Last update: 18/05/22

Tutorial 0 - 13/02/22

Introduction to Matlab and stability Matlab examples.

• Files: equat2.m, equat2_stab.m, round_err.m

Lecture 1 - 16/02/22

Introduction. Well-posedness and Condition Number of a Problem and of a Method. Sources of Error in Computational Models. Rounding of a Real Number in its Machine Representation.

• [Q] Sections 2.1, 2.2, 2.4, 2.5.5

• Files: Lecture1.pptx, illcondsyst.m

Tutorial 1 - 21/02/22

Conditioning Number exercises.

• exercises01.pdf: Ex 1.1, 1.2, 1.3

Lecture 2 - 23/02/22

Stability Analysis of Linear Systems. Solution of Triangular Systems. Implementation of Substitution Methods. The Gaussian Elimination Method (GEM). GEM as a Factorization Method.

• [Q] Sections 3.1 (intro + 3.1.1, 3.1.2), 3.2 (intro + 3.2.1, 3.2.2), 3.3 (intro)

Tutorial 2 - 28/02/22

Gauss Elimination exercises.

• exercises02.pdf: Ex 1, 2, 3

Lecture 3 - 02/02/22

GEM as a Factorization Method. The Effect of Rounding Errors. Implementation of LU Factorization. Pivoting. Cholesky Factorization. QR Factorization.

• [Q] Sections 3.3, 3.4.2, 3.4.3, 3.5, and 1.9

Tutorial 3 - 07/03/22

Gauss Elimination and LU factorization exercises.

• exercises03.pdf: Ex 1, 2, 3

Lecture 4 - 09/03/22

Stationary iterative methods, SOR, Jacobi, Gauss-Seidel methods. Conjugate gradient.

• [Q] Sections 4.1, 4.2.1-4.2.3, 4.3.3, 4.3.4

Tutorial 4 - 14/03/22

Gauss Elimination and LU factorization exercises.

• exercises04_MatLab.pdf: Ex 1, 2

Lecture 5 - 16/03/22

Power method. Ideas of Krylov subspace methods.

• [Q] Sections 5.3.1 (Power methods)

• See the notes discussed in class for Krylov subspace methods. Also: [Q] Section 4.5

Tutorial 5 - 21/03/22

Krylov Subspace Methods vs Stationary Iterative Methods. The power method and Google matrix

• exercises05_MatLab.pdf: Ex 1.1, 1.2, 2.1

Lecture 6 - 23/03/22

Rootfinding for Nonlinear Equations. Conditioning of a Nonlinear Equation. The Bisection Method, Methods of Chord, Secant, and Newton’s Method. Fixed-point methods. Convergence of Secan and Newton's methods. Stopping criteria.

• [Q] Sections 6.1, 6.2.1, 6.2.2, 6.3, 6.5

Lecture 7 - 06/04/22

Newton’s Method for nonlinear systems. The Horner Method. 6.4.2 The Newton-Horner Method.

• [Q] Sections 6.4.1, 6.4.2. 7.1.1

Tutorial 6 - 10/04/22

Rootfinding.

• exercises06.pdf: Ex 1, 2, 3.

Lecture 8 - 13/04/22

Polynomial Interpolation. The Interpolation Error. Drawbacks of Polynomial Interpolation on Equally Spaced Nodes and Runge’s Counterexample. Stability of Polynomial Interpolation. Piecewise Lagrange Interpolation. Approximation by Splines. Interpolatory Cubic Splines

• [Q] Sections 8.1.1, 8.1.2, 8.3, 8.6.1.

Lecture 9 - 20/04/22

Bézier Curves and De Casteljau algorithm.

• [Q] Section 8.7.1

Approximation of Functions by Generalized Fourier Series. 10.1.1 The Chebyshev Polynomials. 10.1.2 The Legendre Polynomials

• [Q] Section 10.1

Tutorial 7 - 25/04/22

Lagrange Interpolation.

• exercises07.pdf: Ex 1, 2, 3.

Lecture 10 - 27/04/22

Quadrature Formulae. Interpolatory Quadratures. The Trapezoidal Formula. The Cavalieri-Simpson Formula. Newton-Cotes Formulae. Composite Newton-Cotes Formulae. Gauss quadrature formulas.

• [Q] Sections 9.1, 9.2, 9.3, 9.4, 10.2

Tutorial 8 - 02/05/22

Cubic splines.

• exercises07.pdf: Ex 1, 2.

Lecture 11 - 04/05/22

The Cauchy Problem. One-Step numerical methods. Runge-Kutta (RK) methods.

• [Q] Sections 11.1, 11.2, 11.3 (excluded 11.3.1, 11.3.2), 11.8 (excluded 1.8.2-1.8.4).

Tutorial 9 - 09/05/22

Orthogonal polynomials and Gauss quadrature.

• exercises08.pdf: Ex 1, 2.

Tutorial 10 - 16/05/22

Orthogonal polynomials and Gauss quadrature. ODEs.

• exercises08.pdf: Ex 3, 4, 5.

• exercises09.pdf: Ex 1, 2.

Lecture 11 - 18/05/22

Questions and discussion.