In this manuscript, we propose a stable algorithm for computing the zeros of Althammer polynomials. These polynomials are orthogonal with respect to a Sobolev inner product, and are even if their degree is even, odd otherwise. Furthermore, their zeros are real, distinct, and located inside the interval (−1, 1). The Althammer polynomial p_n(x) of degree n satisfies a long recurrence relation, whose coefficients can be arranged into a Hessenberg matrix of order n, with eigenvalues equal to the zeros of the considered polynomial. Unfortunately, the eigenvalues of this Hessenberg matrix are very ill–conditioned, and standard balancing procedures do not improve their condition numbers. Here, we introduce a novel algorithm for computing the zeros of p_n(x), which first transforms the Hessenberg matrix into a similar symmetric tridiagonal one, i.e., a matrix whose eigenvalues are perfectly conditioned, and then computes the zeros of p_n(x) as the eigenvalues of the latter tridiagonal matrix. Moreover, we propose a second algorithm, faster but less accurate than the former one, which computes the zeros of p_n(x) as the eigenvalues of a truncated Hessenberg matrix, obtained by properly neglecting some diagonals in the upper part of the original matrix. The computational complexity of the proposed algorithms are, respectively, O( n^3/ 6 ), and O(l^2 n), with l<

In this work, we focus on the computation of the zeros of a monic Laguerre–Sobolev orthogonal polynomial of degree n. Taking into account the associated four–term recurrence relation, this problem can be formulated as a generalized eigenvalue problem, involving a lower bidiagonal matrix and a 2–banded lower Hessenberg matrix of order n. Unfortunately, the considered generalized eigenvalue problem is very ill–conditioned, and classical balancing procedures do not improve it. Therefore, customary techniques for solving the generalized eigenvalue problem, like the QZ method, yield unreliable results. Here, we propose a novel balancing procedure that drastically reduces the ill–conditioning of the eigenvalues of the involved matrix pencil. Moreover, we propose a fast and reliable algorithm, with O(n2) computational complexity and O(n) memory, exploiting the structure of the considered matrix pencil.

In this paper we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight on the real semiaxis via product rules. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three-term recurrence relation of the system of orthonormal Laguerre polynomials.It is shown that the proposed algorithm computes the modified moments with high relative accuracy and linear complexity.Numerical examples illustrate the effectiveness of the proposed method.

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials. Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to different measures, with Moreover, they satisfy -term recurrence relations. In this manuscript, without loss of generality, is considered equal to The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the Matlab function balance.m does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a Matlab package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.

In this paper, we derive a new method to compute the nodes and weights of simultaneous n-point Gaussian quadrature rules. The method is based on the eigendecomposition of the banded lower Hessenberg matrix that contains the coefficients of the recurrence relations for the corresponding multiple orthogonal polynomials. The novelty of the approach is that it uses the property of total nonnegativity of this matrix associated with the particular considered multiple orthogonal polynomials, in order to compute its eigenvalues and eigenvectors in a numerically stable manner. The overall complexity of the computation of all the nodes and weights is O(n^2).

The computation of n-point Gaussian quadrature rules for symmetric weight functions is considered in this paper. It is shown that the nodes and the weights of the Gaussian quadrature rule can be retrieved from the singular value decomposition of a bidiagonal matrix of size n/2. The proposed numerical method allows to compute the nodes with high relative accuracy and a computational complexity of O(n). We also describe an algorithm for computing the weights of a generic Gaussian quadrature rule with high relative accuracy. Numerical examples show the effectiveness of the proposed approach.

