§28.34 Methods of Computation

§28.34(i) Characteristic Exponents

Methods available for computing the values of $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ needed in (28.2.16) include:

• (a)

Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

• (b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

§28.34(ii) Eigenvalues

Methods for computing the eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$, and $\lambda_{\nu}\left(q\right)$, defined in §§28.2(v) and 28.12(i), include:

• (a)

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

• (b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

• (c)

Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

• (d)

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

• (e)

Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

• (f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

§28.34(iii) Floquet Solutions

• (a)

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

• (b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)28.8(iv)).

Also, once the eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$, and $\lambda_{\nu}\left(q\right)$ have been computed the following methods are applicable:

• (c)

Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

• (d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

§28.34(iv) Modified Mathieu Functions

For the modified functions we have:

• (a)

Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

• (b)

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

• (c)

Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.