Last update: 22/05/24

Lecture 1 - 21/02/24

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 - 21/02/24

Conditioning Number exercises.

• exercises01.pdf: Ex 1.1, 1.2, 1.3

Lecture 2 - 28/02/24

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

• [Q] Section 1.11 (matrix norm);

• [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 - 28/02/24

Gauss Elimination exercises.

• execises01.pdf: Ex 2.

• exercises02.pdf: Ex 1, 2, 3

Lecture 3 - 06/03/24

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

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

Tutorial 3 - 06/03/24

Gauss Elimination exercises.

• exercises03.pdf: Ex 1, 2, 3

Lecture 4 - 13/03/24

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

• [Q] Sections 4.1, 4.2.1-4.2.3

Tutorial 4 - 13/03/24

MatLab exercise on the stationary iterative methods

• NM2223_lab5.pdf: Ex 1, 2, 3

• Lab5.m

Stationary iterative methods.

• exercises04.pdf: Ex 1

Lecture 5 - 20/03/24

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 - 20/03/24

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 - 27/03/24

Conjugate gradient.

• [Q] Sections 4.3.3, 4.3.4

Rootfinding for Nonlinear Equations. Conditioning of a Nonlinear Equation. The Bisection Method, Methods of Chord, Secant, and Newton’s Method. 

• [Q] Sections 6.1, 6.2.1, 6.2.2 

Tutorial 6 - 27/03/24

Rootfinding.

• exercises07.pdf: Ex 1, 2

Lecture 7 - 03/04/24

Convergence of Secant and Newton's methods. Stopping criteria.

• [Q] Sections 6.3, 6.5

The Horner Method. 6.4.2 The Newton-Horner Method. 

• [Q] Sections 6.4.1, 6.4.2

Tutorial 7 - 03/04/24

Rootfinding.

• exercises07.pdf: Ex 3, 4

Matlab: Newton's method

• matlab07.pdf

Lecture 8 - 10/04/24

Newton’s Method for nonlinear systems. 

• [Q] Sections 7.1.1 

Polynomial Interpolation. The Interpolation Error. Drawbacks of Polynomial Interpolation on Equally Spaced Nodes and Runge’s Counterexample. Stability of Polynomial Interpolation. Piecewise Lagrange Interpolation. 

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

Tutorial 8 - 10/04/24

Lagrange Interpolation.

• exercises08.pdf: Ex 1, 2, 3.

Lecture 9 - 17/04/24

Approximation by Splines. Interpolatory Cubic Splines 

• [Q] Sections 8.7 (excluded 8.7.2).

Bézier Curves and De Casteljau algorithm.

• [Q] Section 8.8.1

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

• [Q] Section 10.1

Tutorial 9 - 17/04/24

Cubic splines.

• exercises09.pdf: Ex 1,  2, 3.

Lecture 10 - 24/04/24

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 10 - 24/04/24

Orthogonal polynomials.

• exercises10.pdf: Ex 1, 2, 3.

Lecture and Tutorial 11 - 15/05/24

Test

Lecture 12 - 22/05/24

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 12 - 22/05/24

Gauss quadrature, ODEs

• exercises11.pdf: Ex 1, 2, 3.