In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an $n\times n$ matrix $A$ in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue $\lambda_0$. The obtained form in fact reveals the dimensions of the null spaces of $(A-\lambda_0 I)^i$ at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at $\lambda_0$ is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of $A$. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in ${\cal O}(mn^2)$ floating point operations, where $m$ is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples.

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.

The proposed algorithm is based on the block anti-triangular form of the original matrix M, as introduced by the authors in [11]. Via successive orthogonal similarity transformations this form is then updated to a new form A = QMQ(T), whereby the first k rows and columns of M have elements bounded by a given threshold tau and the remaining bottom right part of M is maintained in block anti-triangular form. The updating transformations are all orthogonal, guaranteeing the backward stability of the algorithm, and the algorithm is very economical when the near rank deficiency is detected in some of the anti diagonal elements of the block anti-triangular form. Numerical results are also given showing the reliability of the proposed algorithm. 2015 Elsevier Inc. All rights reserved. We present an algorithm for computing a symmetric rank revealing decomposition of a symmetric n x n matrix A, as defined in the work of Hansen & Yalamov [9]: we factorize the original matrix into a product A = QMQ(T), with Q orthogonal and M symmetric and in block form, with one of the blocks containing the dominant information of A, such as its largest eigenvalues. Moreover, the matrix M is constructed in a form that is easy to update when adding to A a symmetric rank-one matrix or when appending a row and, symmetrically, a column to A: the cost of such an updating is O(n(2)) floating point operations.

We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $ax^2 + bx + c = 0$ with real or complex coefficients $a, b, c$ can be computed in an element-wise mixed stable manner, measured in a relative sense. We also show that this is a stronger property than norm-wise backward stability but weaker than element-wise backward stability. We finally show that there does not exist any method that can compute the roots in an element-wise backward stable sense, which is also illustrated by some numerical experiments.

The equality constrained indefinite least squares problem involves the minimization of an indefinite quadratic form subject to a linear equality constraint. In this paper, we study this problem and present a numerical method that is proved to be backward stable in a strict sense, i.e., that the computed solution satisfies a slightly perturbed equality constrained indefinite least squares problem. We also perform a sensitivity analysis of this problem and derive bounds for the accuracy of the computed solution. We give several numerical experiments to illustrate these results.

We consider the problem of finding a square low-rank correction (?C - B)F to a given square pencil (?E - A) such that the new pencil ?(E - CF) - (A - BF) has all its generalised eigenvalues at the origin. We give necessary and sufficient conditions for this problem to have a solution and we also provide a constructive algorithm to compute F when such a solution exists. We show that this problem is related to the deadbeat control problem of a discrete-time linear system and that an (almost) equivalent formulation is to find a square embedding that has all its finite generalised eigenvalues at the origin.

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.

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.

An algorithm for computing the solution of indefinite least squares problems and of indefinite least squares problems with equality constrained is presented. Such problems arise when solving total least squares problems and in H infinity-smoothing. The proposed algorithm relies only on stable orthogonal transformations reducing recursively the associated augmented matrix to proper block anti-triangular form. Some numerical results are reported showing the properties of the algorithm.

Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations, and variational problems. In these applications one is often interested in computing a factorization of the indefinite matrix that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this paper we propose an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block antitriangular form using orthogonal similarity transformations. We also show that the algorithm is backward stable and has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.

Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations and variational problems. In these applications one is often interested in computing a factorization of the indefinite matrix that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this paper we propose an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block anti-triangular form using orthogonal similarity transformations. We also show that the algorithm is backward stable and has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.

This paper investigates the use of time series of ALOS/PALSAR-1 and COSMO-SkyMed data for the soil moisture retrieval (mv) by means of the SMOSAR algorithm. The application context is the exploitation of mv maps at a moderate spatial and temporal resolution for improving flood/drought monitoring at regional scale. The SAR data were acquired over the Capitanata plain in Southern Italy, over which ground campaigns were carried out in 2007, 2010 and 2011. The analysis shows that the mv retrieval accuracy is 5%-7% m^3/m^3 at L- and X band, although the latter is restricted to a use over nearly bare soil only.

The problem of reconstructing signals and images from degraded ones is considered in this paper. The latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image. Moreover, the coefficient matrix is very ill-conditioned, requiring an additional regularization term. Different boundary conditions can be proposed. In this paper antireflective boundary conditions are considered. Since both sides of the linear system have uncertainties and the coefficient matrix is highly structured, the Regularized Structured Total Least Squares approach seems to be the more appropriate one to compute an approximation of the true signal/image. With the latter approach the original problem is formulated as an highly nonconvex one, and seldom can the global minimum be computed. It is shown that Regularized Structured Total Least Squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform. Such subproblems are then transformed into one-dimensional unimodal realvalued minimization problems which can be solved globally. Some numerical examples show the effectiveness of the proposed approach.

We consider here the problem of tracking the dominant eigenspace of an indefinite matrix by updating recursively a rank k approximation of the given matrix. The tracking uses a window of the given matrix, which increases at every step of the algorithm. Therefore, the rank of the approximation increases also, and hence a rank reduction of the approximation is needed to retrieve an approximation of rank k. In order to perform the window adaptation and the rank reduction in an efficient manner, we make use of a new antitriangular decomposition for indefinite matrices. All steps of the algorithm only make use of orthogonal transformations, which guarantees the stability of the intermediate steps. We also show some numerical experiments to illustrate the performance of the tracking algorithm.

Many important kernel methods in the machine learning area, such as kernel principal component analysis, feature approximation, denoising, compression and prediction require the computation of the dominant set of eigenvectors of the symmetric kernel Gram matrix. Recently, an efficient incremental approach was presented for the fast calculation of the dominant kernel eigenbasis. In this manuscript we propose faster algorithms for incrementally updating and downsizing the dominant kernel eigenbasis. These methods are well-suited for large scale problems since they are both efficient in terms of complexity and data management.