Is Gauss quadrature better than Clenshaw-Curtis?

We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of $\log((z+1)/(z-1))$ in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at $z=\infty$. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $[-1,1]$

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $q\ge 3$ is completely different from comonotone and coconvex cases. Additionally, we show that, for each function $f$ from $\Delta^{(1)}$, the set of all monotone functions on $[-1,1]$, and every $\alpha>0$, we have $$\limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n\cap\Delta^{(1)}} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\| \le c(\alpha) \limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\|$$ where $\mathbb P_n$ denotes the set of algebraic polynomials of degree $<n$, $\varphi(x):=\sqrt{1-x^2}$, and $c=c(\alpha)$ depends only on $\alpha$

Calibrating Option Pricing Models with Heuristics

Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.

The Square Root Function of a Matrix

Having origins in the increasingly popular Matrix Theory, the square root function of a matrix has received notable attention in recent years. In this thesis, we discuss some of the more common matrix functions and their general properties, but we specifically explore the square root function of a matrix and the most efficient method (Schur decomposition) of computing it. Calculating the square root of a 2×2 matrix by the Cayley-Hamilton Theorem is highlighted, along with square roots of positive semidefinite matrices and general square roots using the Jordan Canonical Form