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Analysing and bounding numerical error in spiking neural network simulations
This study explores how numerical error occurs in simulations of spiking neural network models, and also how this error propagates through the simulation, changing its observed behaviour. The issue of non-reproducibility in parallel spiking neural network simulations is illustrated, and a method to bound all possible trajectories is discussed. The base method used in this study is known as mixed interval and affine arithmetic (mixed IA/AA), but some extra modifications are made to improve the tightness of the error bounds.
I introduce Arpra, my new software, which is an arbitrary precision range analysis library, based on the GNU MPFR library. It improves on other implementations by enabling computations in custom floating-point precisions, and reduces the overhead rounding error of mixed IA/AA by computing in extended precision internally. It also implements a new error trimming technique, which reduces the error term whilst preserving correct boundaries. Arpra also implements deviation term condensing functions, which can reduce the number of floating-point operations per function significantly. Arpra is tested by simulating the Hénon map dynamical system, and found to produce tighter ranges than those of INTLAB, an alternative mixed IA/AA implementation.
Arpra is used to bound the trajectories of fan-in spiking neural network simulations. Despite performing better than interval arithmetic, the mixed IA/AA method used by Arpra is shown to be inadequate for bounding the simulation trajectories, due to the highly nonlinear nature of spiking neural networks. A stability analysis of the neural network model is performed, and it is found that error boundaries are moderately tight in non-spiking regions of state space, where linear dynamics dominate, but error boundaries explode in spiking regions of state space, where nonlinear dynamics dominate
Transcendental equations satisfied by the individual zeros of Riemann , Dirichlet and modular -functions
We consider the non-trivial zeros of the Riemann -function and two
classes of -functions; Dirichlet -functions and those based on level one
modular forms. We show that there are an infinite number of zeros on the
critical line in one-to-one correspondence with the zeros of the cosine
function, and thus enumerated by an integer . From this it follows that the
ordinate of the -th zero satisfies a transcendental equation that depends
only on . Under weak assumptions, we show that the number of solutions of
this equation already saturates the counting formula on the entire critical
strip. We compute numerical solutions of these transcendental equations and
also its asymptotic limit of large ordinate. The starting point is an explicit
formula, yielding an approximate solution for the ordinates of the zeros in
terms of the Lambert -function. Our approach is a novel and simple method,
that takes into account , to numerically compute non-trivial zeros of
-functions. The method is surprisingly accurate, fast and easy to implement.
Employing these numerical solutions, in particular for the -function, we
verify that the leading order asymptotic expansion is accurate enough to
numerically support Montgomery's and Odlyzko's pair correlation conjectures,
and also to reconstruct the prime number counting function. Furthermore, the
numerical solutions of the exact transcendental equation can determine the
ordinates of the zeros to any desired accuracy. We also study in detail
Dirichlet -functions and the -function for the modular form based on the
Ramanujan -function, which is closely related to the bosonic string
partition function.Comment: Matches the version to appear in Communications in Number Theory and
Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and
arXiv:1307.8395 [math.NT
On moduli of rings and quadrilaterals: algorithms and experiments
Moduli of rings and quadrilaterals are frequently applied in geometric
function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are
known only in a few special cases. Previously, the class of planar domains with
polygonal boundary has been studied by many authors from the point of view of
numerical computation. We present here a new -FEM algorithm for the
computation of moduli of rings and quadrilaterals and compare its accuracy and
performance with previously known methods such as the Schwarz-Christoffel
Toolbox of Driscoll and Trefethen. We also demonstrate that the -FEM
algorithm applies to the case of non-polygonal boundary and report results with
concrete error bounds
Certifying floating-point implementations using Gappa
High confidence in floating-point programs requires proving numerical
properties of final and intermediate values. One may need to guarantee that a
value stays within some range, or that the error relative to some ideal value
is well bounded. Such work may require several lines of proof for each line of
code, and will usually be broken by the smallest change to the code (e.g. for
maintenance or optimization purpose). Certifying these programs by hand is
therefore very tedious and error-prone. This article discusses the use of the
Gappa proof assistant in this context. Gappa has two main advantages over
previous approaches: Its input format is very close to the actual C code to
validate, and it automates error evaluation and propagation using interval
arithmetic. Besides, it can be used to incrementally prove complex mathematical
properties pertaining to the C code. Yet it does not require any specific
knowledge about automatic theorem proving, and thus is accessible to a wide
community. Moreover, Gappa may generate a formal proof of the results that can
be checked independently by a lower-level proof assistant like Coq, hence
providing an even higher confidence in the certification of the numerical code.
The article demonstrates the use of this tool on a real-size example, an
elementary function with correctly rounded output
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