numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for e−x, because a computation directly from the polynomial coefficients has poor numerical stability. I finally blundered upon an approach alluded to a paper by Campos and Calderón and a related item mentioned in a paper by … Read more

What does the Von Neumann’s stability analysis tell us about non-linear finite difference equations?

I’ve asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well. I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x – u_{xxt} = 0 \end{equation} using finite difference methods. They also … Read more

Efficiently calculate the trace of the product of two large but symmetric matrices, one of which is an inverse

Sorry about the long title. I need to calculate the trace of M(M+D)−1, where M is a dense symmetric matrix, and D is a diagonal matrix. The main issue is the dimension could be large (usually in the hundreds to thousands range) so computing the inverse could be quite expensive especially because the trace needs … Read more

Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul Sutcliffe. “Untangling knots via reaction-diffusion dynamics of vortex strings.” (arXiv abs (2016).)           If anyone could connect the dots (at a high level) … Read more

Difference between Chebyshev first and second degree iterative methods

Consider linear equation Au=f. We want to solve it with iterative method (assuming A is good). First order iterative method is: uk+1=uk−αk+1(Auk−f), The second degree method is: uk+1=uk−αk+1(Auk−f)−β(uk−uk−1). For both methods we can define iteration parameters αk, βk via minimax problem which solution is Chebyshev polynomials. This is good, but it seems to me, that … Read more

Optimisation over SO(3)SO(3): is it safe to use a global parametrisation?

I am a functional analyst by training, but I am doing some numerical experiments which require me to minimise continuous functions f:SO(3)⟶[0,+∞) using a computer (I know that each f will be continuous, but cannot safely assume anything better). I have access to two Nelder-Mead implementations: one which works in SO(3) directly (using slerp and … Read more

For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, and Romberg seem to be mainly intended for cases where one can finely sample the function but not integrate analytically. However, for functions with structures finer than the sampling … Read more

Convergence acceleration of successions with logarithms

I have a numerical question regarding acceleration of a succession. A preliminary: suppose that I have a succession $a_g$ that, for high $g$, asymptotically goes as $$ a_g=s_0+\frac{s_1}g+\frac{s_2}{g^2}+…=\sum_{k=0}^\infty \frac{s_k}{g^k}. $$ I am interested in computing the coefficients $s_k$, but in particular I am interested in computing the leading coefficient $s_0$ (that would also be the … Read more

Numerics for continuity equation with Sobolev vector field

Has any work been done about numerical methods for the continuity equation ∂tρ(x,t)+div(b(x,t)ρ(x,t))=0,t∈[0,T],x∈RN, where b∈L1tW1,px? A related equation has been asked on Mathematica StackExchange. Answer While the standard numerical methods (at least some of them) work even in the Sobolev regularity setting, the analysis of convergence is far from being trivial (due to non-smoothness of … Read more

Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold M. The general form of such an equation is ˙x(t)=V(x(t)),x(0)=x0∈M, where V is a vector field in M. I have spent many hours checking the literature and I cannot find any intrinsic general method. I have found many good methods for … Read more