Two body problem on a sphere in the presence of a uniform magnetic field

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles, interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria

The SVD of the linearized EIT problem on a disk

Abstract: In this paper we calculate the right singular functions to the linearized EIT problem on homogeneous disk. We note the similarity to Zernike disk functions and the dependence on the mesh

Stochastic Rounding: Implementation, Error Analysis, and Applications

Stochastic rounding randomly maps a real number to one of the two nearest values in a finite precision number system. First proposed for use in computer arithmetic in the 1950s, it is attracting renewed interest. If used in floating-point arithmetic in the computation of the inner product of two vectors of length n, it yields an error bounded by \sqrt(n)u with high probability, where u is the unit roundoff, which is not necessarily the case for round to nearest. A particular attraction of stochastic rounding is that, unlike round to nearest, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity are lost. We survey stochastic rounding, covering its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, including deep learning and the numerical solution of differential equations

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrary precision environment. We develop an algorithm for computing the matrix cosine that takes the unit roundoff of the working precision as input, and so works in an arbitrary precision. The algorithm employs a Taylor approximation with scaling and recovering and it can be used with a Schur decomposition or in a decomposition-free manner. We also derive a framework for computing the \fd, construct an efficient evaluation scheme for computing the cosine and its Fr\'echet derivative simultaneously in arbitrary precision, and show how this scheme can be extended to compute the matrix sine, cosine, and their \fd s all together. Numerical experiments show that the new algorithms behave in a forward stable way over a wide range of precisions. The transformation-free version of the algorithm for computing the cosine is competitive in accuracy with the state-of-the-art algorithms in double precision and surpasses existing alternatives in both speed and accuracy in working precisions higher than double

Optimizing and Factorizing the Wilson Matrix

The Wilson matrix, $W$, is a $4\times 4$ unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940s, owing to its mild ill-conditioning. We ask how close $W$ is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations $W = Z^TZ$ with $Z$ having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that $W$ has an integer factor $Z$ and two rational factors, up to signed permutations. This little $4 \times 4$ matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results

The dynamical functional particle method for multi-term linear matrix equations

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of generalized Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size

A comparison of LSTM and GRU networks for learning symbolic sequences

We explore relations between the hyper-parameters of a recurrent neural network (RNN) and the complexity of string sequences it is able to memorize. We compare long short-term memory (LSTM) networks and gated recurrent units (GRUs). We find that an increase of RNN depth does not necessarily result in better memorization capability when the training time is constrained. Our results also indicate that the learning rate and the number of units per layer are among the most important hyper-parameters to be tuned. Generally, GRUs outperform LSTM networks on low complexity sequences while on high complexity sequences LSTMs perform better

Quadratic Realizability of Palindromic Matrix Polynomials: the Real Case

Let \cL = (\cL_1,\cL_2) be a list consisting of structural data for a matrix polynomial; here \cL_1 is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field \RR, and \cL_2 is a sublist of nonnegative integers. For an arbitrary such \cL, we give easy-to-check necessary and sufficient conditions for \cL to be the list of elementary divisors and minimal indices of some real $T$-palindromic quadratic matrix polynomial. For a list \cL satisfying these conditions, we show how to explicitly build a real $T$-palindromic quadratic matrix polynomial having \cL as its structural data; that is, we provide a $T$-palindromic quadratic realization of \cL over \RR. A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list \cL

Computational graphs for matrix functions

Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective way to study existing techniques, improve them, and eventually derive new ones. As the accuracy of these matrix techniques is determined by the accuracy of their scalar counterparts, the design of algorithms for matrix functions can be viewed as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion akin to backpropagation. The Julia package GraphMatFun.jl offers the tools to generate and manipulate computational graphs, to optimize their coefficients, and to generate Julia, MATLAB, and C code to evaluate them efficiently. The software also provides the means to estimate the accuracy of an algorithm and thus obtain numerically reliable methods. For the matrix exponential, for example, using a particular form (degree-optimal) of polynomials produces algorithms that are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online

The dynamical functional particle method for the generalized Sylvester equation

Recent years have seen a renewal of interest in generalized Sylvester equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all generalized Sylvester equations with Hermitian positive definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of generalized Sylvester equations. For the Sylvester equation AX-XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size