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AbstractAbstract

[en] A new method for approximating magnetostatic field problems is given in this paper. The new method approximates the scalar potential for the magnetic intensity and is based on a volume integral formulation. The derivation of the new computational method uses the spectral properties of the relevant integral operator. The corresponding algorithm is similar to that obtained from coupled differential and boundary integral approaches. Convergence and stability theorems are proven. Finally, convergence results in actual computations are compared with results for the usual volume integral method used in GFUN3D

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Mathematics of Computation; ISSN 0025-5718; ; v. 43(168); p. 433-446

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[en] The proposed one-point method for finding the limit of a slowly converging linear sequence features an Anderson-Bjoerck extrapolation step that had previously been applied to the Regula Falsi problem. Converegence is of order 1.839 as compared to √2 for the well-known Aitken-Steffensen delta

^{2}-process, and to 1.618 for another one-point extrapolation procedure of King. There are examples for computing for computing a polynomial's multiple root with Newton's method and for finding a fixed point of a nonlinear functionPrimary Subject

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Mathematics of Computation; ISSN 0025-5718; ; v. 41(164); p. 591-783

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[en] The first forty-one coefficients of a continued fraction for 1n GAMMA(z)+z-(z-1/2) 1n z-1n√2π, are given. The computation, based on Wall's algorithm for converting a function's power series representation to a continued fraction representation, was run on the algebraic manipulation system MACSYMA

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Mathematics of Computation; ISSN 0025-5718; ; v. 34(154); p. 547-551

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[en] Roots of the three transcendental equations j/sub l/(αlambda) y/sub l/(lambda) = j/sub l/(lambda) y/sub l/(αlambda), [xj/sub l/(x)]'/sub x/=αeta[xy/sub l/(x)]'/sub x/=eta =xj/sub l/(x)]'/sub x/=eta[xy/sub l/(x)]'/sub x/=αeta, and [j/sub l/(x)]'/sub x/=αμ[y/sub l/(x)]'/sub x/=μ =j/sub l/(x)]'/sub x/=μ[y/sub l/(x)]'/sub x/=αμ that are degenerate for certain values of the parameter αelement of (0,1) are presented. The symbols j/sub l/ and y/sub l/ denote the spherical Bessel functions of the first and second kind. Root degeneracies are discussed for each equation individually as well as for pairs of equations. Only positive roots are considered, sincelthe equations are invariant under the transformations lambda→-lambda, eta→-eta, and μ→-μ. When l=0, only the third equation has nontrivial roots. These roots are identical with the roots of the first equation for l=1, i.e.μ/sub 0n/=lambda/sub 1n/ (n=1,2,...). Various graphs of lambda/sub l/n, eta/sub l/n, and μ/sub l/n display root-degeneracies as intersections of curves. Accurate values of degenerate roots with the corresponding values of α are exhibited in tables.Roots of the third equation for l=1(1)30, α=0.1(0.1)0.7, together with their minima and associated values of the parameter α, are given in the microfiche supplement accompanying this issue. Roots of the first and the second equation and minima of roots of the second equation are published in v. 31 and v. 32 of this journal.The roots of the first two equations determine the eigenfrequencies of the transverse electric and the transverse magnetic normal modes of an ideal cavity resonator bounded by two concentric spheres (r=αR and r=R). The roots of the third equation determine the frequencies of the irrotational magnetic eigenfields

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Mathematics of Computation; ISSN 0025-5718; ; v. 33(147); p. 1041-1048

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[en] A Monte Carlo technique for Knudsen flow, light transmission in tubes, the flow of cold neutrons, phonon transmission, heat transfer calculations and other problems involving free streaming in cylindrical enclosures is described. The coefficients of a functional expansion of the view factors are estimated by Monte Carlo, are then perturbed in a least squares minimum sense to satisfy some consistency conditions, and finally are used to calculate the collision density on the boundary and the transmission of the enclosure. Without importance sampling or other variance reduction techniques, the calculational work required for a given accuracy is reduced by a factor frequently near 10 when compared to a simple calculation in which particles are followed from entrance to exit

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Mathematics of Computation; ISSN 0025-5718; ; v. 33(146); p. 765-777

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[en] We modify a Galerkin method for nonlinear hyperbolic equations so that it becomes a simpler method of lines, which may be viewed as a collocation method. The high order of accuracy is preserved. We present a linear wave analysis of the scheme and discuss some aspects of nonlinear problems. Our numerical experiments indicate that the addition of a proper artificial viscosity makes the method competitive and the common difference schemes, even when the solution has discontinuities

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Mathematics of Computation; ISSN 0025-5718; ; v. 33(146); p. 647-658

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[en] A special concept of stiffness is appropriate for implict A-stable formulas. It is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ODE. Using this development, a variety of effective ODE solvers could be made insensitive to the type of problem, i.e. the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

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Mathematics of Computation; ISSN 0025-5718; ; v. 36(154); p. 499-523

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[en] The method of artificial viscosity was originally designed by von Neumann and Richtmyer for calculating the propagation of waves in materials that were hydrodynamic and rate-independent (e.g., ideal gas law). However, hydrocodes (such as WONDY) based on this method continue to expand their repertoire of material laws even unto material laws that are rate-dependent (e.g., Maxwell's material law). Restrictions on the timestep required for stability with material laws that are rate-dependent can be considerably more severe than restrictions of the Courant-Friedrichs-Lewy (CFL) type that are imposed in these hydrocodes. These very small timesteps can make computations very expensive. An alternative is to go ahead and integrate the conservation laws with the usual CFL timestep while subcycling (integrating with a smaller timestep) the integration of the stress-rate equation. If the subcycling is done with a large enough number of subcycles (i.e., with a small enough subcycle timestep), then the calculation is stable. Specifically, the number of subcycles must be one greater than the ratio of the CFL timestep to the relaxation time of the material

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Mathematics of Computation; ISSN 0025-5718; ; v. 36(155); p. 69-78

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[en] Iterative methods for solving linear systems arising from the discretization of elliptic/parabolic partial differential equations require the use of preconditioners to gain increased rates of convergence. Preconditioners arising from incomplete factorizations have been shown to be very effective. However, the recursiveness of these methods can offset these gains somewhat on a vector processor. In this paper, an incomplete factorization based on block cyclic reduction is developed. It is shown that under block diagonal dominance conditions the off-diagonal terms decay quadratically, yielding more effective algorithms

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Mathematics of Computation; ISSN 0025-5718; ; v. 42(166); p. 549-565

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[en] This paper describes a technique for solving the large sparse symmetric linear systems that arise from the application of finite element methods. The technique combines an incomplete factorization method called the shifted incomplete Cholesky factorization with the method of generalized conjugate gradients. The shifted incomplete Cholesky factorization produces a splitting of the matrix A that is dependent upon a parameter α. It is shown that if A is positive definite, then there is some α for which this splitting is possible and that this splitting is at least as good as the Jacobi splitting. The method is shown to be more efficient on a set of test problems than either direct methods or explicit iteration schemes

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Mathematics of Computation; ISSN 0025-5718; ; v. 34(154); p. 473-497

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