Area Scientifico Disciplinare: Scienze Matematiche

Dario Bini is full professor of numerical analysis since 1986. He is author of about 70 papers, published in international journals and in proceedings of international conferences, and of several books in numerical analysis and computational mathematics. He is member of the editorial board of the international journal "Calcolo" and has a wide and long expertise in the research fields of structured matrix computations, design and analysis of numerical algorithms and polynomial computations. He has organized international conferences on these research topics. He has been advisor of several PhD thesis. His main research interests include, Toeplitz matrix computations, displacement operators, polynomial computations, numerical solution of Markov chains, solution of matrix equations.

1. BINI D.A., GEMIGNANI L., MEINI B. (2001). Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations. NUMERISCHE MATHEMATIK. vol. 89 (2001), no. 1, pp. 49--82.
2. BINI D.A., L. GEMIGNANI, B. MEINI. (2001). Computations with infinite Toeplitz matrices and polynomials. LINEAR ALGEBRA AND ITS APPLICATIONS. vol. 343-344, pp. 21-61.
3. BINI D.A., MEINI B. (1999). Effective methods for solving banded Toeplitz systems. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. vol. 20, pp. 700-719.
4. BINI D.A., B. MEINI. (1998). Inverting block Toeplitz matrices in block Hessenberg form by means of displacement operators: application to queueing problems. LINEAR ALGEBRA AND ITS APPLICATIONS. vol. 272, pp. 1-16.
5. BINI D.A., B. MEINI. (1996). On the solution of a nonlinear matrix equation arising in queueing problems. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. vol. 17, pp. 906-926.

1.7.2 Personale universitario di altre Università

Nº	Cognome	Nome	Università	Dipart./Istituto	Qualifica	Settore scient.	Mesi uomo
							2002	2003
Personale docente:
1	DI FIORE	CARMINE	ROMA "Tor Vergata"	MATEMATICA	Ricercatore	MAT/08	11 (ore: 1507)	7 (ore: 959)
2	FANELLI	STEFANO	ROMA "Tor Vergata"	MATEMATICA	Prof. associato	MAT/08	11 (ore: 1507)	7 (ore: 959)
3	ZELLINI	PAOLO	ROMA "Tor Vergata"	MATEMATICA	Prof. ordinario	MAT/08	11 (ore: 1507)	7 (ore: 959)
Altro personale:

1.7.3 Titolari di assegni di ricerca

Nº	Cognome	Nome	Dipart./Istituto	Anno del titolo	Mesi uomo
					2002	2003

1.7.4 Titolari di borse per Dottorati di Ricerca e ex L. 398/89 art.4 (post-dottorato e specializzazione)

Nº	Cognome	Nome	Dipart./Istituto	Anno del titolo	Mesi uomo
1.	BACCIARDI	ALESSIO	INFORMATICA	2003	11  (ore: 1507)

1.7.5 Personale a contratto da destinare a questo specifico programma

Nº	Qualifica	Costo previsto	Mesi uomo
1.	Laureato	18000	18  (ore: 2475)

1.7.6 Personale extrauniversitario dipendente da altri Enti

Nº	Cognome	Nome	Ente	Qualifica	Mesi uomo
1.	MASTRONARDI	NICOLA	IRMA, CNR, Bari	Primo Ricercatore	16  (ore: 2200)

Analysis of displacement structured matrices with applications

TOEPLITZ MATRICES ; DISPLACEMENT RANK ; STRUCTURED MATRICES ; NUMERICAL SOLUTION OF MARKOV CHAINS ; POLYNOMIAL COMPUTATIONS ; MATRIX EQUATIONS

