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Metodi di Approssimazione 2022 23
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CDL Matematica
A.A. 2022 - 2023
Metodi di Approssimazione 2022 23
Topic 1
arnoldi.m
arnoldi.m
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◄ sqrtm_db.m
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Course notes (work in progress)
Articoli per il seminario di esame
2023-02-28: vettorizzazione, prodotti di Kronecker; equazioni di Sylvester: risolubilità e algoritmo di Bartels-Stewart.
sylv_triangular.m
2023-03-02: condizionamento di equazioni di Sylvester; stabilità di Bartels-Stewart. Sottospazi invarianti. Riordinare forme di Schur.
2023-03-07: sensitività dei sottospazi invarianti. Introduzione alle funzioni di matrici.
2023-03-09: primi esempi di funzioni di matrici. Interpolazione di Hermite. Proprietà delle funzioni di matrici.
2023-03-13: funzioni di matrici e integrali di Cauchy. Derivate di Fréchet e condizionamento di funzioni di matrici.
2023-03-14: eigenvalues of Fréchet derivatives. First numerical methods for matrix functions: eigenvalue decomposition and Taylor series.
2023-03-16: An example of instability in the Taylor series method. The Schur-Parlett method.
2023-03-27: Numerical differentiation: forward differences, complex-step method. Automatic differentiation via matrix functions. Forward-mode automatic differentiation via dual numbers and type overloading.
Taylor.m: Matlab implementation of a simple "Taylor series" object for automatic differentiation
2023-03-28: The matrix exponential. Conditioning, remarks on the norm and the hump phenomenon. Padé approximants.
2023-03-30: Backward error of the matrix exponential. Scaling and squaring. The sign method: introduction and Schur-Parlett method.
sign_schurparlett.m
2023-04-04: perturbation theory for the sign function. The Newton method for the matrix sign.
2023-04-06: convergence of the Newton iteration for the matrix sign. Scaling in the matrix sign. The matrix square root: introduction and relation to the matrix sign.
sqrt_comparison.m
sqrtm_schurparlett.m
2023-04-18: Schur-Parlett variant for the square root. Backward-(almost)-stability of the method. True and modified Newton method for the square root: theoretical properties and an example in practice where MN diverges.
2023-04-20: local stability of TN and MN. The Denman-Beavers iteration. Introduction to methods for large and sparse matrix functions.
sqrtm_db.m
2023-04-27: the Arnoldi algorithm. Formula for p(A)b.
2023-05-02: approximation properties of Arnoldi for matrix functions. Quick remarks to the non-normal case (Crouzeix-Palencia theorem) and to rational Arnoldi.
2023-05-16: Lyapunov equations. Introduction to control theory.
2023-05-18: Controllability criteria. Computational example.
2023-05-23: Stabilizability. Optimal control: existence of solutions to algebraic Riccati equations
2023-05-24: quick overview of methods to solve algebraic Riccati equations: Newton method, matrix sign method, Schur method. Remarks on structure preservation and stability.
adi_single_shift.m
2023-05-25: overview of methods for large-scale Lyapunov equations.
2023-04-27: the Arnoldi algorithm. Formula for p(A)b. ►