Department of Mathematics
University of Pisa

 Numerical Analysis  and Computational Mathematics

Luca Gemignani
Research activity:
 
  The common feature of the research activities is the development of mathematical tools for the design, analysis and implementation of algorithms for the efficient solution of computational problems. Particular interest is devoted to the analysis of structures, to their exploitation and to the complexity analysis of problems in both sequential and parallel model of computation. A special attention is addressed to numerical issues and to problems arising from applications.  Specific topics are: Papers:
  1. On the Euclidean scheme for polynomials having interlaced real zeros, Proceedings of 2-nd Annual ACM Symposium on Parallel Algorithms and Architectures, Crete, 1990, 254--258 (with D. Bini).

    2.  
  2. Improved parallel computations with matrices and polynomials, Proc. 18--th Intern. Symposium on Automata, Languages and Programming, Lectures Notes in Computer Science, 510, 520-531, Springer  1991 (with D. Bini and V. Pan).

    3.  
  3. Computing the inertia of Bezout and Hankel Matrices, CALCOLO 28 (1991) 267--274.

    4.  
  4. On the complexity of polynomial zeros, SIAM J. Comput. 21 (1992), 781--799 (with D. Bini).

    5.  
  5. Fast inversion of Hankel and Toeplitz matrices, Information Processing Letters 41 (1992) 119--123.

    6.  
  6. Rational interpolation via orthogonal polynomials, Computers Math. Appl. 5 (1993) 27--34.

    7.  
  7. Iteration schemes for the divide-and-conquer eigenvalue solver, Numer. Math. 67 (1994), 403--425 (with D. Bini).

    8.  
  8. Solving Hankel systems over the integers, J. Symbolic Computation 18 (1994) 403--425.

    9.  
  9. Fast parallel computation of the polynomial remainder sequence via Bezout and Hankel matrices, SIAM J. Comput., 24 (1995), 63--77 (with D. Bini).

    10.  
  10. Computationally efficient applications of the Euclidean algorithm to zero location, Linear Algebra Appl. 249 (1996) 79--91.

    11.  
  11. A fast iterative Method for determining the stability of a polynomial, J. Comput. Appl. Math. 76 (1996) 1--11.

    12.  
  12. Manipulating polynomial in generalized form , TR-96/14, Computer Science Department, University of Pisa, (1996).

    13.  
  13. Fast and stable computation of the barycentric representation of rational interpolants, CALCOLO 33 (1996) 371--388.

    14.  
  14. Polynomial root computation by means of the LR algorithm, BIT 37 (1997), 333--345.

    15.  
  15. Chebyshev rational interpolation, Numerical Algorithms 15 (1997) 91--110.

    16.  
  16. Schur complements of Bezoutians with applications to the inversion of block Hankel and Toeplitz matrices, Linear Algebra Appl. 253 (1997) 39--59.

    17.  
  17. A fast algorithm for generalized Hankel matrices arising in finite moment problems, Linear Algebra Appl. 267 (1997) 41--52.

    18.  
  18. GCD of polynomials and Bezout matrices, Proc of the ACM International Symposium on Symbolic and Algebraic Computations, Maui, Hawaii, U.S.A., 1997.

    19.  
  19. Computing a factor of a polynomial by means of multishift LR algorithms , SIAM J. Matrix Anal. Appl. 19 (1998) 161--181.

    20.  
  20. Fast fraction-free triangularization of Bezoutians with applications to sub-resultant chain computation.  Linear Algebra Appl., 284 (1998) 19--39 (with D. Bini).

    21.  
  21. A hybrid approach to the computation of the inertia of a parametric family of Bezoutians with applications to some stability problems for bivariate polynomials.  Linear Algebra Appl., 283 (1998) 221--238.

    22.  
  22. Fast QR factorization of low-rank changes of Vandermonde-like matrices, CALCOLO, 36 (1999) 1--15.

    23.  
  23. Computing a Hurwitz factorization of a polynomial. J. Comput. Appl. Math. , 126 (2000) 369--380.

    24.  
  24. Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations. Numer. Math. , 89 (2001) 49--82, (with  D. Bini and B. Meini).

    25.  
  25. Efficient and stable solution of structured Hessenberg linear systems arising from difference equations. Numer. Linear Algebra Appl., 7 (2000) 319--335.

