Section: Topic 1 | Metodi di Approssimazione 2024 25 | MAT - e-learning

Section outline

- Select activity 2025-02-27: Introduction. Vectorization, Kronecker product. Sylvester equations; the Bartels-Stewart algorithm.
  
  2025-02-27: Introduction. Vectorization, Kronecker product. Sylvester equations; the Bartels-Stewart algorithm. File PDF
  
  4.5 MB
- Select activity sylv triangular
  
  sylv triangular File M
  
  337 bytes
- Select activity 2025-02-28: conditioning of Sylvester equations, stability of the Bartels-Stewart algorithm
  
  2025-02-28: conditioning of Sylvester equations, stability of the Bartels-Stewart algorithm File PDF
  
  726.6 KB
- Select activity 2025-03-06: Invariant subspaces. Definition and examples. Computing invariant subspaces by reordering Schur forms.
  
  2025-03-06: Invariant subspaces. Definition and examples. Computing invariant subspaces by reordering Schur forms. File PDF
  
  4.6 MB
- Select activity 2025-03-07: Sensitivity of invariant subspaces; the Stewart-Sun theorem. Evaluating polynomials in a matrix in Jordan form; definition of functions of matrices.
  
  2025-03-07: Sensitivity of invariant subspaces; the Stewart-Sun theorem. Evaluating polynomials in a matrix in Jordan form; definition of functions of matrices. File PDF
  
  6.3 MB
- Select activity 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.
  
  2025-03-13: Examples of matrix functions. Examples of multi-branch and non-primary functions. Some properties of matrix functions that descend from their expression as polynomials. File PDF
  
  4.5 MB
- Select activity 2025-03-20: more properties of matrix function. Convergence of Taylor expansions. Cauchy's integral formula. Fréchet derivatives with examples.
  
  2025-03-20: more properties of matrix function. Convergence of Taylor expansions. Cauchy's integral formula. Fréchet derivatives with examples. File PDF
  
  4.6 MB
- Select activity 2025-03-21: block formula for the Fréchet derivative. Eigenvalues of Fréchet derivatives.
  
  2025-03-21: block formula for the Fréchet derivative. Eigenvalues of Fréchet derivatives. File PDF
  
  6.3 MB
- Select activity 2025-03-27: example of low accuracy using Taylor series. The Parlett recurrence. The (blocked) Schur-Parlett method and funm.
  
  2025-03-27: example of low accuracy using Taylor series. The Parlett recurrence. The (blocked) Schur-Parlett method and funm. File PDF
  
  421.5 KB
- Select activity funm parlett
  
  funm parlett File M
  
  298 bytes
- Select activity 2025-03-28: automatic differentiation: the main ideas through matrix functions, Taylor expansions. Some quick remarks on the multivariate case and reverse mode.
  
  2025-03-28: automatic differentiation: the main ideas through matrix functions, Taylor expansions. Some quick remarks on the multivariate case and reverse mode. File PDF
  
  4.5 MB
- Select activity Taylor
  
  Taylor File M
  
  947 bytes
- Select activity 2025-04-03: the matrix exponential: properties; backward stability of Padé approximants; scaling and squaring.
  
  2025-04-03: the matrix exponential: properties; backward stability of Padé approximants; scaling and squaring. File PDF
  
  2.6 MB
- Select activity 2025-04-04: the matrix sign function. Schur-Parlett method; the Newton iteration for the matrix sign: scalar version and convergence.
  
  2025-04-04: the matrix sign function. Schur-Parlett method; the Newton iteration for the matrix sign: scalar version and convergence. File PDF
  
  2.7 MB
- Select activity 2025-04-10: numerical experiments with the matrix sign. Scaling. The matrix square root: Schur-Parlett variant, Newton iteration, modified Newton method.
  
  2025-04-10: numerical experiments with the matrix sign. Scaling. The matrix square root: Schur-Parlett variant, Newton iteration, modified Newton method. File PDF
  
  4.5 MB
- Select activity 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.
  
  2025-04-11: (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. File PDF
  
  4.7 MB
- Select activity 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.
  
  2025-04-17: 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. File PDF
  
  572.2 KB
- Select activity 2025-05-02: Lyapunov and Stein equations. Introduction to control theory: the inverted pendulum
  
  2025-05-02: Lyapunov and Stein equations. Introduction to control theory: the inverted pendulum File PDF
  
  4.4 MB
- Select activity 2025-05-07: example: heating. Controllability; definition and criteria.
  
  2025-05-07: example: heating. Controllability; definition and criteria. File PDF
  
  4.8 MB
- Select activity 2025-05-15: Matlab examples of control systems. Linear-quadratic optimal control: the optimality theorem and the Riccati equation.
  
  2025-05-15: Matlab examples of control systems. Linear-quadratic optimal control: the optimality theorem and the Riccati equation. File PDF
  
  4.5 MB
- Select activity 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.
  
  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. File PDF
  
  8.3 MB
- Select activity CAREX benchmark suite
  
  CAREX benchmark suite URL
  
  Link per scaricare il pacchetto Matlab con esempi di equazioni di Riccati che abbiamo usato nei nostri esempi
- Select activity 2025-05-22: Methods for dense algebraic Riccati equations: Newton method; matrix-sign method; the Schur method. The importance of preserving the Hamiltonian structure.
  
  2025-05-22: Methods for dense algebraic Riccati equations: Newton method; matrix-sign method; the Schur method. The importance of preserving the Hamiltonian structure. File PDF
  
  8.1 MB
- Select activity 2025-05-23: Methods for sparse Lyapunov equations: the ADI algorithm. Quick remarks on the pole selection problems and on rational Arnoldi.
  
  2025-05-23: Methods for sparse Lyapunov equations: the ADI algorithm. Quick remarks on the pole selection problems and on rational Arnoldi. File PDF
  
  6.5 MB