M-matrices are extensively employed in numerical analysis. These matrices can be generalized by corresponding operators on a partially ordered normed space. We extend results which are well-known for M-matrices to this more general setting. We investigate two different notions of an M-operator, where we focus on two questions: 1. For which types of partially ordered normed spaces do the both notions coincide? This leads to the study of positive-off-diagonal operators. 2. Which conditions on an M-operator ensure that its (positive) inverse satisfies certain maximum principles? We deal with generalizations of the "maximum principle for inverse column entries". / M-Matrizen werden in der numerischen Mathematik vielfältig angewandt. Eine Verallgemeinerung dieser Matrizen sind entsprechende Operatoren auf halbgeordneten normierten Räumen. Bekannte Aussagen aus der Theorie der M-Matrizen werden auf diese Situation übertragen. Für zwei verschiedene Typen von M-Operatoren werden die folgenden Fragen behandelt: 1. Für welche geordneten normierten Räume sind die beiden Typen gleich? Dies führt zur Untersuchung außerdiagonal-positiver Operatoren. 2. Welche Bedingungen an einen M-Operator sichern, dass seine (positive) Inverse gewissen Maximumprinzipien genügt? Es werden Verallgemeinerungen des "Maximumprinzips für inverse Spalteneinträge" angegeben und untersucht.

M-matrices are extensively employed in numerical analysis. These matrices can be generalized by corresponding operators on a partially ordered normed space. We extend results which are well-known for M-matrices to this more general setting. We investigate two different notions of an M-operator, where we focus on two questions: 1. For which types of partially ordered normed spaces do the both notions coincide? This leads to the study of positive-off-diagonal operators. 2. Which conditions on an M-operator ensure that its (positive) inverse satisfies certain maximum principles? We deal with generalizations of the "maximum principle for inverse column entries". / M-Matrizen werden in der numerischen Mathematik vielfältig angewandt. Eine Verallgemeinerung dieser Matrizen sind entsprechende Operatoren auf halbgeordneten normierten Räumen. Bekannte Aussagen aus der Theorie der M-Matrizen werden auf diese Situation übertragen. Für zwei verschiedene Typen von M-Operatoren werden die folgenden Fragen behandelt: 1. Für welche geordneten normierten Räume sind die beiden Typen gleich? Dies führt zur Untersuchung außerdiagonal-positiver Operatoren. 2. Welche Bedingungen an einen M-Operator sichern, dass seine (positive) Inverse gewissen Maximumprinzipien genügt? Es werden Verallgemeinerungen des "Maximumprinzips für inverse Spalteneinträge" angegeben und untersucht.

In this work, we design a linear, two step implicit finite difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. Three separate models are analyzed, all of which can be described as systems of partial differential equations (PDE)s with nonlinear diffusion and reaction, where the biological colony grows and decays based on the substrate bioavailability. The systems under investigation are all complex models describing the dynamics of biological films. In view of the difficulties to calculate analytical solutions of the models, we design here a numerical technique to consistently approximate the system evolution dynamics, guaranteeing that nonnegative initial conditions will evolve uniquely into new, nonnegative approximations. This property of our technique is established using the theory of M-matrices, which are nonsingular matrices where all the entries of their inverses are positive numbers. We provide numerical simulations to evince the preservation of the nonnegative character of solutions under homogeneous Dirichlet and Neumann boundary conditions. The computational results suggest that the method proposed in this work is stable, and that it also preserves the bounded character of the discrete solutions.

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.

Nous établissons des conditions nécessaires et suffisantes d’existence et d’infinie divisibilité pour des processus ponctuels alpha-déterminantaux et, lorsque alpha est positif, pour leur intensité sous-jacente (en tant que processus de Cox). Dans le cas où l’espace est fini, ces distributions correspondent à des lois binomiales, négatives binomiales et gamma multidimensionnelles. Nous étudions de façon approfondie ces deux derniers cas avec un noyau non nécessairement symétrique. / We establish necessary and sufficient conditions for the existence and infinite divisibility of alpha-determinantal processes and, when alpha is positive, of their underlying intensity (as Cox process). When the space is finite, these distributions correspond to multidimensional binomial, negative binomial and gamma distributions. We make an in-depth study of these last two cases with a non necessarily symmetric kernel.