Course: Metodi di Approssimazione 2022 23 | MAT - e-learning

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Informazioni logistiche

Il corso inizia il 28 febbraio 2023.

Orario delle lezioni:

Martedì 9-11, aula X1

Giovedì 11-13, aula X1

Cambiamenti sono possibili per venire incontro alle necessità degli studenti partecipanti. The course can be taught in English on request.

Docente

Federico Poloni federico.poloni@unipi.it

Programma di massima

funzioni di matrici: definizione, proprietà teoriche, e metodi numerici per il loro calcolo

metodi computazionali per equazioni di matrici, con riferimento particolare a quelle che compaiono in teoria dei controlli (equazioni di Sylvester e di Riccati).

Libri

N. Higham, "Functions of Matrices: Theory and Computation", SIAM, 2008. Disponibile per il download gratuitamente (un pdf per capitolo) dall'interno della rete Unipi (o via VPN).

B. Datta, "Numerical Methods for Linear Control Systems", Elsevier, 2004.

Verranno messe a disposizione di volta in volta anche slides preparate dal docente sugli argomenti di ogni singola lezione.

Esame

A seminario, presentazione su un articolo sui temi del corso.
- Select activity Annunci
  
  Annunci Forum
- Select activity Presentazione del corso
  
  Presentazione del corso File PDF
  
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  132.9 KB
- Select activity Course notes (work in progress)
  
  Course notes (work in progress) File PDF
  
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  663.9 KB
- Select activity Articoli per il seminario di esame
  
  Articoli per il seminario di esame Page
  
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Select section Topic 1

Topic 1
- Select activity 2023-02-28: vettorizzazione, prodotti di Kronecker; equazioni di Sylvester: risolubilità e algoritmo di Bartels-Stewart.
  
  2023-02-28: vettorizzazione, prodotti di Kronecker; equazioni di Sylvester: risolubilità e algoritmo di Bartels-Stewart. File PDF
  
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  806.7 KB
- Select activity sylv_triangular.m
  
  sylv_triangular.m File M
  
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  295 bytes
- Select activity 2023-03-02: condizionamento di equazioni di Sylvester; stabilità di Bartels-Stewart. Sottospazi invarianti. Riordinare forme di Schur.
  
  2023-03-02: condizionamento di equazioni di Sylvester; stabilità di Bartels-Stewart. Sottospazi invarianti. Riordinare forme di Schur. File PDF
  
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  904.3 KB
- Select activity 2023-03-07: sensitività dei sottospazi invarianti. Introduzione alle funzioni di matrici.
  
  2023-03-07: sensitività dei sottospazi invarianti. Introduzione alle funzioni di matrici. File PDF
  
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  792.1 KB
- Select activity 2023-03-09: primi esempi di funzioni di matrici. Interpolazione di Hermite. Proprietà delle funzioni di matrici.
  
  2023-03-09: primi esempi di funzioni di matrici. Interpolazione di Hermite. Proprietà delle funzioni di matrici. File PDF
  
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  869.1 KB
- Select activity 2023-03-13: funzioni di matrici e integrali di Cauchy. Derivate di Fréchet e condizionamento di funzioni di matrici.
  
  2023-03-13: funzioni di matrici e integrali di Cauchy. Derivate di Fréchet e condizionamento di funzioni di matrici. File PDF
  
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  829.7 KB
- Select activity 2023-03-14: eigenvalues of Fréchet derivatives. First numerical methods for matrix functions: eigenvalue decomposition and Taylor series.
  
  2023-03-14: eigenvalues of Fréchet derivatives. First numerical methods for matrix functions: eigenvalue decomposition and Taylor series. File PDF
  
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  969.3 KB
- Select activity 2023-03-16: An example of instability in the Taylor series method. The Schur-Parlett method.
  
  2023-03-16: An example of instability in the Taylor series method. The Schur-Parlett method. File PDF
  
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  837.2 KB
- Select activity 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.
  
  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. File PDF
  
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  577.3 KB
- Select activity Taylor.m: Matlab implementation of a simple "Taylor series" object for automatic differentiation
  
  Taylor.m: Matlab implementation of a simple "Taylor series" object for automatic differentiation File M
  
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  1.3 KB
- Select activity 2023-03-28: The matrix exponential. Conditioning, remarks on the norm and the hump phenomenon. Padé approximants.
  
  2023-03-28: The matrix exponential. Conditioning, remarks on the norm and the hump phenomenon. Padé approximants. File PDF
  
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  694.4 KB
- Select activity 2023-03-30: Backward error of the matrix exponential. Scaling and squaring. The sign method: introduction and Schur-Parlett method.
  
  2023-03-30: Backward error of the matrix exponential. Scaling and squaring. The sign method: introduction and Schur-Parlett method. File PDF
  
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  703.9 KB
- Select activity sign_schurparlett.m
  
  sign_schurparlett.m File M
  
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  848 bytes
- Select activity 2023-04-04: perturbation theory for the sign function. The Newton method for the matrix sign.
  
  2023-04-04: perturbation theory for the sign function. The Newton method for the matrix sign. File PDF
  
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  2.6 MB
- Select activity 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.
  
  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. File PDF
  
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  774.6 KB
- Select activity sqrt_comparison.m
  
  sqrt_comparison.m File M
  
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  383 bytes
- Select activity sqrtm_schurparlett.m
  
  sqrtm_schurparlett.m File M
  
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  319 bytes
- Select activity 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.
  
  2023-04-18: 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. File PDF
  
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  1.5 MB
- Select activity 2023-04-20: local stability of TN and MN. The Denman-Beavers iteration. Introduction to methods for large and sparse matrix functions.
  
  2023-04-20: local stability of TN and MN. The Denman-Beavers iteration. Introduction to methods for large and sparse matrix functions. File PDF
  
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  2.6 MB
- Select activity sqrtm_db.m
  
  sqrtm_db.m File M
  
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  248 bytes
- Select activity arnoldi.m
  
  arnoldi.m File M
  
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  346 bytes
- Select activity 2023-04-27: the Arnoldi algorithm. Formula for p(A)b.
  
  2023-04-27: the Arnoldi algorithm. Formula for p(A)b. File PDF
  
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  2.9 MB
- Select activity 2023-05-02: approximation properties of Arnoldi for matrix functions. Quick remarks to the non-normal case (Crouzeix-Palencia theorem) and to rational Arnoldi.
  
  2023-05-02: approximation properties of Arnoldi for matrix functions. Quick remarks to the non-normal case (Crouzeix-Palencia theorem) and to rational Arnoldi. File PDF
  
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  2.5 MB
- Select activity 2023-05-16: Lyapunov equations. Introduction to control theory.
  
  2023-05-16: Lyapunov equations. Introduction to control theory. File PDF
  
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  2.7 MB
- Select activity 2023-05-18: Controllability criteria. Computational example.
  
  2023-05-18: Controllability criteria. Computational example. File PDF
  
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  2.8 MB
- Select activity 2023-05-23: Stabilizability. Optimal control: existence of solutions to algebraic Riccati equations
  
  2023-05-23: Stabilizability. Optimal control: existence of solutions to algebraic Riccati equations File PDF
  
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  2.8 MB
- Select activity 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.
  
  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. File PDF
  
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  679.7 KB
- Select activity adi_single_shift.m
  
  adi_single_shift.m File M
  
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  275 bytes
- Select activity 2023-05-25: overview of methods for large-scale Lyapunov equations.
  
  2023-05-25: overview of methods for large-scale Lyapunov equations. File PDF
  
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  716.6 KB
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Topic 10

Section outline

Informazioni logistiche

Docente

Programma di massima

Libri

Esame