Robert M. Corless

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We show how to construct linearizations of matrix polynomials $z\mathbf{a}(z)\mathbf{d}_0 + \mathbf{c}_0$, $\mathbf{a}(z)\mathbf{b}(z)$, $\mathbf{a}(z) + \mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right) < \mathrm{deg}\left(\mathbf{a}(z)\right)$), and $z\mathbf{a}(z)\mathbf{d}_0\mathbf{b}(z) + \mathbf{c_0}$ from linearizations of the component parts, $\mathbf{a}(z)$ and $\mathbf{b}(z)$. This allows the extension to matrix polynomials of a new companion matrix construction.

Robert M. Corless, C. Yalcin Kaya, Robert H. C. Moir

We show how to compute the \emph{optimal relative backward error} for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces new insight into the numerical solution of stiff problems.

Robert M. Corless, Leili Rafiee Sevyeri

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.

Eunice Y. S. Chan, Robert M. Corless

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.

Ao Li, Robert M. Corless

In the paper "A Chaotic Search for $i$"~(\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\cot{ \left( 2^{n} θ_{0} \right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\cot{ \left( θ_{n-1}+θ_{n-2} \right) }$ with $θ_{n-1}=\mathrm{arccot}{\left(x_{n-1}\right)}$ and $θ_{n-2}=\mathrm{arccot}{\left(x_{n-2}\right)}$, and $x_{n}=\cot{ \left( (k+1)^{n} θ_{0} \right) }$ with $θ_{0} = \mathrm{arccot}(x_{0})$, are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schröder iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's \textsl{Fractals[Newton]} package to visualize general one-step iterations by disguising them as Newton iterations.

Robert M. Corless, Robert H. C. Moir, Marc Moreno Maza et al.

We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the stable yet efficient computation of symbolic antiderivatives while avoiding issues of ill-conditioning to which numerical methods are susceptible. We propose two alternative methods for exact input that compute the rational part of the integral using Hermite reduction and then compute the transcendental part two different ways using a combination of exact integration and efficient numerical computation of roots. The symbolic computation is done within BPAS, or Basic Polynomial Algebra Subprograms, which is a highly optimized environment for polynomial computation on parallel architectures, while the numerical computation is done using the highly optimized multiprecision rootfinding package MPSolve. We show that both methods are forward and backward stable in a structured sense and away from singularities tolerance proportionality is achieved by adjusting the precision of the rootfinding tasks.

Wenqiang Yang, Wenyuan Wu, Robert M. Corless

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.

Robert M. Corless, Deepak Singh Kalhan, Stephen M. Watt

Previous work has made use of a parameterized plane curve polynomial representation for mathematical handwriting, with the polynomials represented in a Legendre or Legendre-Sobolev graded basis. This provides a compact geometric representation for the digital ink. Preliminary results have also been shown for Chebyshev and Chebyshev-Sobolev bases. This article explores the trade-offs between basis choice and polynomial degree to achieve accurate modeling with a low computational cost. To do this, we consider the condition number for polynomial evaluation in these bases and bound how the various inner products give norms for the variations between symbols.

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $\pm1$. Many properties remain after these specializations, some of which surprised us. We find two recursive formulae for the characteristic polynomials of upper Hessenberg matrices. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix. We count stable matrices, normal matrices, and neutral matrices, and tabulate the results of our experiments. We prove a theorem about the only possible kinds of normal matrices amongst a specific family of Bohemian upper Hessenberg matrices.

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many properties remain after these specializations, some of which surprised us. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix.

Amirhossein Amiraslani, Robert M. Corless, Madhusoodan Gunasingham

We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite interpolational bases.

Robert M. Corless, Nicolas Fillion

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.