- 3MMBMI

Goal(s)

The objective of the course is to develop a solid command of the notions of norms, differential calculus in finite-dimensional spaces, and numerical matrix analysis.

This is achieved through the acquisition of foundational theoretical concepts and the study of various applications.

Core techniques related to normed vector spaces and finite-dimensional differential calculus serve as fundamental tools extensively used in many areas of applied mathematics, including optimization, image processing, variational methods, and artificial intelligence.

Responsible(s)

Guillaume JAMES, Stefanie HAHMANN

Content(s)

Differential Calculus in \mathbb{R}^n

• Differentiable and derivable functions
• Partial derivatives, Jacobian matrix, gradient
• Functions of class \mathcal{C}^1
  Normed Vector Spaces
• Finite-dimensional normed spaces
• Linear mappings
  Fixed Point Theory and Differential Equations
• Fixed-point theorems
• Cauchy–Lipschitz theorem (existence and uniqueness of solutions to ODEs)
  Cauchy Sequences and Completeness
  Numerical Methods
• Matrix computations: LU factorization (Gaussian elimination with pivoting), Cholesky decomposition, Gram–Schmidt orthonormalization
• Interpolation and quadrature formulas

The first-year mathematics curriculum, in particular: matrix algebra, finite-dimensional vector spaces, sequences, and real-valued functions of a real variable (differentiation and integration).

Test

Evaluation : Examen écrit (3h)

Resit : Examen écrit (2h)

Calendar

The course exists in the following branches:

• Curriculum - Core curriculum - Semester 5

Additional Information

Course ID : 3MMBMI
Course language(s):

You can find this course among all other courses.

Bibliography

• J.M. Arnaudiès, H. Fraysse. Cours de mathématiques - 2 Analyse, Classes préparatoires 1er cycle universitaire, Dunod Université, 1988

• J.M. Arnaudiès, H. Fraysse. Cours de mathématiques - 3 Compléments d'analyse, Classes préparatoires 1er cycle universitaire, Dunod Université, 1989

• J.-P. Demailly, Analyse numérique et équations différentielles, EDP Sciences, 2006

• D. Serre. Les Matrices Théorie et pratique, Dunod, 2001