Download Computational Methods for Physicists: Compendium for by Simon Sirca, Martin Horvat PDF

By Simon Sirca, Martin Horvat

This ebook is helping complicated undergraduate, graduate and postdoctoral scholars of their day-by-day paintings by means of providing them a compendium of numerical equipment. the alternative of equipment will pay major recognition to errors estimates, balance and convergence matters in addition to to the how one can optimize software execution speeds. Many examples are given in the course of the chapters, and every bankruptcy is via no less than a handful of extra finished difficulties that could be handled, for instance, on a weekly foundation in a one- or two-semester direction. In those end-of-chapter difficulties the physics history is reported, and the most textual content previous them is meant as an advent or as a later reference. much less rigidity is given to the reason of person algorithms. it's attempted to urge within the reader an personal self sufficient pondering and a specific amount of scepticism and scrutiny rather than blindly following available advertisement tools.

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If λ is complex, we rename x → x/λ; thus g(x) = (1 + x)−1 and h(λ, x) = Then for Re λ > 0, the functions gn , gn+1 and Hn+1 correspond to the second criterion of item 2 above, and we have |Rn | ≤ 2|sn |. At any rate, the remainder Rn is on the order of the first omitted term, Rn = O(sn ). The approach described above is particularly useful when h can be integrated easily, like in the case of h(λ, x) = exp(λx) when Hm (λ, x) = h(x)/λm . This is the foundation of the asymptotic expansion of Laplace and Fourier integrals λ−1 exp(−x/λ).

The extension of the Taylor series to the complex plane and inclusion of negative powers of the arguments is called the Laurent expansion: f (z) = ak (z − z0 )k , k∈Z ak = 1 2πi γ f (z) dz. (z − z0 )k+1 Here γ is an arbitrary closed contour encircling z0 in the positive sense. According to the coefficients {a−k }k∈N of the negative powers, we have three distinct cases. The function f is analytic if all these coefficients are zero; it is meromorphic if there exists a minimal N such that a−k = 0 for ∀k ≥ N ; if such N does not exist, we say that f has an essential singularity at z0 .

Left] Aitken’s method for the sum Sn = nk=0 (−1)k /(k + 1) with the limit S = limn→∞ Sn = log 2. Shown is the acceleration of this series with very slow (logarithmic) convergence by three-fold repetition of the Aitken’s method. 58) suffices. [Right] Kummer’s acceleration of the sums Sn = nk=1 1/k 2 with the limit S = limn→∞ Sn = π 2 /6 by using the auxiliary series ∞ k=1 1/(k(k + 1)) Richardson’s procedure is an example of a linear extrapolation method X, in which for partial sums Sn and Tn of two series we have X(λSn + μTn ) = λX(Sn ) + μX(Tn ).

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