Section: Topic 1 | Metodi di Approssimazione 2022 23 | MAT - e-learning

Section outline

- 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