§30.16 Methods of Computation

Permalink:: http://dlmf.nist.gov/30.16

§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

Notes:: See Meixner and Schäfke (1954, §3.93), Volkmer (2004a).
Keywords:: spheroidal differential equation
Referenced by:: ¶ ‣ §3.10(iii)
Permalink:: http://dlmf.nist.gov/30.16.i

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

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

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

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

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

30.16.2 $\alpha _{{j,d+1}}\leq\alpha _{{j,d}},$

Symbols:: $d$ : dimension and $\alpha _{{j,k}}$ : real eigenvalues
Permalink:: http://dlmf.nist.gov/30.16.E2
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and

30.16.3 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)=\lim _{{d\to\infty}}\alpha _{{p,d}},$ $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$ .

Symbols:: $\left\lfloor x\right\rfloor$ : floor of $x$ , $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : 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

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

30.16.4 $\alpha _{{p,d}}-\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)=\mathop{O\/}\nolimits\!\left(\frac{\gamma^{{4d}}}{4^{{2d+1}}((m+2d-1)!(m+2d+1)!)^{2}}\right),$ $d\to\infty$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $!$ : $n!$ : factorial, $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $d$ : dimension, $\alpha _{{j,k}}$ : real eigenvalues and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.16.E4
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¶ Example

For $m=2$ , $n=4$ , $\gamma^{2}=10$ ,

30.16.5

$\alpha _{{2,2}}=14.18833\; 246,$

$\alpha _{{2,3}}=13.98002\; 0 13,$

$\alpha _{{2,4}}=13.97907\; 459,$

$\alpha _{{2,5}}=13.97907\; 345,$

$\alpha _{{2,6}}=13.97907\; 345,$

Symbols:: $\alpha _{{j,k}}$ : real eigenvalues
Permalink:: http://dlmf.nist.gov/30.16.E5
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which yields $\mathop{\lambda^{{2}}_{{4}}\/}\nolimits\!\left(10\right)=13.97907\; 345$ . If $n-m$ is odd, then (30.16.1) is replaced by

30.16.6

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

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

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

Symbols:: $m$ : 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
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§30.16(ii) Spheroidal Wave Functions of the First Kind

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

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

If $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ is known, then we can compute $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ (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 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ is known, then $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ 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 $n-m$ is even, or by (30.16.6) if $n-m$ 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 $\sum _{{j=1}}^{d}e_{{j,d}}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}=\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$

Symbols:: $!$ : $n!$ : factorial, $m$ : nonnegative integer, $n\geq m$ : integer degree and $d$ : dimension
Permalink:: http://dlmf.nist.gov/30.16.E7
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Then

30.16.8 $a_{{n,k}}^{m}(\gamma^{2})=\lim _{{d\to\infty}}e_{{k+p,d}},$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $d$ : dimension, $\gamma^{2}$ : real parameter and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
Permalink:: http://dlmf.nist.gov/30.16.E8
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30.16.9 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)=\lim _{{d\to\infty}}\sum _{{j=1}}^{d}(-1)^{{j-p}}e_{{j,d}}\mathop{\mathsf{P}^{{m}}_{{n+2(j-p)}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $d$ : dimension and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.16.E9
Encodings:: TeX, pMML, png

For error estimates see Volkmer (2004a).

§30.16(iii) Radial Spheroidal Wave Functions

Keywords:: radial spheroidal wave functions
Referenced by:: Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/30.16.iii
Amendment (effective with 1.0.1):: The citation to Van Buren and Boisvert (2004) was added.
Reported 2010-05-31 by Lee Van Buren

The coefficients $a_{{n,k}}^{m}(\gamma^{2})$ calculated in §30.16(ii) can be used to compute $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ , $j=1,2,3,4$ 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).

§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