Large part of important problems in pure and applied mathematics involves structured matrices. The analysis of theoretical and computational properties of these structures is a fundamental step in the design of efficient algorithms.
One of the most frequent structures is the class of Toeplitz matrices. A matrix is Toeplitz if its entries depend on the difference of the two subscripts, i.e., they are constant along the diagonals. This structure reflects the shift invariance properties shared by the mathematical entities which appear in the model. Toeplitz matrices are a particular instance of the wider class of matrices having a displacement structure. Related structures which reflect the specificity of the problem are frequently encountered, say, Pick, Cauchy, Lowener, Frobenius, semiseparable matrices, or matrices defined recursively (tree structures). For multidimensional problems the matrices show a multilevel (multi-index) structure, say block Toeplitz matrices with Toeplitz blocks.
The interest on structured matrices, already alive in the 60's [GS] and in the 70's [Tr], [GoS] with the direct methods for Toeplitz systems and implicitly with the fast Poisson solvers, has increased much its importance in the years due to the work of some active researchers. Most recently a very strong interest has been devoted to several research problems related to structured analysis and computations. A wide literature has been consolidated on different related mathematical problems. Many international research groups have grown around these problems and many international conferences specially devoted to structured matrix analysis and its applications have been organized.
In order to trace the main advances in this field it is worth to cite some books related to this research area, some of which are milestones in this field: the books of Iohvidov [I] and of Heinig and Rost [HR] on Toeplitz and Hankel structures, the book of Bultheel and Van Barel [BvB] on the relations between structures and orthogonal polynomials, the book of Bini and Pan [BP] on the relations between structures and polynomial computations and on more computational issues, the book of Grenander and Szego [GS] and of Boettcher and Silbermann [BS] on asymptotic spectral properties of Toeplitz matrices, the books of Gohberg, Goldberg and Kaashoek [GGK] and of Gohberg Lancaster and Rodman [GLR] on more theoretical issues, the books edited by Kailath and Sayed [KS], by Bini, Tyrtyshnikov and Yalamov [BTY], and by V. Olshevsky [O] which contain very recent advances in the field, and the review paper by Chan and Ng [CN] on Toeplitz preconditioning.
 

Concerning the more specific research subjects of this Research Unit, we cite as starting basis the theory of displacement operators by Kailath et al [KS] and the large literature originated around it. Specifically, the analysis of displacement operators and representation formulae for Toeplitz-like matrices [DZ], [DF],[BZ],[Bo1] and the problems related to structured matrix algebras (trigonometric algebras) [BB],[Bo2],[Bo3],[BD],[BF]. We cite the interplay of structured computations and polynomial computations [BP], and the wide existing literature, among which [B], [BP1], [BP2], [G1-G7]. We cite the spectral theory of Toeplitz matrices [BS], [GS] with the many papers on preconditioning theory and the many applications. All these tools which constitute the "Toeplitz matrix technology" have a fundamental importance in the solution of many problems of interest for this Unit, among wich
 

-- Polynomial computations
-- Factorization and inversion of finite and infinite Toeplitz matrices
-- Numerical solution of problems in queueing theory
-- Solving matrix equations
-- Minimum problems and least squares
 

for which recently we had impressive achievements.
In fact, polynomial computations are strongly related to structured matrices [BP]. In the papers [G1-2], [BP], [BP1], [B], the interplay between structured matrices and polynomials is pointed out and exploited for computational purposes. Problems concerning the location and the approximation of polynomial zeros are solved in terms of LU factorization of Hankel and Bezout matrices over (finite) fields or integral domains [G2-4]. These methods rely on the Schur complement properties of certain structured matrices. These properties are used in [BG1] for the effective computation of the generalized Euclidean algorithm. Methods for approximating polynomial zeros are obtained by reinterpreting the LR algorithm applied to suitable structured matrices [G5-7]. The most recent results in this field concern the problem of factoring polynomials or analytic functions with respect to the unit circle, and the computation of spectral factorizations [BGM] by means of a different formulation of Koenig's theorem given in terms of Toeplitz matrices.
 

Concerning the computation of Wiener-Hopf factorization, Laurent polynomials and bi-infinite Toeplitz matrix inversion, the most recent results are in [BGM1] where the Graeffe method is used for inverting Laurent polynomials. Another set of problems for which there is a solid basis concerns the theory of displacement operators and the concept of approximate displacement rank recently introduced in [BM6], [BM7]. By using this tool, the algorithms for multilevel Toeplitz systems have been designed based on projected Newton's iteration and cyclic reduction. Successful applications to image restoration problems have been shown [BFFM]. Preconditioning techniques have a very wide basis [KS], [FS].
 

