ترغب بنشر مسار تعليمي؟ اضغط هنا

Computational Discovery with Newton Fractals, Bohemian Matrices, & Mandelbrot Polynomials

80   0   0.0 ( 0 )
 نشر من قبل Robert Corless
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

The authors have been using a largely algebraic form of ``computational discovery in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. This paper describes what we mean by ``computational discovery, what good it does for the students, and some specific techniques that we used.

قيم البحث

اقرأ أيضاً

We derive explicit formulas for calculating $e^A$, $cosh{A}$, $sinh{A}, cos{A}$ and $sin{A}$ for a given $2times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3times3$ matrix $A$. These formulas are expressed exclusively in te rms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients.
238 - Alexander Aycock 2015
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.
We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sar nak random matrix model) between the distributions of Lp(T) and of characteristic polynomials of random matrices in the compact Lie group USp(2g). We then formulate an analogue of the Sato-Tate conjecture for curves of genus 2, in which the generic distribution is augmented by 22 exceptional distributions, each corresponding to a compact subgroup of USp(4). In every case, we exhibit a curve closely matching the proposed distribution, and can find no curves unaccounted for by our classification.
122 - Bahman Kalantari 2020
Newtons method for polynomial root finding is one of mathematics most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge locally. Based on the {it Geometric Modulus Principle} for a complex polynomial $p(z)$, together with a {it Modulus Reduction Theorem} proved here, we develop the {it Robust Newtons method} (RNM), defined everywhere with a step-size that guarantees an {it a priori} reduction in polynomial modulus in each iteration. Furthermore, we prove RNM iterates converge globally, either to a root or a critical point. Specifically, given $varepsilon $ and any seed $z_0$, in $t=O(1/varepsilon^{2})$ iterations of RNM, independent of degree of $p(z)$, either $|p(z_t)| leq varepsilon$ or $|p(z_t) p(z_t)| leq varepsilon$. By adjusting the iterates at {it near-critical points}, we describe a {it modified} RNM that necessarily convergence to a root. In combination with Smales point estimation, RNM results in a globally convergent Newtons method having a locally quadratic rate. We present sample polynomiographs that demonstrate how in contrast with Newtons method RNM smooths out the fractal boundaries of basins of attraction of roots. RNM also finds potentials in computing all roots of arbitrary degree polynomials. A particular consequence of RNM is a simple algorithm for solving cubic equations.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا