### Preprints

This section contains preprints that are currently submitted to a peer-reviewed journal and/or theses and other papers that are not published. Some notes that have not (yet?) submitted to journals can be on this list. Most of the following items are expected to migrate to the next section at some point in time.

- Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions,

Negri Porzio, G. M, Noferini, V., R. L, arXiv preprint arXiv:2107.07982, 2021.

### Book(s)

- Core-Chasing Algorithms for the Eigenvalue problem

Aurentz J, L., Mach T., R. L., Vandebril R., Watkins D. S, SIAM, Fundamentals of Algorithms series, July 2018.

### Papers

The following papers, listed in reverse chronological order, are published (or accepted for publication) in a journal.

- Computing eigenvalues of semi-infinite quasi-Toeplitz matrices,

Bini, D. A., Iannazzo, B., Meini, B., Meng, J., R. L., to appear on Numerical Algorithms, 2022. - Hierarchical adaptive low-rank format with applications to discretized PDEs,

Massei, S., R. L., Kressner, D., Numerical Linear Algebra with Applications, 2022. - Mixed precision recursive block diagonalization for bivariate functions of matrices,

Massei, S., R. L., SIAM Journal on Matrix Analysis and Applications, 2022. - Rank-structured QR for Chebyshev rootfinding,

Casulli, A., R. L., SIAM Journal on Matrix Analysis and Applications, 2021 – DOI: 10.1137/20M1375115 - Structured backward errors in linearizations,

Noferini, V., R. L., Vandebril, R., ETNA, 2021- DOI: 10.1553/etna_vol54s420. - Sampling the eigenvalues of random orthogonal and unitary matrices

Fasi, M., R. L., Linear Algebra and Its Applications, 2021 – DOI: 10.1016/j.laa.2021.02.031. - Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Massei S., R. L., BIT Numerical Mathematics, 2020 – DOI: 10.1007/s10543-020-00826-z. - A computational framework for two-dimensional random walks with restarts, Bini, D. A., Massei S., Meini, B., R. L., SIAM Journal on Scientific Computing, 2020 – DOI: 10.1137/19M1304362.
- Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations, R. L., Linear Algebra and its Applications, 2020 – DOI: 10.1016/j.laa.2020.06.013.
- hm-toolbox: Matlab software for HODLR and HSS matrices

Kressner D., Massei S., R. L., SIAM Journal on Scientific Computing, 2020 – DOI: 10.1137/19M1288048. - Finite element model updating for structural applications

Girardi M., Padovani C., Pellegrini D., Porcelli M., R. L., Journal of Computational and Applied Mathematics, 2020 – DOI: 10.1016/j.cam.2019.112675. - Computing performability measures in Markov chains by means of matrix functions

Masetti G., R. L., Journal of Computational and Applied Mathematics, 2020, DOI: 10.1016/j.cam.2019.112534. - When is a matrix unitary or Hermitian plus low rank?

Del Corso G., Poloni F., R. L., Vandebril R., Numerical Linear Algebra with Applications, 2019 – DOI: 10.1002/nla.2266. - Nonsingular systems of generalized Sylvester equations: an algorithmic approach

De Terán F., Iannazzo B., Poloni F., and R. L., Numerical Linear Algebra with Applications, 2019. – DOI: 10.1002/nla.2261. - Fast solvers for 2D fractional diffusion equations using rank structured matrices

Mazza M., Massei S., R. L., SIAM Journal on Scientific Computing, 2019 – DOI: 10.1137/18M1180803. - Low-rank updates and a divide-and-conquer method for linear matrix equations

Kressner D., Massei S., and R. L., SIAM Journal on Scientific Computing, 41(2), 2019, DOI: 10.1137/17M1161038. - Fast and backward stable computation of the eigenvalues of matrix polynomials

Aurentz J. L., Mach T., R. L., Vandebril R., Watkins D. S., Mathematics of Computation, 2019 – DOI: 10.1090/mcom/3338 - Factoring block Fiedler Companion Matrices

Del Corso G. M., Poloni F., R. L., Vandebril R., chapter in Structured Matrices in Numerical Linear Algebra: Analysis, Algorithms and Applications, Springer INdAM Series, 2019 – DOI: 10.1007/978-3-030-04088-8_7. - Solving rank structured Sylvester and Lyapunov equations

Massei S., Palitta D., and R. L., SIAM Journal on Matrix Analysis and Applications, 39(4), 2018 – DOI: 10.1137/17M1157155. - Fast and backward stable computation of roots of polynomials, Part II: backward error analysis; companion matrix and companion pencil

Aurentz J. L., Mach T., R. L., Vandebril R., Watkins D. S., SIAM Journal on Matrix Analysis and Applications, 2018 – DOI: 10.1137/17M1152802 - Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Bini D. A., Massei S., and R. L., to appear in Numerical Algorithms, 2018 – DOI: 10.1007/s11075-018-0571-6 - On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

Bini D. A., Massei S., Meini B., R. L., Numerical Linear Algebra with Applications, 2018 – DOI: 10.1002/nla.2128 - Solvability and uniqueness criteria for generalized Sylvester-type equations

De Terán F., Iannazzo B., Poloni F., and R. L., Linear Algebra and its Applications, 2018 – DOI: 10.1016/j.laa.2017.07.010 - Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

R. L., and Vandebril R., Linear Algebra and its Applications, 2018 – DOI: 10.1016/j.laa.2017.05.010 - Fast Hessenberg reduction of some rank structured matrices

Gemignani L., R. L., SIAM Journal on Matrix Analysis and Applications, 2017 – DOI: 10.1137/16M1107851 - On the decay of the off-diagonal singular values in cyclic reduction

Bini D. A., Massei S. and R. L., Linear Algebra and its Applications, 2017 – DOI: 10.1016/j.laa.2016.12.027 - Decay bounds for the numerical quasiseparable preservation in matrix functions

Massei S. and R. L., Linear Algebra and its Applications, 2017 – DOI: 10.1016/j.laa.2016.11.041 - A framework for structured linearizations of matrix polynomials in various bases

R. L., Vandebril R. and Van Dooren P., SIAM Journal on Matrix Analysis and Applications, 2017 – DOI: 10.1137/16M106296X - Efficient cyclic reduction for Quasi-Birth-Death problems with rank structured blocks

Bini D. A. and Massei S. and R. L., Applied Numerical Mathematics, 2017 – DOI: 10.1016/j.apnum.2016.06.014. - On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial,

Bini D. A. and R. L., Linear Algebra and Its Applications, 2016 – DOI: 10.1016/j.laa.2015.07.017. - Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications,

Bini D. A. and R. L., Linear Algebra and Its Applications, 2016 – DOI: 10.1016/j.laa.2015.08.026. - Solving secular and polynomial equations: A multiprecision algorithm,

Bini D. A. and R. L., Journal of Computational and Applied Mathematics, 2014 – DOI: 10.1016/j.cam.2013.04.037.

### Theses

My PhD and master theses.

- Exploiting rank structures for the numerical treatment of matrix polynomials,

Robol, L. My PhD thesis that I defended in November 2015, under the supervision of prof. Dario A. Bini, 2015. - A rootfinding algorithm for polynomials and secular equations,

Robol, L. – My master thesis on polynomial rootfinding at arbitrary precision, 2012. Alternative link through ETD.

## Software

- MPSolve is an open source package that approximates roots of polynomials with arbitrary precision. The package can solve polynomials represented in different basis as well as secular equations.
**News**: MPSolve is also available on Android. You can check it out on the Play Store.