Many problems in queueing theory modeled by Markov chains of type MG1, GM1, QBD, BMAP, TREE, (see [Ne], [LR]) are described by Toeplitz matrices which often have additional structures [BMC], [BMR]. The results obtained in [BM1], [BM2], [BM3], [BM4], based on structure analysis, lead to new algorithms which are much faster than the ones available in the literature with a speedup factor of several hundreds. In [M1] the Fast Ramaswami Formula is introduced which allows a strong acceleration on the computation of certain important parameters needed in the analysis of Markov chains, based on the use of Toeplitz computations and FFT. In [B3] the cyclic reduction algorithm is applied and extended to the solution of block tridiagonal block Toeplitz systems and to systems in generalized Hessenberg form. Other methods, based on divide-and-conquer strategies are presented in [BM2].
 

Another important problem, strictly related to queueing models which is encountered in many other applications [HK] and where structured matrices play an important role, concerns the solution of nonlinear matrix equations. In [BM4], [M2] new effective methods based on structure analysis are designed; specific analysis and new solution algorithms are presented in [M2],[M3].
 

Using structured matrices in minimum problems for general functions has been done in [DFFZ], structured total least squares problems are analyzed in [MLV], [LMV].
 

This research Unit has a long tradition in structured matrix analysis and in polynomial and matrix computations. The expertise of the members of the Unit includes theoretical and computational issues, design and analysis of algorithms and software. Some members are very active in certain applications concerning the numerical solution of Markov chains and of matrix equations.

