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Journal ArticleDOI

Algorithm 352: characteristic values and associated solutions of Mathieu's differential equation [S22]

01 Jul 1969-Communications of The ACM (ACM)-Vol. 12, Iss: 7, pp 399-407
TL;DR: Algor i thm 352 is a package of double-precision FORTRAN rout ines which consists of the following p r imary Rout ines : MFCVAL, M A T H, and BESSEL.
About: This article is published in Communications of The ACM.The article was published on 1969-07-01. It has received 37 citations till now. The article focuses on the topics: First-order partial differential equation & Differential equation.
Citations
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Journal ArticleDOI
TL;DR: With these expressions, the algorithms for computing MCNs can handle very high orders, up to $n = 1000$ and beyond.
Abstract: A complete method and expressions for computing Mathieu characteristic numbers (MCNs) of integer orders are given. The paper reviews available algorithms and their shortcomings, and then suitable methods are selected on the basis of simplicity and accuracy. For small orders $(n < 4)$, new approximate formulas for MCNs are developed; these have much larger range than available expressions. A new normalization technique for the MCNs is proposed. With the aid of this normalization a new approximate recurrence relation for MCNs is obtained. Using the same normalization, general formulas are obtained for regional limits where the approximate and asymptotic expressions can be used to compute MCNs. With these expressions, the algorithms for computing MCNs can handle very high orders, up to $n = 1000$ and beyond.

63 citations

Journal ArticleDOI
TL;DR: In this article, the three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain.
Abstract: The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

58 citations

01 Jan 2008

36 citations


Cites background from "Algorithm 352: characteristic value..."

  • ...Clemm published several pioneering Fortran routines to evaluate the Mathieu functions and their derivatives [17]....

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Journal ArticleDOI
TL;DR: Two algorithms for calculating the eigenvalues and solutions of Mathieu's differential equation for noninteger order are described, one faster and optimized to obtain accuracy of one part in 1014, but has only been implemented for orders less than 10.5.
Abstract: Two algorithms for calculating the eigenvalues and solutions of Mathieu's differential equation for noninteger order are described. In the first algorithm, Leeb's method is generalized, expanding the Mathieu equation in Fourier series and diagonalizing the symmetric tridiagonal matrix that results. Numerical testing was used to parameterize the minimum matrix dimension that must be used to achieve accuracy in the eigenvalue of one part in 1012. This method returns a set of eigenvalues below a given order and their associated solutions simultaneously. A second algorithm is presented which uses approximations to the eigenvalues (Taylor series and asymptotic expansions) and then iteratively corrects the approximations using Newton's method until the corrections are less than a given tolerance. A backward recursion of the continued fraction expansion is used. The second algorithm is faster and is optimized to obtain accuracy of one part in 1014, but has only been implemented for orders less than 10.5.

35 citations


Cites methods from "Algorithm 352: characteristic value..."

  • ...The first integer order algorithm was that of Clemm [7]....

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  • ...Leeb [18], Clemm [7], the Numerical Analysis Group at Delft [34], Dolbeeva [11], Toyama and Shogen [36], Arscott [2], Canosa [6], and Ikebe [16], and Sui and Ding [3’1] have all presented algorithms for integer order....

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  • ...Recurrences of the form used by Clemm will converge to one eigenvalue in some regions of q and another eigenvalue in other regions of q....

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  • ...The first integer order algorithm was that of Clemm [7]....

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  • ...The eigenvalues and corresponding solutions to (1.1) with period ir or 2 n are infinite in number and can be ordered by the magnitude of a and whether the solution is even (a., n = 0,1,2 ,... )orodd(b~, n==l,2,3,... ), where n is the order [5] and indicates how many zeros the solution has for O x rr. Leeb [18], Clemm [7], the Numerical Analysis Group at Delft [34], Dolbeeva [11], Toyama and Shogen [36], Arscott [2], Canosa [6], and Ikebe [16], and Sui and Ding [3 1] have all presented algorithms for integer order....

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Journal ArticleDOI
TL;DR: In this paper, the behavior of TM wave scattering from hollow and dielectric-filled semielliptic channels in a perfectly conducting substrate is investigated, where the scattered field is represented in terms of an infinite series of Mathieu functions with unknown coefficients.
Abstract: The behavior of TM wave scattering from hollow and dielectric-filled semielliptic channels in a perfectly conducting substrate is investigated. The scattered field is represented in terms of an infinite series of Mathieu functions with unknown coefficients. By applying the separation of variables and employing the partial orthogonality of the first-kind angular Mathieu functions, the unknown coefficients are obtained. Numerical results are given for the scattered-field patterns by the channels with different eccentricities and permittivities.

28 citations


Cites methods from "Algorithm 352: characteristic value..."

  • ...The normalization factors adopted by Ince [6] are used, and Mathieu functions are computed using the algorithms in [7]....

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References
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78 citations

Journal ArticleDOI
01 Jan 1933
TL;DR: In fact, until 1924 all attempts to reduce the Mathieu equation to a form suitable for calculation had failed as discussed by the authors, and in that year a successful method of attack was discovered by the present writer, who then undertook the computation of the tables which are here published.
Abstract: Although the physical problem which originally led to the Mathieu equation is now well over sixty years old, and although other problems studied in more recent years, such as the scattering of electric waves by an elliptic cylinder, depend essentially on the same equation, no thorough investigation of detail has hitherto been possible owing to the lack of tables of the functions defined by the equation. In fact, until 1924 all attempts to reduce the functions to a form suitable for calculation had failed. In that year a successful method of attack was discovered by the present writer, who then undertook the computation of the tables which are here published.

40 citations