The Numerical Analysis Group includes people affiliated to the departments of Mathematics and of Computer Science of Pisa University, and by some postdoc.

The people in each department are as follows:

Department of Mathematics:

- Lidia Aceto
- Dario A. Bini
- Paola Boito
- Maurizio Ciampa
- Paolo Ghelardoni
- Cecilia Magherini
- Beatrice Meini
- Sergio Steffé

Department of Computer Science

Postdoc:

- Stefano Massei (firstname.lastname@epfl.ch) EPFL, Lausanne;
- Leonardo Robol, ISTI, CNR, Pisa (firstname.lastname@isti.cnr.it)

Former members of the group:

Roberto Bevilacqua, Ornella Menchi, Fabio Di Benedetto, Enrico Bozzo, Dario Fasino, Giuseppe Fiorentino, Bruno Iannazzo, Vanni Noferini, Stefano Serra Capizzano.

The research is aimed at the design, analysis and implementation of numerical algorithms for solving classes of problems from the theory and applications. Specific properties of the problems, which often reveal themselves in terms of matrix structures, are investigated with the aim of devising theoretical and computational tools for the design and analysis of algorithms.

The research activity is performed within the following research projects:

-Progetto di ricerca GNCS 2017, resp. Beatrice Meini: "Metodi numerici avanzati per equazioni e funzioni di matrici con struttura"

-Progetto di Ricerca di Ateneo PRA_2017, resp. Luca Gemignani: "Modelli ed algoritmi innovativi per problemi strutturati e sparsi di grandi dimensioni"

Current interests are Numerical Linear Algebra issues, polynomial computations, some models from applications, numerical treatment of ODE, theory of circuits and networks. Below is a short description of the topics and further information can be found at the page of our group NumPi

**Numerical Methods for Polynomial Root-Finding**

This research area concerns the design, analysis and implementation of numerical algorithms for the guaranteed approximation of the roots of a polynomial up to any number of digits. The motivation of this kind of numerical tools comes mainly from computer algebra systems where the symbolic treatment of polynomial systems leads to solving polynomials with very large degrees and with huge coefficients (exact or approximate). Other motivations come from problems in combinatorics, problems in dynamics of holomorphic functions and problems in celestial mechanics. The package MPSolve, designed in this framework, is the fastest available software for the guaranteed computation of polynomial zeros up to any precision.

Involved scholars: D.A. Bini, G. Fiorentino, L. Robol

Collaborations: Dhagash Mehta (University of Notre Dame, Indiana USA)

**Matrix polynomials, linearizations and quasiseparable matrices**

Matrix polynomials are encountered in many applications in Engineering and in Scientific Computing, in particular in the analysis of vibrations of complex systems. Generalized eigenvalue problems associated with matrix polynomials are usually reduced to standard problems by means of linearization. The research concerns the design and analysis of linearizations which have useful computational properties, and the design of methods for their implementation. This kind of approach relies on the analysis of quasiseparable matrices and of classes of matrices endowed with rank structure properties.

Involved scholars: D.A. Bini, R. Bevilacqua, P. Boito, G. Del Corso, L. Gemignani, F. Poloni, L. Robol

Collaborations: Yuli Eidelman, Israel Gohberg (Tel Aviv), Raf Vandebril (Leuven), Francoise Tisseur (Manchester), Vanni Noferini (Essex University), V. Pan (New York).

**Matrix equations and matrix functions**

Many problems from the Real World and from Scientific Computing are modeled by matrix equations and matrix functions. Typical examples are the Lyapunov and Sylvester equations, or the algebraic Riccati equation concerning the stability of dynamical systems, and the unilateral quadratic equations. Other equations encountered in in queueing models are given in terms of analytic matrix functions. Here the goal is to analyze theoretical properties useful for the design of effective numerical algorithms.

Involved scholars: D.A. Bini, B. Meini, F. Poloni, S. Massei, L. Robol

Collaborations: B. Iannazzo (Perugia), Guy Latouche (Bruxelles), Sophie Hautphenne (Bruxelles), V. Ramaswami (AT&T Labs), G. Sbrana (Rouen), C. Shroeder (Berlin), V. Mehrmann (Berlin), G.T. Nguyen (Adelaide), T. Reiss (Hamburg), C.-H. Guo (Regina).

**Geometric Matrix Means**

The concept of geometric mean of positive numbers can be extended to positive definite matrices. This extension is not trivial and, except for the case of two matrices, there exist infinitely many extensions. The current definitions are given in terms of geodesics in a suitable Riemannian geometry. There are several open problems both from the theoretical and the algorithmic points of view. Matrix geometric means play an important role in the applications as in the radar detection problems. Here, we aim to have a better understanding of the different concepts and to provide the most suited definitions for the applications together with effective numerical algorithms.

Involved scholars: D.A. Bini, B. Meini, F. Poloni

Collaborations: B. Iannazzo (Perugia), R. Vandebril (Leuven), B. Jeuris (Leuven), M. Fasi (Manchester).

**Numerical solution of Markov chains**

Many problems from the applications are modeled by Markov chains. Often the number of states is huge, sometimes infinite. In these cases the standard solution algorithms are not able to compute the solution. Here the goal is to devise effective algorithms for solving specific problems from the applications where the classical algorithms fail. The used techniques rely on tools from complex analysis, numerical analysis and numerical linear algebra.

Involved scholars: D.A. Bini, B. Meini, S. Massei, F. Poloni, S. Steffe

Collaborations: P. Favati (CNR Pisa), G. Latouche (Bruxelles), Van Houdt (Antwerp), V. Ramaswami (AT&T Labs), C. H. Foh (Surrey), M. Zukerman (Hong Kong)

**Analysis of complex networks**

The analysis of complex networks has been applied to the assessment of the quality of research including scholars, institutions and journals. Another area concerns models for patent evaluation.

Involved scholars: D. Bini, F. Romani, G. Del Corso

Collaborations: M. Shaffer (Tucson)

Patents: Patent Pending n. 61/494,821, filed 6/8/2011, US Patent Office; Patent Pending n.,61/933,812, filed 1/20/2014, US Patent Office.

**Numerical methods for ordinary and fractional differential equations**

Initial and boundary value problems for ordinary and fractional ODEs are analyzed with the aim of designing new and more effective algorithms for their numerical solution. Direct and inverse eigenvalue problems for the Sturm-Liouville type operators are investigated both in the regular and in the singular case.

Involved scholars: L. Aceto, P. Ghelardoni, C. Magherini

Collaborations: L. Brugnano (Firenze), F. Iavernaro (Bari), M. Marletta (Cardiff), P. Novati (Padova), E. Weinmueller (Vienna)

**Theory of circuits**

The aim is to analyze the dynamic behavior of linear networks containing models which depend polynomially on a set of parameters. The existence and uniqueness of the solution of such networks is investigated in the case of op-amp. Distributional methods for the analysis of continuous-time and time-invariant linear systems are considered.

Involved scholars: Maurizio Ciampa

Collaborations: Marco Franciosi, M. Poletti