The aim of this work is to propose a fast and reliable algorithm for computingintegrals of the type$$\int_{-\infty}^{\infty} f(x) e^{\scriptstyle -x^2 -\frac{\scriptstyle 1}{\scriptstyle x^2}} dx,$$where $f(x)$ is a sufficiently smooth function, in floating point arithmetic.The algorithm is based on a product integration rule, whose rate of convergencedepends only on the regularity of $f$, since the coefficients of the rule are ``exactly'' computed by means of suitable recurrence relations here derived.We prove stability and convergence in the space of locally continuous functions on $\RR$ equipped with weighted uniform norm.By extensive numerical tests, the accuracy of the proposed product rule is compared with that of the Gauss--Hermite quadrature formula w.r.t. the function $f(x) e^{-\frac{\scriptstyle 1}{\scriptstyle x^2}}$. The numerical results confirm the effectiveness of the method, supporting the proven theoretical estimates.

The computation of the eigenvalue decomposition of matrices is one of the most investigated problems in numerical linear algebra. In particular, real nonsymmetric tridiagonal eigenvalue problems arise in a variety of applications. In this paper the problem of computing an eigenvector corresponding to a known eigenvalue of a real nonsymmetric tridiagonal matrix is considered, developing an algorithm that combines part of a QR sweep and part of a QL sweep, both with the shift equal to the known eigenvalue. The numerical tests show the reliability of the proposed method.

The paper proposes a decision support system (DSS) for the supply chain of packaged fresh and highly perishable products. The DSS combines a unique tool for sales forecasting with order planning which includes an individual model selection system equipped with ARIMA, ARIMAX and transfer function forecasting model families, the latter two accounting for the impact of prices. Forecasting model parameters are chosen via two alternative tuning algorithms: a two-step statistical analysis, and a sequential parameter optimisation framework for automatic parameter tuning. The DSS selects the model to apply according to user-defined performance criteria. Then, it considers sales forecasting as a proxy of expected demand and uses it as input for a multi-objective optimisation algorithm that defines a set of non-dominated order proposals with respect to outdating, shortage, freshness of products and residual stock. A set of real data and a benchmark - based on the methods already in use - are employed to evaluate the performance of the proposed DSS. The analysis of different configurations shows that the DSS is suitable for the problem under investigation; in particular, the DSS ensures acceptable forecasting errors and proper computational effort, providing order plans with associated satisfactory performances.

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the QR and LU factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the QR factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices.

We address the problem of forecasting sales for fresh and highly perishable products, in the general context of supply chain management. The forecasting activity refers to the single item in a given store and started from a pre-processing phase for data analysis and normalization. Then data was used as input for a forecasting algorithm designed to be user interactive. We implemented three forecasting methods: ARIMA, ARIMAX and transfer function models. The exogenous components of the forecasting models took the impact of prices into account. The best configuration of these models is dynamically chosen via two alternative methods: (i) a two-step procedure, based on properly selected statistical indicators, (ii) a Sequential Parameter Optimization approach for automatic parameter tuning. The user or the decision maker at the store level should not be exposed to the complexity of the forecasting system which - for this reason - is designed to adaptively select the best model configuration at every forecast session, to be used for each item/store combination. A set of real data based on 19 small and medium sized stores and 156 fresh products was employed to evaluate both quality of forecasting results and their effects on the order planning activity, where sales forecasting is considered as a proxy of the expected demand. Some examples are reported and discussed. Our results confirm that there is no 'one-size-fits-all' forecasting model, whose performance strictly depends on the specific characteristics of the underlying data. This supports the adoption of a data-driven tool to automate the dynamic selection of the most appropriate forecasting model. (C) 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

An algorithm for computing the antitriangular factorization of symmetric matrices, relying only on orthogonal transformations, was recently proposed. The computed antitriangular form straightforwardly reveals the inertia of the matrix. A block version of the latter algorithm was described in a different paper, where it was noticed that the algorithm sometimes fails to compute the correct inertia of the matrix.In this paper we analyze a possible cause of the failure of detecting the inertia and propose a procedure to recover it. Furthermore, we propose a different algorithm to compute the antitriangular factorization of a symmetric matrix that handles most of the singularities of the matrix at the very end of the algorithm.Numerical results are also given showing the reliability of the proposed algorithm.