    26.  
  26. Polinomial factors, lemniscates and structured matrix technology . In "Structured matrices: recent developments in theory and computation", D. A. Bini, P. Yalamov and E. Tyrtyshnikov editors, NOVA Science, 2001.

    27.  
  27. On a generalization of Poincare`'s theorem for matrix difference equations arising from root-finding problems . In V. Olshevsky, editor, "Structured Matrices in Mathematics, Computer Science, and Engineering II", Contemporary Mathematics (AMS) Vol. 280-281, 2001.

    28.  
  28. Efficient and stable solution of M-matrix linear systems of (block) Hessenberg form. SIAM J. Matrix Anal. Appl., 24 , (2003) 852--876, (with G. Lotti).

    29.  
  29. A numerical approach to the solution of stable resultant linear systems. Riv. Mat. Univ. Parma , 6 , (2000) 259--291.

    30.  
  30. Computations with infinite Toeplitz matrices and polynomials Linear Algebra Appl. , 343--344 , (2002) 21--61, (with  D. Bini and B. Meini).

    31.  
  31. A generalized Graeffe's iteration for evaluating polynomials and rational functions , 2001. Proceedings of the ACM International Symposium on Symbolic and Algebraic Computations, ISSAC 2001.

    32.  
  32. A Lanczos-type algorithm for the QR factorization of regular Cauchy matrices, Numer. Linear Algebra Appl., 9 , (2002) 305--319, (with  D. Fasino).

    33.  
  33. Structural and computational properties of possibly singular semiseparable matrices, Linear Algebra Appl. , 340 , (2002) 183--198, (with  D. Fasino).

    34.  
  34. A superfast solver for Sylvester's resultant matrices generated by a stable and an anti-stable polynomial. Linear Algebra Appl. , 366 , (2003) 233--255.

    35.  
  35. Direct and inverse eigenvalue problems for diagonal-plus-semiseparable matrices. Numerical Algorithms, , 34 , (2003) 313--324, (with D. Fasino).

    36.  
  36. Effective fast algorithms for polynomial spectral factorization. Numerical Algorithms, , 34 , (2003) 217--227 (with D. A. Bini, G. Fiorentino and B. Meini).

    37.  
  37. Solving certain matrix equations by means of Toeplitz computations: algorithms and applications. Contemporary Mathematics, vol. 323, 2003, 151--167, (with D. A. Bini and B. Meini).

    38.  
  38. A Lanczos-type Algorithm for the $QR$ factorization of Cauchy-like matrices. Contemporary Mathematics, vol. 323, 2003, 91--103, (with D. Fasino).

    39.  
  39. Iterative refinement techniques for the spectral factorization of Laurent polynomials , 2002. Proceedings of the 47th Annual SPIE Meeting, Seattle, WA, USA. (with A. Bacciardi).

    40.  
  40. Orthogonal rational functions and diagonal-plus-semiseparable matrices , 2002. Proceedings of the 47th Annual SPIE Meeting, Seattle, WA, USA. (with M. Van Barel, D. Fasino, N. Mastronardi).

    41.  
  41. Fast and stable solution of banded-plus-semiseparable linear systems. CALCOLO , 39 , (2002) 201--217, (with D. Fasino).

    42.  
  42. Bernstein-Bezoutian matrices. Theoretical Computer Science , 315 , (2004) 319--333, (with D. A. Bini).

    43.  
  43. Solving quadratic matrix equations and factoring polynomials: new fixed point iterations based on Schur complements of Toeplitz matrices. Numerical Linear Algebra with Applications, , 12 , (2005) 181--189, (with D. A. Bini).

    44.  
  44. Orthogonal rational functions and structured matrices. SIAM J. Matrix Anal. Appl., 26 , (2005) 310--329, (with M. Van Barel, D. Fasino, N. Mastronardi).

    45.  
  45. The Ehrlich-Aberth method for the nonsymmetric tridiagonal eigenvalue problem. SIAM J. Matrix Anal. Appl. 27 , (2005) 153--175, (with D. A. Bini, F. Tisseur).

    46.  
  46. Rounding error analysis in solving $\B M$-matrix linear systems of block Hessenberg form. Numerical Algorithms , 36 , (2004) 157--168, (with G. Lotti).

    47.  
  47. $QR$-like algorithms for generalized semiseparable matrices Tech. Report no. 1470, Department of Mathematics, University of Pisa, 2003, (with D. A. Bini and V. Y. Pan).

    48.  
  48. On the shifted QR iteration applied to Frobenius matrices. Electron. Trans. Numer. Anal., 18(2004), 137--152. (with D. A. Bini and F. Daddi).

    49.  
  49. Structured matrix methods for CAGD: An application to computing the resultant of polynomials in Bernstein basis. Numer. Linear Algebra Appl., 12(2005), 685--698. (with D. A. Bini and J. R. Winkler).

    50.  
  50. Improved initialization of the accelerated and robust $QR$-like polynomial root-finding. Electron. Trans. Numer. Anal., 17 (2004), 195--205. (with D. A. Bini and V. Y. Pan).

    51.  
  51. Inverse power and Durand-Kerner iterations for univariate polynomial root-finding. Comput. Math. Appl. , 47 , (2004) 447--459, (with D. A. Bini and V. Y. Pan).

    52.  
  52. A unitary Hessenberg QR-based algorithm via semiseparable matrices. J. Comput. Appl. Math. , 184 , (2005) 505--517.

    53.  
  53. Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations Numer. Math., , 100 , (2005) 373--408, (with D. A. Bini and V. Y. Pan).

    54.  
  54. Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices SIAM Matrix Anal. Appl. , 29 , (2007), (with D. A. Bini, Y. Eidelman and I. Gohberg).

    55.  
  55. Quasiseparable Structures of Companion Pencils under the QZ-Algorithm CALCOLO, , 42 , (2005) 215--226.

    56.  
  56. On the Fast Reduction of a Quasiseparable Matrix to Hessenberg and Tridiagonal Forms. Linear Algebra Appl., 420, 2007, (with Y. Eidelman and I. Gohberg).

    57.  
  57. Fast QR Factorization of Cauchy-like Matrices. Linear Algebra Appl., 428, 2008, (with M. Van Barel and S. Delvaux)

    58.  
  58. Structured eigenvalue problems for rational Gauss quadrature. Numerical Algorithms, 45 , (2007) (with D. Fasino).

    59.  
  59. Structured matrix methods for polynomial root-finding. Proceedings of ISSAC 2007, (2007).

    60.  
  60. The unitary completion and QR iterations for a class of structured matrices. Mathematics of Computation, 77 , (2008), (with D. A. Bini, Y. Eidelman and I. Gohberg).

    61.  
  61. QR-factorization of displacement structured matrices using a rank structured matrix approach. (with S. Delvaux and M. Van Barel). To appear in Numerical Methods for Structured Matrices and Applications: Georg Heinig memorial volume.

    62.  
  62. Efficient Eigenvalue Computation for Quasiseparable Hermitian Matrices Under Low Rank Perturbations. Numerical Algorithms, 47 , (2008), (with Y. Eidelman and I. Gohberg).

    63.  
  63. Neville elimination for rank-structured matrices. Linear Algebra Appl., 428, (2008).

    64.  
  64. Numerical Behaviour of the DQR Method for Rank Structured Matrices. (with F. Uhlig). Submitted for publication.

    65.  
  65. On the Fast Reduction of Symmetric Rationally Generated Toeplitz Matrices to Tridiagonal Form. ETNA, 35 , (2009), (with K. Frederix and M. Van Barel ).

    66.  
  66. A Fast Implicit QR Eigenvalue Algorithm for Companion Matrices. (with D. A. Bini, P. Boito, Y. Eidelman and I. Gohberg). To appear in Linear Algebra and its Application, doi:10.1016/j.laa.2009.08.003.

    67.  
  67. From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks. (with C. Conti and L. Romani). To appear in Linear Algebra and its Applications, doi:10.1016/j.laa.2009.06.037.

    68.  
  68. From approximating to interpolatory non-stationary subdivision schemes with the same reproduction properties. (with C. Conti and L. Romani). Submitted.

    69.  
  69. On some classes of structured matrices with algebraic trigonometric eigenvalues. Submitted.

    70.  
  70. Solving Bezout-like polynomial equations for the design of interpolatory subdivision schemes. (with C. Conti and L. Romani). Submitted.

    71.  
     
     
     
     
Software:

     
  1. Efficient and stable solution of Hessenberg linear systems: hess.cpp

    2.  
  2. A Unitary Hessenberg QR-based algorithm via semiseparable matrices: hesseigstru.m , giv.m , norm1est.m , matrixex.m

    3.  
  3. Fast QR iteration for companion matrices: compqr.m

    4.  
  4. DQR iteration for certain rank structured matrices: matdir.tar