)},$

$A_{j,j+1}=-\gamma^{2}\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)},$

$A_{j,j-1}=-\gamma^{2}\frac{(2j-3)(2j-2)}{(2m+4j-7)(2m+4j-5)},$

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Symbols:: : 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 $\alpha_{1,d}$ , $\alpha_{2,d}$ , $\dots$ , $\alpha_{d,d}$ , arranged in ascending order of magnitude. Then

30.16.2 $\alpha_{j,d+1}\leq\alpha_{j,d},$

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Symbols:: : 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 $\lambda^{m}_{n}\left(\gamma^{2}\right)=\lim_{d\to\infty}\alpha_{p,d},$ $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$ .

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, : nonnegative integer, $n\geq m$ : integer degree, : 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 $\alpha_{p,d}-\lambda^{m}_{n}\left(\gamma^{2}\right)=O\left(\frac{\gamma^{4d}}{% 4^{2d+1}((m+2d-1)!(m+2d+1)!)^{2}}\right),$ $d\to\infty$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : factorial (as in ), $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, : nonnegative integer, $n\geq m$ : integer degree, : 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

Example

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See also:: Annotations for §30.16(i), §30.16 and Ch.30

For , , $\gamma^{2}=10$ ,

30.16.5

$\alpha_{2,2}=14.18833\;246,$

$\alpha_{2,3}=13.98002\;013,$

$\alpha_{2,4}=13.97907\;459,$

$\alpha_{2,5}=13.97907\;345,$

$\alpha_{2,6}=13.97907\;345,$

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Symbols:: $\alpha_{j,k}$ : real eigenvalues
Permalink:: http://dlmf.nist.gov/30.16.E5
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png
See also:: Annotations for §30.16(i), §30.16(i), §30.16 and Ch.30

which yields $\lambda^{2}_{4}\left(10\right)=13.97907\;345$ . If is odd, then (30.16.1) is replaced by

30.16.6

$A_{j,j}=(m+2j-1)(m+2j)-2\gamma^{2}\*\frac{(m+2j-1)(m+2j)-1+m^{2}}{(2m+4j-3)(2m% +4j+1)},$

$A_{j,j+1}=-\gamma^{2}\frac{(2m+2j)(2m+2j+1)}{(2m+4j+1)(2m+4j+3)},$

$A_{j,j-1}=-\gamma^{2}\frac{(2j-2)(2j-1)}{(2m+4j-5)(2m+4j-3)}.$

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Symbols:: : nonnegative integer, $A_{j,k}$ : tridiagonal matrix elements and $\gamma^{2}$ : real parameter
Referenced by:: §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.16.E6
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §30.16(i), §30.16(i), §30.16 and Ch.30

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

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Keywords:: computation, spheroidal wave functions
Notes:: See Van Buren et al. (1972) and Volkmer (2004a).
Referenced by:: §30.16(iii)
Permalink:: http://dlmf.nist.gov/30.16.ii
See also:: Annotations for §30.16 and Ch.30

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

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

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

30.16.7 $\sum_{j=1}^{d}e_{j,d}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}% =\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$

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Symbols:: : factorial (as in ), : nonnegative integer, $n\geq m$ : integer degree and : 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 $a_{n,k}^{m}(\gamma^{2})=\lim_{d\to\infty}e_{k+p,d},$

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Symbols:: : nonnegative integer, $n\geq m$ : integer degree, : dimension, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : 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 $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\lim_{d\to\infty}\sum_{j=1}^{d}(-% 1)^{j-p}e_{j,d}\mathsf{P}^{m}_{n+2(j-p)}\left(x\right).$

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : nonnegative integer, $n\geq m$ : integer degree, : 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).

§30.16(iii) Radial Spheroidal Wave Functions

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Keywords:: computation, radial spheroidal wave functions
Referenced by:: Erratum (V1.0.1) for References
Permalink:: http://dlmf.nist.gov/30.16.iii
Amendment (effective with 1.0.1):: The citation to Van Buren and Boisvert (2004) was added.
Suggested 2010-05-31 by Lee Van Buren
See also:: Annotations for §30.16 and Ch.30

The coefficients $a_{n,k}^{m}(\gamma^{2})$ calculated in §30.16(ii) can be used to compute $S^{m(j)}_{n}\left(z,\gamma\right)$ , 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).

30.15 Signal Analysis 30.17 Tables

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§30.16 Methods of Computation

Contents

§30.16(i) Eigenvalues

Example

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

§30.16(iii) Radial Spheroidal Wave Functions