arXiv subjects

Robert M. Corless

Publications and source records attributed to Robert M. Corless.

45 records · Page 3Linked to original sources

arXiv2018

Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method

Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions represented by points. However, few work about optimal solution to minimize the residual can be found in the literatures. In this paper, we first use Hermit cubic spline interpolation at mesh points to represent the solution, then we define the residual error as the square of the L2 norm of the residual obtained by substituting the interpolation solution back to ODEs. Thus, solving ODEs is reduced to an optimization problem in curtain solution space which can be solved by conjugate gradient method with taking advantages of sparsity of the corresponding matrix. The examples of IVP and BVP in the paper show that this method can find a solution with smaller global error without additional mesh points.

math.NA↗

arXiv2018

The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding

We look at two classical examples in the theory of numerical analysis, namely the Runge example for interpolation and Wilkinson's example (actually two examples) for rootfinding. We use the modern theory of backward error analysis and conditioning, as instigated and popularized by Wilkinson, but refined by Farouki and Rajan. By this means, we arrive at a satisfactory explanation of the puzzling phenomena encountered by students when they try to fit polynomials to numerical data, or when they try to use numerical rootfinding to find polynomial zeros. Computer algebra, with its controlled, arbitrary precision, plays an important didactic role.

math.NA↗

arXiv2018

A random walk through experimental mathematics

We describe our adventures in creating a new first-year course in Experimental Mathematics that uses active learning. We used a state-of-the-art facility, called The Western Active Learning Space, and got the students to "drive the spaceship" (at least a little bit). This paper describes some of our techniques for pedagogy, some of the vignettes of experimental mathematics that we used, and some of the outcomes. EYSC was a student in the simultaneously-taught senior sister course "Open Problems in Experimental Mathematics" the first time it was taught and an unofficial co-instructor the second time. Jon Borwein attended the Project Presentation Day (the second time) and gave thoughtful feedback to each student. This paper is dedicated to his memory.

math.HO↗

arXiv2017

Minimal height companion matrices for Euclid polynomials

We define Euclid polynomials $E_{k+1}(λ) = E_{k}(λ)\left(E_{k}(λ) - 1\right) + 1$ and $E_{1}(λ) = λ+ 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\mathbb{E}_k$, so $E_k(λ) = \operatorname{det}\left(λ\mathbf{I} - \mathbb{E}_{k}\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(λ)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties.

math.NA↗

arXiv2017

Gamma and Factorial in the Monthly

The Monthly has published roughly fifty papers on the $Γ$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $Γ$.

math.HO↗

arXiv2016

Backward Error Analysis for Perturbation Methods

We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate its utility in deciding when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations (as scripts that can be modified). We also give a generic backward error based method that is equivalent to iteration (but we believe useful as an organizational viewpoint) for regular perturbation.

math.NA↗

arXiv2016

Narayana, Mandelbrot, and A New Kind of Companion Matrix

We demonstrate a new kind of companion matrix, for polynomials of the form $c(λ) = λa(λ)b(λ) + c_0$ where upper Hessenberg companions are known for the polynomials $a(λ)$ and $b(λ)$. This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence's Mandelbrot companion matrix. We motivate the construction by use of Narayana-Mandelbrot polynomials, which are also new to this paper.

math.CO↗

arXiv2011

Stieltjes, Poisson and other integral representations for functions of Lambert $W$

We show that many functions containing $W$ are Stieltjes functions. Explicit Stieltjes integrals are given for functions $1/W(z)$, $W(z)/z$, and others. We also prove a generalization of a conjecture of Jackson, Procacci & Sokal. Integral representations of $W$ and related functions are also given which are associated with the properties of their being Pick or Bernstein functions. Representations based on Poisson and Burniston--Siewert integrals are given as well.

math.CV↗

arXiv1995

On the inversion of $y^αe^y$ in terms of associated Stirling numbers

The function $y=Φ_α(x)$, the solution of $y^αe^y=x$ for $x$ and $y$ large enough, has a series expansion in terms of $\ln x$ and $\ln\ln x$, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for $x>(αe)^α$ for $α\ge 1$. It is also shown that new expansions can be obtained for $Φ_α$ in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.

math.CA↗