M221 Introduction to Linear Algebra
My main interest are Matrix Analysis and Numerical Linear Algebra and its applications. These fields are a fundamental part of Numerical Analysis, Scientific Computing, and Computational Mathematics. More specifically my interests include (i) polynomial eigenvalue problems, matrix polynomials, and their linearizations; (ii) conditioning and backward error analyses; (iii) nonlinear eigenvalue problems.
My main research interest is Numerical Linear Algebra and, more specifically solving Nonlinear Eigenvalue Problems. A nonlinear eigenvalue problem consits in finding scalars λ and nonzero vectors x such that
Here A(λ) is an n×n matrix whose entries depend analytically on λ. The scalar λ is called an eigenvalue of A, and x is the corresponding eigenvector. This type of problems arise in a wide variety of science and engineering applications, such as the dynamic analysis of mechanical systems, the linear stability of flows in fluid mechanics, the stability analysis of time-delay systems, and electronic band structure calculations for photonic crystals, etc.
In Automatic rational approximation and linearization of nonlinear eigenvalue problems, we have developed a new nonlinear eigenvalue solver that uses the recent AAA algorithm to approximate the nonlinear eigenvalue problem by a rational eigenvalue problem. The resulting rational eigenvalue problem is solved by using linearization and compact rational Krylov methods, allowing to efficiently solve large scale nonlinear eigenvalue problems. Our algorithm is competitive with NLEIGS, and obtains the same accuracy but with less effort for the user.
- A note on eigenvector error bounds for polynomial eigenvalue problems.
- The Conditioning of block Kronecker ℓ-ifictions of matrix polynomials,
- Block Minimal bases ℓ-ifictions of matrix polynomials, with Froilán M. Dopico and Paul Van Dooren.
- Automatic rational approximation and linearization of nonlinear eigenvalue problems, with Pieter Lietaert, Bart Vandereycken, and Karl Meerbergen.
- Explicit block-structures for block-symmetric Fiedler-like pencils, with Maribel Bueno, M. Martin, A. Song, and I. Viviano. To appear in Electronic Journal of Linear Algebra, 2018.
- Mixed forward-backward stability of the two-level orthogonal Arnoldi method for quadratic problems, with Karl Meerbergen. Linear Algebra and its Applications, 553, pp. 1-15, 2018.
- Structured backward error analysis of linearized structured polynomial eigenvalue problems, with Froilán M. Dopico and Paul Van Dooren. To appear in Mathematics of Computation, 2017.
- A simplified approach to Fiedler-like pencils via strong block minimal bases pencils, with Maribel Bueno, Froilán M. Dopico, R. Saavedra and B. Zykoski. Linear Algebra and its Applications, 547, pp. 45-104, 2018.
- Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors, with Piers W. Lawrence, Froilán M. Dopico, and Paul Van Dooren. To appear in Numerische Mathematik, 2017.
- Symmetric and skew-symmetric block Kronecker linearizations, with Heike Fassbender, and Nikta Shayanfar. Technical report. Available as arXiv:1606.01766.
- Constructing strong linearizations for matrix polynomials in the Chebyshev bases, with Piers W. Lawrence. SIAM Journal on Matrix Analysis and Applications, 37(4), pp. 1600-1624, 2016.
- Pseudospectra and eigenvalue condition numbers of Fiedler matrices, with Fernando De Terán and Froilán M. Dopico. Calcolo, 54(1), pp. 319-365, 2017.
- Fiedler-comrade and Fiedler-Chebyshev pencils, with Vanni Noferini. SIAM Journal on Matrix Analysis and Applications, 37(4), pp. 1600-1624, 2016.
- Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable? with Vanni Noferini. Mathematics of Computations, 86(306), pp. 1741-1767, 2016.
- Backward stability of polynomial root-finding using Fiedler companion matrices, with Fernando De Terán, and Froilán M. Dopico. IMA Journal of Numerical Analysis, 36, pp. 133-173, 2015.
- Technical report on backward stability of polynomial root-finding using Fiedler companion matrices, with Fernando De Terán, and Froilán M. Dopico. Technical report. Available as MIMS EPrint 2014.38, 2014.
- New bounds for roots of polynomials based on Fiedler companion matrices, with Fernando De Terán, and Froilán M. Dopico. Linear Algebra and its Applications, 451, pp. 197-230, 2014.
- Condition numbers for inversion of Fiedler companion matrices, with Fernando De Terán, and Froilán M. Dopico. Linear Algebra and its Applications, 439, pp. 944-981, 2013