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Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously , allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the… (More)

When an algorithm in dimension one is extended to dimension d, in nearly every case its computational cost is taken to the power d. This fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality. For numerical analysis in dimension d, we propose to use a representation for… (More)

Spherical Harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions fe ik g k2Z arise on the circle. Spherical Harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast… (More)

We show that the problems of approximating tensors and multivariate functions as a sums of (tensor) products of vectors/functions can be considered in a unified framework, thus exposing their common multilinear structure. We study the alternating least squares algorithm within this framework from the orthogonal projection and gradient perspectives. We then… (More)

The wavefunction for the multiparticle Schrödinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation… (More)

We present an algorithm for learning (or estimating) a function of many variables from scattered data. The function is approximated by a sum of separable functions, following the paradigm of separated representations. The central fitting algorithm is linear in both the number of data points and the number of variables, and thus is suitable for large data… (More)

For what value(s) of α, β, γ does the equality sin(x + y + z) = sin(x) sin(y + β − α) sin(β − α) sin(z + γ − α) sin(γ − α) + sin(x + α − β) sin(α − β) sin(y) sin(z + γ − β) sin(γ − β) (1) + sin(x + α − γ) sin(α − γ) sin(y + β − γ) sin(β − γ) sin(z) hold for all values of x, y, and z?

Students at Ohio University may print, copy and use for class and as reference material. Adoption of these notes for classroom purposes is encouraged, but instructors are asked to notify the authors of such use. Preface These notes were developed by the first author in the process of teaching a course on applied numerical methods for Civil Engineering… (More)

This pamphlet is intended for the scientist who is considering using Spherical Harmonics for some application. It is designed to introduce the Spherical Harmonics from a theoretical perspective and then discuss those practical issues necessary for their use in applications. I expect great variability in the backgrounds of the readers. In order to be… (More)

- Gregory Beylkin, Nicholas Coult, Martin J. Mohlenkamp
- 1998

We present a fast algorithm for the construction of a spectral projector. This algorithm allows us to compute the density matrix, as used in, e.g., the Kohn–Sham iteration, and so obtain the electron density. We compute the spectral projector by constructing the matrix sign function through a simple polynomial recursion. We present several matrix… (More)