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Metodi di Approssimazione 2024 25
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CDL Matematica
A.A. 2024 - 2025
Metodi di Approssimazione 2024 25
Topic 1
Taylor
Taylor
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◄ 2025-03-28: automatic differentiation: the main ideas through matrix functions, Taylor expansions. Some quick remarks on the multivariate case and reverse mode.
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Course notes (last update: 2025-04-17)
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Informazioni sull'esame e articoli suggeriti
2025-02-27: Introduction. Vectorization, Kronecker product. Sylvester equations; the Bartels-Stewart algorithm.
sylv triangular
2025-02-28: conditioning of Sylvester equations, stability of the Bartels-Stewart algorithm
2025-03-06: Invariant subspaces. Definition and examples. Computing invariant subspaces by reordering Schur forms.
2025-03-07: Sensitivity of invariant subspaces; the Stewart-Sun theorem. Evaluating polynomials in a matrix in Jordan form; definition of functions of matrices.
2025-03-13: Examples of matrix functions. Examples of multi-branch and non-primary functions. Some properties of matrix functions that descend from their expression as polynomials.
2025-03-20: more properties of matrix function. Convergence of Taylor expansions. Cauchy's integral formula. Fréchet derivatives with examples.
2025-03-21: block formula for the Fréchet derivative. Eigenvalues of Fréchet derivatives.
2025-03-27: example of low accuracy using Taylor series. The Parlett recurrence. The (blocked) Schur-Parlett method and funm.
funm parlett
2025-03-28: automatic differentiation: the main ideas through matrix functions, Taylor expansions. Some quick remarks on the multivariate case and reverse mode.
2025-04-03: the matrix exponential: properties; backward stability of Padé approximants; scaling and squaring.
2025-04-04: the matrix sign function. Schur-Parlett method; the Newton iteration for the matrix sign: scalar version and convergence.
2025-04-10: numerical experiments with the matrix sign. Scaling. The matrix square root: Schur-Parlett variant, Newton iteration, modified Newton method.
2025-04-11: (non-)convergence of the Newton method for the matrix square root. Variants. Functions of large and sparse matrices: general strategies, the Arnoldi approximation for matrix functions.
2025-04-17: Arnoldi for matrix functions: error bound for normal matrices. Brief overview of other recent results: numerical radius, Crouzeix-Palencia, and results for non-normal A; shift-and-invert Arnoldi and rational Arnoldi.
2025-05-02: Lyapunov and Stein equations. Introduction to control theory: the inverted pendulum
2025-05-07: example: heating. Controllability; definition and criteria.
2025-05-15: Matlab examples of control systems. Linear-quadratic optimal control: the optimality theorem and the Riccati equation.
2025-05-16: Algebraic Riccati equations: equivalence to invariant subspace; structural properties of Hamiltonian matrices; existence of the stable invariant subspace and of the stabilizing Riccati solution.
CAREX benchmark suite
2025-05-22: Methods for dense algebraic Riccati equations: Newton method; matrix-sign method; the Schur method. The importance of preserving the Hamiltonian structure.
2025-05-23: Methods for sparse Lyapunov equations: the ADI algorithm. Quick remarks on the pole selection problems and on rational Arnoldi.
2025-04-03: the matrix exponential: properties; backward stability of Padé approximants; scaling and squaring. ►