Understanding the complexity of a problem; ability in decomposing in into smaller and easier subproblems, by exploiting interdisciplinary tools, deriving from Numerical Analysis, Matrix Theory, Linear Algebra, and Approximation techniques in Analysis and Numerical Analysis
Prerequisiti
Programming, Computational Mathematics, Numerical Analysis, Linear Algebra, Calculus
Metodi didattici
Classroom teaching; practical exercises (on blackboard)
Verifica Apprendimento
Oral exam (possibly accompanied by a seminar and intermediate exams)
Contenuti
Definition of Structured Matrices Examples of Structured Matrices (Vandermode, Toeplitz, Hankel, Circulants etc) Vandermonde matrix, the interpolation problem, necessary and sufficient conditions for invertibility Vandermonde matrix and its (asymptotic) conditioning as a function of the distribution of points Generalized Vandermonde matrices, the special case of the Fourier Matrix Fourier Matrix and quadrature formulae for the Fourier coefficients Discrete Fourier Transform and the computational challenge of a fast algorithm Algebraic properties of the Fourier Matrix, basics of the tensor calculus Fast Fourier Transform in special dimensions: the recursive algorithm and its computational cost Fast Fourier Transform and the direct tensor decomposition: the direct algorithm and its computational cost Circulant matrices, algebra of matrices via the Cayley Hamilton Theorem Circulant matrices and Fast Fourier Transform Fast matrix vector product with Toeplitz, Hankek, g-Toeplitz, g-Hankel Fast Fourier Transform for every matrix size Spectral Analysis of Circulants, Toeplitz Approximation of elliptic differential operators via Finite Differences Approximation of elliptic differential operators via Finite Elements Spectral analysis of Locally Toeplitz Sequences Spectral analysis of Generalized Locally Toeplitz Sequences Applications of approximation of differential and integral operators