[B] D.Bini, Using FFT-based techniques in polynomial and matrix computations: recent advances and applications. Numer.Funct.Anal.Optim. 21(2000),47-66.
[BB] R.Bevilacqua, E.Bozzo, On algebras of symmetric Loewner matrices. Lin.Alg.Appl. 248(1996),241-251.
[BD] E.Bozzo,C.Di Fiore, On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decomposition. SIMAX 16(1995),312-326.
[BF] D.Bini,P.Favati, On a matrix algebra related to the discrete Hartley transform. SIMAX 14(1993),500-507.
[BFi] D.A.Bini,G.Fiorentino, Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algo. 23(2000),127-173.
[BFFM] D.Bini,A.Farusi,G.Fiorentino,B.Meini, On the regularized solution of block banded block Toeplitz systems, Proc. of SPIE, Vol. 4116, F. T. Luk Editor, 135-146, 2000.
[BG1] D.Bini,L.Gemignani, Fast fraction-free triangularization of Bezoutians with applications to sub-resultant chain computation. Lin.Alg.Appl. 284(1998).
[BGHN] A.Bultheel, et Al, Orthogonal Rational Functions, Cambridge Univ. Press, 1999.
[BGM] D.Bini,L.Gemignani,B.Meini, Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations, Numer. Math.89,2001.
[BGM1] D.Bini,L.Gemignani,B.Meini, Computations with infinite Toeplitz matrices and polynomials, Lin.Alg.Appl., 343,2002.
[BiBo] D.Bini,A.Boettcher, Polynomial factorization through Toeplitz matrix computations. T.R. 1326, 2001 Dip.to di Matematica, Univ. Pisa.
[BM1] D.Bini,B.Meini, Effective methods for solving banded Toeplitz systems. SIMAX. 20(1999),700-719.
[BM2] D.Bini,B.Meini, Inverting block Toeplitz matrices in block Hessenberg form by means of displacement operators: application to queueing problems. Lin.Alg.Appl. 272(1998), 1-16.
[BM3] D.Bini,B.Meini, Improved cyclic reduction for solving queueing problems. Numer.Algo. 15(1997),57-74.
[BM4] D.Bini,B.Meini, On the solution of a nonlinear matrix equation arising in queueing problems. SIMAX 17 (1996),906-926.
[BM5] D.Bini,B.Meini, Solving Block banded block Toeplitz systems with structured blocks: algorithms and applications, in Structured Matrices: Recent Developments in Theory and Computation, D. Bini, E. Tyrtyshnikov and P. Yalamov Editors, Nova Science Inc., New York, 2001.
[BM6] D.Bini,B.Meini, Approximate displacement rank and applications, Contemporary Mathematics, 281, AMS 2002.
[BMC] D.Bini,B.Meini, S.Chakravarthy, Control of the BMAP/PH/1/K queue with group services, in Advances in Algorithmic Methods for Stochastic Models, G. Latouche and P. Taylor eds., Notable Publications, 2000, 57-72.
[BMR] D.Bini,B.Meini, V.Ramaswami, Analyzing M/G/1 paradigms through QBDs: the role of the block structure in computing the matrix G, in Adv. in Alg. Methods for Stoch. Models, Proc. G. Latouche and P. Taylor eds., Notable Publications, 2000, 73-86.
[Bo1] E.Bozzo, A note on matrix displacement representation. Int. Eq. Op. Th. 29 (1997)368-372.
[Bo2] E.Bozzo, Algebras in matrix spaces with displacement structure. Lin. and Mult.Alg. 42(1997),23-41.
[Bo3] E.Bozzo, Algebras of higher dimension for displacement decompositions and computations with Toeplitz plus Hankel matrices. Lin.Alg.Appl. 230(1995),127-150.
[BP] D.Bini,V.Pan. Polynomial and matrix computations. Vol. 1. Birkhäuser Boston, MA, 1994
[BP1] D.Bini,V.Pan, Graeffe's, Chebyshev-like, and Cardinal's processes for splitting a polynomial into factors. J. Complexity 12 (1996),492-511.
[BP2] D.Bini,V.Pan, Improved parallel computations with Toeplitz-like and Hankel-like matrices. Lin.Alg.Appl. 188/189 (1993), 3-29.
[BS] A.Boettcher, B.Silbermann, Introduction to Large Truncated Toeplitz Matrices, Universitext, Springer-Verlag, New York 1999.
[BTY] D.Bini,E.Tyrtyshnikov,P.Yalamov, Structured Matrices: Recent Developments in Theory and Computation, Nova Science Inc. N.Y. 2001
[BvB] A.Bultheel,M.Van Barel, Linear algebra, rational approximation and orthogonal polynomials. North-Holland, Amsterdam, 1997.
[BZ] R.Bevilacqua,P.Zellini, Structure of algebras of commutative matrices. Lin.Alg.Appl. 233(1996), 5-41.
[CN] R.Chan, M.Ng, Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38(1996), 427-482
[Da] K.Datta, Numerical Linear Algebra and Applications. 1995, Brooks/Cole Publ.
[DF] C.DiFiore, Matrix algebras in displacement decomposition. SIMAX 2000.
[DFFZ] C.DiFiore,S.Fanelli,P.Zellini, Matrix algebras in quasi-Newtonian algorithms for optimal learning in multi-layer perceptrons. Proc. ICONIP '99, Dunedin, New Zeeland, 1999.
[DZ]C.DiFiore,P.Zellini, Matrix displacement decomposition and applications to Toeplitz linear systems. Lin.Alg.Appl. 268(1998)
[FS] G.Fiorentino,S.Serra, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions SIAM J. Sci Comp 17(1996).
[G1] L.Gemignani, A hybrid approach to the computation of the inertia of a parametric family of Bezoutians with application to some stability problems for bivariate polynomials. Lin.Alg.Appl. 283(1998),221-238.
[G2] L.Gemignani, Computing a factor of a polynomial by means of multishift LR algorithms. SIMAX 19(1998), 161-181
[G3] L.Gemignani, Schur complements of Bezoutians and the inversion of block Hankel and block Toeplitz matrices. Lin.Alg.Appl. 253(1997),39-59.
[G4] L.Gemignani, Computationally efficient applications of the Euclidean algorithm to zero location. Lin.Alg.Appl. 249(1996),79-91.
[G5] L.Gemignani, Polynomial root computation by means of the LR algorithm. BIT 37(1997),333-345.
[G6] L.Gemignani, Computing a factor of a polynomial by means of multishift LR algorithms. SIMAX 19(1998),161-181
[G7] L.Gemignani, Computing a Hurwitz factorization of a polynomial. J. Comput.Appl.Math. 126(2000),369-380.
[GLR] I.Gohberg,P.Lancaster,L.Rodman, Matrix polynomials. Academic Press, Inc. New York, 1982.
[GoS] I.Gohberg,A.Semencul, The inversion of finite Toeplitz matrices and their continual analogues. Mat. Issled. 7 (1972), 201-223.
[GGK] I.Gohberg, S.Goldberg, M.Kaashoek. Classes of linear operators. Vol. I, Birkhäuser, Basel, 1990.
[GS] U.Grenander,G.Szegö, Toeplitz forms and their applications. University of California Press, Berkeley-Los Angeles 1958.
[HK] N.Higham,H.Kim, Numerical analysis of a quadratic matrix equation, IMA, J. Num.Anal. 2000.
[HR] G.Heinig,K.Rost, Algebraic methods for Toeplitz-like matrices and operators. Birkhäuser, Basel, 1984.
[I] I.Iohvidov, Hankel and Toeplitz matrices and forms. Algebraic theory. Birkhäuser, Boston, Mass., 1982.
[KS] T.Kailath.A.Sayed Eds., Fast Reliable Methods for Matrices with Structure, SIAM Philadelphia 1999.
[KS1] T.Kailath,A.Sayed, Displacement structure: Theory and Applications, SIAM Review 37, 1995.
[LR] G.Latouche,V.Ramaswami, Introduction to matrix analytic methods in stochastic modeling. SIAM, Philadelphia, 1999.
[LMV] P.Lemmerling,N.Mastronardi,S. Van Huffel, Fast algorithm for Solving the Hankel/Toeplitz Structured Total Least Squares Problem, Numer. Algo, 23,371-392,2000.
[M1] B.Meini An improved FFT-based version of Ramaswami's formula. Comm. Statist. Stoch. Models, 13:223-238, 1997.
[M2] B.Meini, Matrix equations and structures: efficient solution of special discrete algebraic Riccati equations, Proc, of the WLSSC00, Bulgaria, 2000.
[M3] B.Meini, Efficient computation of the extreme solutions of X+A^* X^{-1}A=Q and X-A^*X^{-1}A=Q, Math. of Comp., 2001.
[MLV] N.Mastronardi,P.Lemmerling,S. Van Huffel, Fast structured total least squares algorithm for solving the basic deconvolution problem, SIMAX 22,533-553,2000.
[Ne] M.Neuts, Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, Inc., New York, 1989.
[O] V.Olshevsky Ed., Structured Matrices in Mathematics, Computer Science, and Engineering II, Contemporary Mathematics 281, 2001.
[Tr] W.Trench. An algorithm for the inversion of finite Toeplitz matrices. J. Soc. Indust. Appl. Math. 12 1964 515-522.

