The group of Numerical Analysis covers research topics from Numerical Linear Algebra, such as lowrank approximation of matrices and tensors, the numerical solution of matrix equations, the computation of matrix functions, polynomial rootfinding, and the theory of circuits. Details on these topics can be found below.
The group works in close collaboration with the Numerical Analysis group at the Department of Computer Science of the University of Pisa (G. Del Corso, L. Gemignani, F. Poloni), and at Scuola Normale Superiore (M. Benzi).
The group runs a Seminar Series: further information can be found at this link.
Research Topics
Lowrank Approximation and Structured Matrices
This research area deal with the problem of efficiently finding lowrank factorization of matrices $A$ in the form $$A \approx UV^T,$$ where $U,V$ are tall and skinny matrices. In most cases of interest, one assumes to either be able to evaluate $X \mapsto AX$ and $Y \mapsto A^T Y$ or to sample selected entries of $A$ at a reduced cost. Several strategies may be adopted for retrieving matrices $U,V$ such as randomized linear algebra methods, or crossapproximation techniques.
A natural evolution of this idea is to consider structured matrices, which are – in general – not lowrank, but may have lowrank blocks, or obtained as lowrank perturbations of particularly structured matrices. The aim is to represent such matrices using $\mathcal O(n)$ or even $O(1)$ storage and perform operations with a similar complexity (up to logarithmic factors in the dimension).
In particular, the following structures are considered:
 Rankstructured matrices: Matrices with offdiagonal blocks of lowrank (HODLR), which may be efficiently representable using hierarchically structured bases (HSS).
 Lowrank perturbations of Toeplitz matrices: matrices with this structure form an algebra, which has important applications in the study of queues and Markov chains. See also the section on the numerical solution of Markov chains below. Matrices in this class can often be represented with a storage that is independent of the dimensions, enabling the numerical study of infinite and semiinfinite matrices.
For matrices in this form, efficient arithmetic operations are available. MATLAB toolboxes for handling such matrices are available (see hmtoolbox for rankstructured matrices and cqttoolbox for quasiToeplitz ones).
Members
Numerical Methods for Polynomial RootFinding
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 for this kind of numerical tool 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.
Various numerical methods and tools are available to solve this problem and a huge literature exists. Among the classical choices, there are matrix methods, like the QR iteration applied to companion matrices, and functional iteration techniques (often based on the Newton method). In the latter class, an important method is Aberth’s iteration, a global version of the Newton iteration, that allows approximating all the roots of $p(x)$ simultaneously.
In the year 2000, we have produced the package MPSolve in the framework of the European project FRISCO. This package has been improved recently and is the fastest available software for polynomial rootfinding available so far. Just to give an example, the software can solve the Mandelbrot polynomials of degree $2^{20}$ in a few hours over a dual Xeon server. Polynomials of degree $2^{21}$ and $2^{22}$ can be solved in a few days and in a few weeks, respectively. Mandelbrot polynomials have zeros that coincide with the cycles of a given length in the Mandelbrot iteration. These zeros are severely illconditioned.
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Collaborators
Matrix Polynomials, Companion Linearizations, and Quasiseparable Matrices
An $m \times m$ matrix polynomial $A(x)$ is a polynomial in the variable $x$ whose coefficients are matrices. Matrix polynomials are encountered in many applications; an important computational problem is to compute the values of $x$ such that $\det(A(x)) = 0$. This problem, which is related to the analysis of eigenfrequencies of complex dynamical systems, is customarily reduced to solving a generalized linear eigenvalue problem of the kind $Hv = \lambda Kv$ where the $nm \times nm$ pencil $H \lambda K$ provides a linearization of $A(x)$.
Here, the research concerns the design and analysis of linearizations that have nice computational properties and keep the condition number of the eigenvalues as small as possible. The literature in this area is very rich with many theoretical and computational results.
One nice feature is that the known linearization shares the quasiseparable property. That is, for any, $\lambda$ the submatrices of $H – \lambda K$ which are contained in the upper or in the lower triangular part of the matrix have a low rank. This property has been investigated in a different context and is exploitable to a certain extent for designing highly efficient algorithms.
In this research, we aim both to determine new and more effective linearizations and to design efficient algorithms for solving the linear pencil by relying on the quasiseparable structure. We also aim to combine analytic techniques, like the Aberth iteration, and matrix techniques to improve the efficiency of solution algorithms. Other related researches concern the localization of the eigenvalues of matrix polynomials.
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Collaborators
Matrix Equations and Matrix Functions
Many problems from the real world and from Scientific Computing are modeled by matrix equations or by matrix functions. For instance, the celebrated algebraic Riccati equation which in the continuoustime models takes the form $XBX+AX+XD+C=0$, is related to the analysis of the stability of dynamical systems. Here, $A,B,C,D$ are given matrices of compatible sizes and $X$ is the unknown matrix. Quadratic equations like $AX^2+BX+C=0$ model damped vibration problems as well as stochastic models encountered in the analysis of queues. In the case of queues of the M/G/1 type the equations take the form $\sum_{i={1}}^\infty A_i X^{i+1} = 0$ where a matrix analytic function over a suitable domain is involved.
The goal of the research in this area is to develop tools for designing fast and effective algorithms to solve this kind of equation. These equations, as well as similar ones, can be recast as generalized eigenvalue problems; for instance, the latter is related to finding $\lambda\in\mathbb C$ and $x\in \mathbb{C}^n$ such that $(\lambda^2 P+\lambda Q+R)x=0$ (quadratic eigenvalue problem). The techniques needed to combine the ones used for general nonlinear equations (multivariate Newton methods, fixedpoint iterations) and eigenvalue problems (Schur decompositions, orthogonal reductions, rational approximations).
Matrix structures (such as entrywise nonnegativity, symmetry, and symplecticity) play a crucial role in all of this; they are needed for defining the solutions of these equations in the first place, and to ensure feasibility, accuracy, and computational efficiency of the numerical algorithms.
Applications of these equations include control and system theory, queuing theory and structured Markov chain modeling in applied probability, and time series estimation in statistics. It is useful to interface directly with researchers working in these application fields: the different points of view provide useful insight, and the algorithms can be better tailored to the needs of the practitioners.
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Collaborators
Matrix Geometric Means
In several applications, different sets of measurements produce different symmetric positivedefinite matrices as a result; a natural problem is finding the most plausible correct value for the desired matrix. This corresponds to a form of averaging. The plain arithmetic mean $\frac{1}{n} (X_1 + X_2 + \ldots + X_n)$ is not always the best choice from this point of view.
The concept of the geometric mean of a set of positive numbers can be extended to the case of a set of positive definite matrices. However, this extension is not so trivial and, while for the case of two matrices there is a unique definition, in the case of several matrices there exists an infinite number of valid definitions.
The most general setting under which to study this problem is the one of Riemannian geometry: one gives a Riemannian scalar product on the manifold of symmetric positivedefinite matrices and studies the point which minimizes the sum of squared distances from the given matrices (Cartan mean). This gives rise to a mean which is compatible in some sense with matrix inversions, much like the geometric mean in the scalar case.
In addition to the theoretical aspects, practical computation of these means is an interesting problem, a special case of optimization on manifolds. The classical multivariate methods from optimization need to be adapted to work on a generic manifold; one needs to consider the role of its tangent space and construct suitable maps to and from it.
This kind of mean is of great importance in applications, especially in engineering and in problems of radar detection.
Our goal is to provide a better understanding of the different concepts involved in this research, new definitions of mean which are better suited for the applicative models, and faster and more reliable algorithms for their computation.
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Collaborators
Numerical Solution of Markov Chains
Many problems from the applications are modeled by Markov chains. Very often the set of the states is huge or even infinite. In these cases, customary techniques are not suited to solve this kind of problems.
The goal of this research area is the design and analysis of effective solution algorithms for infinite Markov chains, with special attention to the ones coming from queuing models. This goal is reached by developing theoretical tools relying on complex analysis, numerical analysis, and structured matrix computations.
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Collaborators
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 are investigated in the case of opamp. Distributional methods for the analysis of continuoustime and timeinvariant linear systems are considered.
Members
Collaborators
People
Faculty
Name  Surname  Personal Card  

