Optimized computation of the voigt and complex probability functions

This paper presents some work in progress on the development of fast and accurate support for complex floatingpoint arithmetic on embedded processors. Focusing on the case of multiplication, we describe algorithms and implementations for computing both the real and imaginary parts with high relative accuracy. We show that, in practice, such accuracy guarantees can be achieved with reasonable overhead compared with conventional algorithms (which are those offered by current implementations and for which the real or imaginary part of a product can have no correct digit at all). For example, the average execution-time overheads when computing an FFT on the ARM Cortex-A53 and -A57 processors range from 1.04x to 1.17x only, while arithmetic costs suggest overheads from 1.5x to 1.8x.

In this work, we propose a method for the derivation of prolate spheroidal wave functions (PSWFs) and Slepian functions on continuous and disjoint intervals on the real number line. The proposed method uses Fourier series to obtain a closed-form approximation for Slepian functions on the real line. With this closed-form expression, Slepian functions can be evaluated at arbitrary points in the region of interest with high accuracy. The conventional method uses properties of the Slepian concentration problem to evaluate PSWFs on finite number of points in an interval. The conventional method is computationally expensive and does not allow for easy storage. By approximating an interval containing regions of interest as periodic, we express the Slepian concentration problem as a finite dimensional problem using the Fourier series domain. Solutions to the Slepian concentration problem in this form are Fourier series coefficients corresponding to the Slepian functions. Reconstruction in Fourier series basis, scaling and subsequent truncation provides the closed-form expression for the Slepian problem. Upon comparison with PSWFs obtained by the conventional method, we find negligible difference.

Monocular rotational microscopes suffer from shallow depth of field and reduced field of view. Consequently, we propose monocular wide-field optical microscopy with extended depth of field to enable accurate 3D measurements of micro-objects. We use a liquid lens and a 4f system to extend the system's depth of field and two additional rotational reflectors to capture multi-view image sequences. In addition, we increase the number of expansion directions in the patch-based multi-view stereo algorithm and optimize the expansion radius. Our approach outperforms that based on the patch-based multi-view stereo method and achieves a high relative measurement accuracy of 1.9%. RESEARCH HIGHLIGHTS: This study proposes 3D measurements of micro-objects via monocular rotational microscopy. The depth of field is extended by the use of a liquid lens and a 4f system. Our approach outperforms that based on the patch-based multi-view stereo method and achieves a high relative measurement accuracy of 1.9%.

Suppose that f is the characteristic function of a probability measure on the real line R. We deal with the following open problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part Im f? In other words, is it true that for any characteristic function f, there exists a characteristic function g such that Im f = Im g, but f ≠ g? The answer to this question is no. We give a characterization of those characteristic functions, which are uniquely determined by their imaginary parts. Also, several examples of characteristic functions, which are uniquely determined by their imaginary parts, are given.

We propose a new algorithm for the symmetric eigenproblem that computes eigenvalues and eigenvectors with high relative accuracy for the largest class of symmetric, definite and indefinite, matrices known so far. The algorithm is divided into two stages: the first one computes a singular value decomposition (SVD) with high relative accuracy, and the second one obtains eigenvalues and eigenvectors from singular values and vectors. The SVD, used as a first stage, is responsible both for the wide applicability of the algorithm and for being able to use only orthogonal transformations, unlike previous algorithms in the literature. Theory, a complete error analysis, and numerical experiments are presented.

For some families of classical orthogonal polynomials defined on appropriate intervals, it is shown that the corresponding Jacobi matrices are totally positive and their bidiagonal factorizations can be accurately computed. By exploiting these facts, an algorithm to compute with high relative accuracy the eigenvalues of those Jacobi matrices, and consequently the nodes of Gaussian quadrature formulae for those families of orthogonal polynomials, is presented. An algorithm is also presented for the computation of the eigenvectors of these Jacobi matrices, and hence the weights of Gaussian quadrature formulae. Although in this case high relative accuracy is not theoretically guaranteed, the numerical experiments with our algorithm provide very accurate results.

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

Gaussian Markov Random Fields over graphs of paths and high relative accuracy

This paper provides an efficient method to compute an LDU decomposition of the Laplacian matrix of a connected graph with high relative accuracy. Several applications of this method are presented. In particular, it can be applied to efficiently compute the eigenvalues of the mentioned Laplacian matrix. Moreover, the method can be extended to graphs with weighted edges.

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

The self-starter is an efficient approach for solving geoacoustic inverse problems involving a vertical array of receivers located on the order of ten wavelengths from a source [R. J. Cederberg and M. D. Collins, J. Acoust. Soc. Am. 22, 102–109 (1997)]. With this approach, accurate solutions can be obtained by sampling the wave number spectrum at approximately ten points. There have been unsubstantiated claims that the spectral solution provides similar efficiency when implemented with an approach that involves perturbing the integration contour off the real line [F. B. Jensen et al., Computational Ocean Acoustics (American Institute of Physics, New York, 1994), pp. 231–240]. It is demonstrated that this quadrature scheme breaks down at short ranges and that the self-starter solution is about an order of magnitude faster. These conclusions are based on comparisons of the rational approximations associated with the self-starter and the quadrature scheme, comparisons of acoustic fields for particular problems, and a simple analysis of the spectral integral. These conclusions are not surprising since the rational approximation associated with the self-starter is based on the analytic evaluation of the spectral integral while the other rational approximation is based on numerical integration. [Work supported by ONR.]

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

This study aimed to determine Salmonella enterica subspecies enterica serovar Enteritidis (SE) presence by conventional serotyping and SE-specific real-time PCR (SE-rPCR) in poultry-derived food and avian isolates in our laboratory Salmonella spp. collection. Conventional serotyping indicated that 32 (8 chicken meat, 10 egg, 14 avian) out of 56 (57%) isolates were SE, whereas 8 (3 chicken meat, 2 turkey meat, 3 avian) (14%) isolates were serogroup B, 6 (1 chicken meat, 1 egg, 4 avian) (11%) were serogroup C1, 4 (3 chicken meat, 1 turkey meat) (7%) were serogroup C2-C3, 4 (3 chicken meat, 1 turkey meat) (7%) were serogroup E4, and 2 (avian) (4%) isolates were categorized as nonserogrouped/nonserotyped. Thirty-three (8/18 chicken meat, 10/11 egg, 15/23 avian) out of 56 (59%) Salmonella isolates were positive by SE-rPCR. SE was determined as the most prevalent serotype in both of the tests regardless of the sample type. Conventional serotyping and SE-rPCR results were in agreement in all but 1 isolate. Considerably high relative accuracy (98%), sensitivity (100%), and specificity (96%) with almost perfect agreement between the two methods (Cohen's kappa = 0.96) indicated SE-rPCR as a reliable tool in the rapid identification of SE isolates to complement conventional serotyping.