The design and analysis of algorithms for the numerical treatment of structured matrices is a fundamental part of large scale scientific computing. The achievements reached in the design of algorithms based on structured matrix analysis in the solution of certain applied problems are dramatic. Besides the theoretical aspects of the theoretical analysis of structures, the algorithmic and applicative interest has a strong relevance. In fact, matrices endowed with a structure are encountered in many important problems like the analysis of queueing models (telecommunications, internet, performance analysis, stochastic processes) [LR], numerical solution of integral and differential equations, signal and image processing, inverse problems, time series analysis, polynomial computations and computer algebra, matrix equations, orthogonal polynomials, just to quote a few ([BP], [BTY], [BvB], [KS], [Ne], [LR], [Da]).

The main goals of this research Unit are
 

-- The analysis of algebraic, structural, analytic, spectral and computational properties of Toeplitz-like matrices finite, semi-infinite and bi-infinite which intervene in theoretical and applied mathematics (including Hankel-like matrices, Cauchy-like, multilevel matrices having scalar entries or with blocks, matrices recursively defined).

-- The exploitation of the properties obtained in this way for creating general methodologies for the design and analysis of effective solution algorithms, with special attention devoted to applications and large scale scientific computing.

--The implementation with prototype software of the algorithms designed in this research and their experimental validation.

In particular we wish to design and analyze efficient algorithms for the following classes of problems.

1) Polynomial computations. We continue the research of [BGM], [BGM1], in particular concerning the computation of the Wiener-Hopf factorization of Laurent polynomials with scalar and with matrix coefficients; computation of the Laurent series (if it exists) reciprocal of a given Laurent polynomial with scalar/matrix coefficients; application to factoring a polynomial with respect to the unit circle and the imaginary axis; computation of the reciprocal of multivariate polynomials (if it exists). We continue the analysis of evaluation/interpolation methods along the line of [BiBo], and of Wilson's method. We wish to investigate on the relations between the Graeffe method for polynomials and the cyclic reduction method (as developed in [BGM1], [BGM], [BM3]) and the possible extension to matrix polynomials. We wish to investigate on the relations between the problem of polynomial factorization and solving matrix equations (see next item 3) and to analyze methods based on functional iterations. Applications of the techniques of polynomial factorization to the solution of linear systems with a Sylvester-like matrix with scalar/block entries. Design of methods for computing QR factorizations of Cauchy-like matrices pointing out their interplay with direct and inverse spectral problems concerning semi-separable matrices and with constructing set of orthogonal rational functions (part of this research is in collaboration with the Unit 2). This kind of functions arise in generalized moment problems and in rational interpolation [BGHN]
 

