An O(N squared) method for computing the eigensystem of N by N symmetric tridiagonal matrices by the divide and conquer approach

An efficient method is proposed to solve the eigenproblem of N by N Symmetric Tridiagonal (ST) matrices. Unlike the standard eigensolvers which necessitate O(N cubed) operations to compute the eigenvectors of such ST matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N squared) operations. The method is based on serial implementation of the recently introduced Divide and Conquer (DC) algorithm. It exploits the fact that by O(N squared) of DC operations, one can compute the eigenvalues of N by N ST matrix and a finite number of pairs of successive rows of its eigenvector matrix. The rest of the eigenvectors--all of them or one at a time--are computed by linear three-term recurrence relations. Numerical examples are presented which demonstrate the superiority of the proposed method by saving an order of magnitude in execution time at the expense of sacrificing a few orders of accuracy

Towards a re-conceptualisation of risk in early childhood education

© The Author(s) 2019. Children’s engagement in risk-taking has been on the agenda for early childhood education for the past 10–15 years. At a time when some say the minority world has become overly risk averse, early childhood education aims to support confident, competent and resilient children through the inclusion of beneficial risk in early childhood education. The concept of risk is a complex phenomenon. Beneficial risk is engaging in experiences that take a person outside of their comfort zone and include outcomes that may be beneficial to learning, development and life satisfaction. To date, research on beneficial risk in early childhood has focused on children’s risk-taking in outdoor play. This focus has led to a predominant conceptualisation of beneficial risk in early childhood as an outdoor physical play activity for children. In this article, the authors problematise this conceptualisation. Drawing on both broad and early childhood education specific literature, the authors explore the current discourse on risk in both childhood and early childhood education. The authors identify the development of the current conceptualisation of risk as an experience for children within play, outdoors and as a physical activity, and highlight the limitations of this conceptualisation. The authors argue that for risk-taking to be in line with the predominantly holistic approach of early childhood education, a broad view of risk is needed. To achieve this broad view, the authors argue for a re-conceptualisation of risk that encompasses a wide range of risk experiences for both children and educators. The authors suggest further research is needed to expand our understanding of beneficial risk in early childhood education. They propose further research will offer a significant contribution to the early childhood sector

Microbiota and neurologic diseases : potential effects of probiotics

Background: The microbiota colonizing the gastrointestinal tract have been associated with both gastrointestinal and extra-gastrointestinal diseases. In recent years, considerable interest has been devoted to their role in the development of neurologic diseases, as many studies have described bidirectional communication between the central nervous system and the gut, the so-called "microbiota-gut-brain axis". Considering the ability of probiotics (i.e., live non-pathogenic microorganisms) to restore the normal microbial population and produce benefits for the host, their potential effects have been investigated in the context of neurologic diseases. The main aims of this review are to analyse the relationship between the gut microbiota and brain disorders and to evaluate the current evidence for the use of probiotics in the treatment and prevention of neurologic conditions. Discussion: Overall, trials involving animal models and adults have reported encouraging results, suggesting that the administration of probiotic strains may exert some prophylactic and therapeutic effects in a wide range of neurologic conditions. Studies involving children have mainly focused on autism spectrum disorder and have shown that probiotics seem to improve neuro behavioural symptoms. However, the available data are incomplete and far from conclusive. Conclusions: The potential usefulness of probiotics in preventing or treating neurologic diseases is becoming a topic of great interest. However, deeper studies are needed to understand which formulation, dosage and timing might represent the optimal regimen for each specific neurologic disease and what populations can benefit. Moreover, future trials should also consider the tolerability and safety of probiotics in patients with neurologic diseases

An O(N 2 ) method for computing the eigensystem of N × N symmetric tridiagonal matrices by the divide and conquer approach

The QR algorithm computes the eigenvalues of N x N symmetric tridiagonal (ST) matrices with O(N 2) operations. The cost of computing the eigenvectors of such matrices is O(N z) operations. The additional order of magnitude required to compute these eigenvectors is typical of sequential algorithms. Recently, a parallel divide-and-conquer algorithm was introduced [1, 2] for computing the spectral decomposition of ST matrices. A sequential implementation of this algorithm requires the same number of operations. Namely, the eigenvalues which coincide with the roots of the so-called secular equation [3] are computed at the cost of no more than O(N 2) sequential operations; to compute the associated eigenvectors necessitates, as before, O(N 3) sequential operations. Here, we propose an efficient method derived from the DC algorithm, which computes the eigensystem of ST matrices with only O(N 2) sequential operations. The method employs linear three-term recurrence relations which successively compute the rows of the eigenvector matrix. The coefficients of these relations depend on the already computed eigenvalues, and the method hinges on the observation that the initial first two rows for the recurrence relations emerge naturally from the DC computation of these eigenvalues. Thus, the input data for the recurrence relations depend solely on the O(N 2) operations for the DC calculation of the eigenvalues. Together with the additional O(N 2) operations required to carry out these relations, we end up with a most efficient method to compute the whole eigensystem of ST matrices. Due to the sensitivity of the three-term recurrence relations, their input data should be provided with high accuracy. To achieve this, we employ an improved root finderminteresting for its own sake--in order to solve th

AN O(N²) METHOD FOR COMPUTING THE EIGENSYSTEM OF N x N SYMMETRIC TRIDIAGONAL MATRICES BY THE DIVIDE AND CONQUER APPROACH

An efficient method to solve the eigenproblem of N x N symmetric tridiagonal matrices is proposed. Unlike the standard eigensolvers that necessitate O(N3) operations to compute the eigenvectors of such matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N2) operations. The method is based on serial implementation of the recently introduced Divide and Conquer algorithm [3], [1], [4]. It exploits the fact that by O(N2) Divide and Conquer operations one can compute the eigenvalues of an N x N symmetric tridiagonal matrix and a small number of pairs of successive rows of its eigenvector matrix. The rest of the eigenvectors (either all together or one at a time) are computed by linear three-term recurrence relations. The paper is concluded with numerical examples that demonstrate the superiority of the proposed method for a special class of symmetric tridiagonal matrices, by saving an order of magnitude in execution time at the expense of sacrificing a few orders of accuracy, although for symmetric tridiagonal matrices in general, the method appears to be unstable