TooMuchCoffeeMan's numer-an [8 articles]

Abstract

A natural problem at the interface of operator theory and numerical analysis is that of finding a (finite dimensional) matrix whose eigenvalues approximate the spectrum of a given (infinite dimensional) operator. It is well-known that classical work of Pimsner and Voiculescu produces explicit matrix models for an interesting class of nontrivial examples (e.g., many discretized one-dimensional Schrödinger operators). In this paper, we observe that the spectra of their models (often) converge in the strongest possible sensein the Hausdorff metricand demonstrate that ...

Abstract

The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz's equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form or its inverse . To illustrate the theory, we reduce ...

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We describe a subtle error which can appear in numerical calculations involving the spacing statistics of eigenvalues of random unitary matrices. ...

Abstract

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^ω)$ operations, then it can be done stably in $O(n^ω + η)$ operations for any $η > 0$. Here we extend this result to show that many standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, and determinant computation can also be done stably (in ...

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We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f*g, defined for all S⊆ N by (f * g)(S) = ∑_T ⊆ Sf(T) g(S\\ T), where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n^2 2^n) additions and multiplications, substantially improving upon the straightforward O(3^n) algorithm. ...

Abstract

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of ε-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a disk of radius σ. Besides ε and σ, this ...

Abstract

The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a necessary condition for convergence, which improves on several earlier results. ...