2) Factorization of Toeplitz matrices with scalar/block entries. We apply the result of item 1 above and of [BGM], [BGM1], [Goodman et Al., Adv. Comput. Math. 7 (1997)], to the inversion of semi-infinite and bi-infinite Toeplitz matrices with scalar/block entries, with a multi-level structure (in collaboration with the Unit 3).
We continue the analysis of [G7] for the design and analysis of efficient methods for the factorization of Toeplitz-like matrices with entries over rings, with possible applications to problems in computer graphics (intersection of surfaces and curves).
 

3) Solution of matrix equations. We consider equations of the kind Riccati (XAX+BX+XC+D=0), of polynomial type (A_0+A_1X+A_2X^2+...+A_nX^n=0), or defined by a power series (A_0+A_1X+A_2X^2+...=0), in the line of [BM4], [M2], [HK]; we will apply deflation methods for accelerating the convergence of known methods, we will design new iterations based on the properties of Frobenius matrices. We analyze the Ramaswami reduction to transform an equation defined by a power series to a polynomial equation. We will analyze the relations between solving certain matrix equations and computing polynomial factors.
 

4) Solution of structured linear systems with finite size. Analysis of the properties of the inverses of bi-infinite (multilevel) Toeplitz matrices to be used as preconditioners in the solution of finite systems with the PCG technique (in collaboration with the Unit 3). Analysis of multigrid methods for (multi-level) Toeplitz matrices along the line of [FS] (in collaboration with the Unit 2). Analysis and exploitation of the properties of the approximate displacement rank of [BM6] in the framework of the inversion of Toeplitz matrices with Newton's iteration. Analysis of fast and stable algorithms for the factorization of Diagonal plus Semi-Separable matrices and for the solution of related systems.
 

5) Queueing theory problems. We wish to perform the analysis of the structures encountered in the model of the type QBD, BMAP, PH1, NonSkipFree and Tree (see [BMC], [BMR], [Ne], [LR]) where different structures at the different levels generally occur (say, outer structure Toeplitz, inner structure Frobenius or Kronecker) in order to design more effective methods along the line of [BCM], [BMR]. Moreover, related to the previous item 3, we wish to investigate specific matrix equations arising from queueing models.
 

6) Unconstrained minimization problems and structured total least squares. Matrix algebras have been investigated in the Hessian updating formulas of quasi-Newton methods for the unconstrained minimization of general nonlinear functions of n variables [DFFZ]. The novel methods, named LQN, have an O(nlogn) complexity per step and require only O(n) memory allocations, by maintaining a quasi-newtonian rate of convergence. The purpose of the research in this field is to deepen the study of the novel minimization LQN methods of [DFFZ].
In particular, it is reasonable that, by generalizing classical Powell's techniques, a general convergence theorem can be achieved. Furthermore, experimental data suggest that a superlinear convergence result could be obtained for a suitable subclass of LQN-methods. Applications of these results will be addressed to the design of efficient algorithms for optimal learning on supervised neural networks (multi-layer perceptrons and recurrent networks). Analysis of stable and fast algorithms for solving structured total least squares problems.
 

7) Implementation of prototype software. The algorithms obtained in our research will be implemented and numerically tested on meaningful problems with different software environments, either more oriented to experimentation (Mathematica, Matlab) or involving more efficient languages (Fortran, C). We wish to develop some software for the automatic recursive factorization of polynomials based on multiprecision arithmetic in the line developed in [BFi] and based on the results of [BGM1]. For both the tasks we need to create a network of fast workstations, accessible from the researcher of the other units, so that all the software and the data physically stay in the same place and can be easily used and elaborated by all the researchers of the project. We wish to collect a set of test matrices for testing the performance of our algorithms and organizing all the information in web pages with links to the software, benchmarks, and main results.