539 research outputs found
Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm
The discrete-time Toda equation arises as a universal equation for the
relevant Hankel determinants associated with one-variable orthogonal
polynomials through the mechanism of adjacency, which amounts to the inclusion
of shifted weight functions in the orthogonality condition. In this paper we
extend this mechanism to a new class of two-variable orthogonal polynomials
where the variables are related via an elliptic curve. This leads to a `Higher
order Analogue of the Discrete-time Toda' (HADT) equation for the associated
Hankel determinants, together with its Lax pair, which is derived from the
relevant recurrence relations for the orthogonal polynomials. In a similar way
as the quotient-difference (QD) algorithm is related to the discrete-time Toda
equation, a novel quotient-quotient-difference (QQD) scheme is presented for
the HADT equation. We show that for both the HADT equation and the QQD scheme,
there exists well-posed -periodic initial value problems, for almost all
\s\in\Z^2. From the Lax-pairs we furthermore derive invariants for
corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page
Restitution du carnet n°3 de Cholesky
24 Juillet Pierre sur Haute. Mesure des Ă©lĂ©ments de dĂ©centrage. 25 juillet Installation du miroir pour Mt Pilat, sur un pilier auxiliaire, et d'une mire plate dans la verticale du miroir. La baraque est beaucoup plus loin que cela. 28 juillet Le Montellier (chĂąteau). ArrivĂ©e au pilier Ă 8h. Le miroir est orientĂ© Ă gauche et trop haut. Orientation Ă 1cm de la feuille de centrage. Le Pilat Ă©tait visible, il a disparu complĂštement. 9h- Pointage en hauteur par rapport Ă la crĂȘte en avant au S.E..
Géodésie, topographie et cartographie
La gĂ©odĂ©sie sâoccupe de la dĂ©termination mathĂ©matique de la forme de la Terre. Les observations gĂ©odĂ©siques conduisent Ă des donnĂ©es numĂ©riques : forme et dimensions de la Terre, coordonnĂ©es gĂ©ographiques des points, altitudes, dĂ©viations de la verticale, longueurs dâarcs de mĂ©ridiens et de parallĂšles, etc. La topographie est la sĆur de la gĂ©odĂ©sie. Elle sâintĂ©resse aux mĂȘmes quantitĂ©s, mais Ă une plus petite Ă©chelle, et elle rentre dans des dĂ©tails de plus en plus fins pour Ă©tablir des carte..
Breakdowns in the implementation of the LĂĄnczos method for solving linear systems
AbstractThe LĂĄnczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the LĂĄnczos method which came out from them, the only reliable algorithm is LĂĄnczos/Orthodir which can only suffer from true breakdowns. It is shown how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of near-breakdown is treated. The same treatment applies to other methods related to LĂĄnczos'
A solver combining reduced basis and convergence acceleration with applications to non-linear elasticity
International audienceAn iterative solver is proposed to solve the family of linear equations arising from the numerical computation of nonâlinear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two subâsteps. In the first subâstep, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second subâstep, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a wellâknown method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to nonâlinear solvers with few matrix factorizations and few iterations
Acceleration of generalized hypergeometric functions through precise remainder asymptotics
We express the asymptotics of the remainders of the partial sums {s_n} of the
generalized hypergeometric function q+1_F_q through an inverse power series z^n
n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k}
may be recursively computed to any desired order from the hypergeometric
parameters and argument. From this we derive a new series acceleration
technique that can be applied to any such function, even with complex
parameters and at the branch point z=1. For moderate parameters (up to
approximately ten) a C implementation at fixed precision is very effective at
computing these functions; for larger parameters an implementation in higher
than machine precision would be needed. Even for larger parameters, however,
our C implementation is able to correctly determine whether or not it has
converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added
several references, added comparison to other methods, and added discussion
of recursion stabilit
Vector Meson Dominance as a first step in a systematic approximation: the pion vector form factor
Pade Approximants can be used to go beyond Vector Meson Dominance in a
systematic approximation. We illustrate this fact with the case of the pion
vector form factor and extract values for the first two coefficients of its
Taylor expansion. Pade Approximants are shown to be a useful and simple tool
for incorporating high-energy information, allowing an improved determination
of these Taylor coefficients.Comment: 13 pages, 7 figure
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. I. Algebraic Theory
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962
- 968 (2003)] introduced in connection with the summation of the divergent
perturbation expansion of the hydrogen atom in an external magnetic field a new
sequence transformation which uses as input data not only the elements of a
sequence of partial sums, but also explicit estimates
for the truncation errors. The explicit
incorporation of the information contained in the truncation error estimates
makes this and related transformations potentially much more powerful than for
instance Pad\'{e} approximants. Special cases of the new transformation are
sequence transformations introduced by Levin [Int. J. Comput. Math. B
\textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189
- 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and
also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A
\textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations
- explicit expressions, recurrence formulas, explicit expressions in the case
of special remainder estimates, and asymptotic order estimates satisfied by
rational approximants to power series - is formulated in terms of hitherto
unknown mathematical properties of the new transformation introduced by
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. This leads to a considerable
formal simplification and unification.Comment: 41 + ii pages, LaTeX2e, 0 figures. Submitted to Journal of
Mathematical Physic
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