Paola  Boito  paola.boito@unipi.it  
Maurizio  Ciampa  maurizio.ciampa@unipi.it  
Fabio  Durastante  fabio.durastante@unipi.it  
Stefano  Massei  stefano.massei@unipi.it  
Beatrice  Meini  beatrice.meini@unipi.it  
Cecilia  Pagliantini  cecilia.pagliantini@unipi.it  
Leonardo  Robol  leonardo.robol@unipi.it 
Affiliate Members
Name  Surname  Personal Card  

Dario Andrea  Bini  dario.bini@unipi.it  
Sergio  Steffè  sergio.steffe@unipi.it 
Postdoctoral Fellows
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Ph.D. Students at the University of Pisa
Name  Surname  Personal Card  

Alberto  Bucci  alberto.bucci@phd.unipi.it 
Ph.D. Students at other institutions
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Ph.D. Theses supervised by members of the group
awarded by the University of Pisa
Year  Name  Surname  Title of the Thesis  Supervisor(s) 

2012  Vanni  Noferini  Polynomial Eigenproblems: a RootFinding Approach  Luca Gemignani and Dario Andrea Bini 
2007  Bruno  Iannazzo  Numerical Solution of Certain Nonlinear Matrix Equations  Dario Andrea Bini 
2004  Manuela  Bagnasco  Il metodo QR per matrici semiseparabili: aspetti teorici e computazionali  Dario Andrea Bini 
1998  Beatrice  Meini  Fast Algorithms For The Numerical Solution of Structured Markov Chains  Dario Andrea Bini 
1997  Giuseppe  Fiorentino  Tau Matrices and Generating Functions for Solving Toeplitz Systems  Dario Andrea Bini 
1994  Enrico  Bozzo  Matrix Algebras and Discrete Transforms  Dario Andrea Bini 
awarded by another institution
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Grants
Current
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Past

Analisi di reti complesse: dalla teoria alle applicazioni (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Principal Investigator: Federico Giovanni Poloni
Project period: Jul 07, 2020 – Dec 31, 2022

Metodi lowrank per problemi di algebra lineare con struttura datasparse (Progetto Giovani GNCS)
Principal Investigator: Leonardo Robol
Project period: Mar 09, 2020 – Dec 31, 2021
Visitors
Prospective
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Current
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Grouped by year
2023
First Name  Last Name  Affiliation  From  To 

Aleksey Yordanov  Nikolov  Technical University of Sofia  Feb 04, 2023  Mar 04, 2023 
Stoyan  Popov  Technical University of Sofia  Feb 04, 2023  Mar 04, 2023 
2022
First Name  Last Name  Affiliation  From  To 

Gianluca  Ceruti  École polytechnique fédérale de Lausanne  Dec 12, 2022  Dec 17, 2022 
Bruno  Iannazzo  Università di Perugia  Dec 09, 2022  Dec 12, 2022 
Stefano  Massei  Eindhoven University of Technology  Feb 07, 2022  Mar 06, 2022 
Mariarosa  Mazza  Università dell'Insubria  Nov 28, 2022  Dec 01, 2022 
Aaron  Melman  Santa Clara University  Nov 09, 2022  Nov 10, 2022 