Computing Real Roots of Real Polynomials ... and now For Real!

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes' Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. It achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes. In the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm. With an excerpt of our extensive collection of benchmarks, available online at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. These experiments also show that our new implementation outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots. For all other instances, we avoid almost any overhead by integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo, Ontario, Canad

On the Complexity of Computing with Planar Algebraic Curves

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials $f$, $g \in \mathbb{Z}[x,y]$ and an arbitrary polynomial $h \in \mathbb{Z}[x,y]$, each of total degree less than $n$ and with integer coefficients of absolute value less than $2^\tau$, we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded by $\tilde{O}(n^6+n^5\tau)$, where we ignore polylogarithmic factors in $n$ and $\tau$: (1) The computation of isolating regions in $\mathbb{C}^2$ for all complex solutions of the system $f = g = 0$, (2) the computation of a separating form for the solutions of $f = g = 0$, (3) the computation of the sign of $h$ at all real valued solutions of $f = g = 0$, and (4) the computation of the topology of the planar algebraic curve $\mathcal{C}$ defined as the real valued vanishing set of the polynomial $f$. Our bound improves upon the best currently known bounds for the first three problems by a factor of $n^2$ or more and closes the gap to the state-of-the-art randomized complexity for the last problem.Comment: 41 pages, 1 figur

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469

Vermittlung von Teilchenphysik: Outreach und Physikdidaktik im Gespräch

 Auf der DPG Frühjahrstagung in Aachen (25.-29. März 2019) haben die Fachverbände Teilchenphysik und Didaktik der Physik mit der Durchführung eines gemeinsamen Workshops zur Vermittlung der Teilchenphysik ein neues Format erprobt. Eingeleitet durch drei knappe Impulsreferate, die die Perspektiven „Outreach“ (Claudia Behnke), „Schule“ (Klaus Buschhüter) und „Fachdidaktik“ (Oliver Passon) vertreten haben, wurde eine moderierte Diskussion geführt. In diesem Beitrag stellen zunächst die Sprecher die Kernpunkte ihrer Referate dar, bevor die anschließende Diskussion auf Grundlage einer Audioaufzeichnung zusammengefasst wird

Different metabolic responses during incremental exercise assessed by localized 31P MRS in sprint and endurance athletes and untrained individuals

Until recently, assessment of muscle metabolism was only possible by invasive sampling. 31P magnetic resonance spectroscopy ( 31P MRS) offers a way to study muscle metabolism non-invasively. The aim of the present study was to use spatially-resolved 31P MRS to assess the metabolism of the quadriceps muscle in sprint-trained, endurance-trained and untrained individuals during exercise and recovery. 5 sprint-trained (STA), 5 endurance-trained (ETA) and 7 untrained individuals (UTI) completed one unlocalized 31P MRS session to measure phosphocreatine (PCr) recovery, and a second session in which spatially-resolved 31P MR spectra were obtained. PCr recovery time constant (τ) was significantly longer in STA (50±17 s) and UTI (41±9 s) than in ETA (30±4 s), (P\u3c0.05). PCr changes during exercise differed between the groups, but were uniform across the different components of the quadriceps within each group. pH during recovery was higher for the ETA than for the UTI (P\u3c0.05) and also higher than for the STA (P\u3c0.01). Muscle volume was greater in STA than in UTI (P\u3c0.05) but not different from ETA. Dynamic 31P MRS revealed considerable differences among endurance and sprint athletes and untrained people. This non-invasive method offers a way to quantify differences between individual muscles and muscle components in athletes compared to untrained individuals. 2013 Georg Thieme Verlag KG Stuttgart.New York