In this study non-negative matrix factorization (NMF) was hierarchically applied to simulated and in vivo three-dimensional 3 T MRSI data of the prostate to extract patterns for tumour and benign tissue and to visualize their spatial distribution. Our studies show that the hierarchical scheme provides more reliable tissue patterns than those obtained by performing only one NMF level. We compared the performance of three different NMF implementations in terms of pattern detection accuracy and efficiency when embedded into the same kind of hierarchical scheme. The simulation and in vivo results show that the three implementations perform similarly, although one of them is more robust and better pinpoints the most aggressive tumour voxel(s) in the dataset. Furthermore, they are able to detect tumour and benign tissue patterns even in spectra with lipid artefacts. Copyright (C) 2016 John Wiley & Sons, Ltd.

This paper deals with the issue of forecasting energy production of a Photo-Voltaic (PV) plant, needed by the Distribution System Operator (DSO) for grid planning. As the energy production of a PV plant is strongly dependent on the environmental conditions, the DSO has difficulties to manage an electrical system with stochastic generation. This implies the need to have a reliable forecasting of the irradiance level for the next day in order to setup the whole distribution network. To this aim, this paper proposes the use of transfer function models. The assessment of quality and accuracy of the proposed method has been conducted on a set of scenarios based on real data.

Nowadays, Magnetic Resonance Spectroscopy (MRS) represents a powerful nuclear magnetic resonance (NMR) technique in oncology since it provides information on the biochemical profile of tissues, thereby allowing clinicians and radiologists to identify in a non-invasive way the different tissue types characterising the sample under investigation. The main purpose of the pre-sent chapter is to provide a review of the most recent and significant applica-tions of non-negative matrix factorization (NMF) to MRS data in the field of tissue typing methods for tumour diagnosis. Specifically, NMF-based methods for the recovery of constituent spectra in ex vivo and in vivo brain MRS data, for brain tissue pattern differentiation using Magnetic Resonance Spectro-scopic Imaging (MRSI) data, and for automatic detection and visualisation of prostate tumours will be described. Furthermore, since several NMF imple-mentations are available in the literature, a comparison in terms of pattern de-tection accuracy of some NMF algorithms will be reported and discussed, and the NMF performance for MRS data analysis will be compared with that of other blind source separation (BSS) techniques.

We address the problem of supply chain management for a set of fresh and highly perishable products. Our activity mainly concerns forecasting sales. The study involves 19 retailers (small and medium size stores) and a set of 156 different fresh products. The available data is made of three year sales for each store from 2011 to 2013. The forecasting activity started from a pre-processing analysis to identify seasonality, cycle and trend components, and data filtering to remove noise. Moreover, we performed a statistical analysis to estimate the impact of prices and promotions on sales and customers' behaviour. The filtered data is used as input for a forecasting algorithm which is designed to be interactive for the user. The latter is asked to specify ID store, items, training set and planning horizon, and the algorithm provides sales forecasting. We used ARIMA, ARIMAX and transfer function models in which the value of parameters ranges in predefined intervals. The best setting of these parameters is chosen via a two-step analysis, the first based on well-known indicators of information entropy and parsimony, and the second based on standard statistical indicators. The exogenous components of the forecasting models take the impact of prices into account. Quality and accuracy of forecasting are evaluated and compared on a set of real data and some examples are reported.

In this article a nonnegative blind source separation technique, known as nonnegative matrix factorization, is applied to microdiffraction data in order to extract characteristic patterns and to determine their spatial distribution in tissue typing problems occurring in bone-tissue engineering. In contrast to other blind source separation methods, nonnegative matrix factorization only requires nonnegative constraints on the extracted sources and corresponding weights, which makes it suitable for the analysis of data occurring in a variety of applications. In particular, here nonnegative matrix factorization is hierarchically applied to two-dimensional meshes of X-ray diffraction data measured in bone samples with implanted tissue. Such data are characterized by nonnegative profiles and their analysis provides significant information about the structure of possibly new deposited bone tissue. A simulation and real data studies show that the proposed method is able to retrieve the patterns of interest and to provide a reliable and accurate segmentation of the given X-ray diffraction data.