This course aims to illustrate the fundamental principles, techniques, and applications of the most common numerical methods for linear algebra and functional analysis. Particular emphasis will be placed on implementation aspects, with laboratory exercises conducted using Matlab.

1.      Fundamentals of Linear Algebra and Matrix Analysis
1.1. Vector spaces
1.2. Matrices and basic operations
1.2.1. Orthogonal matrices
1.2.2. Block matrices
1.3. Rank, nullity, and inverse
1.4. Eigenvalues and eigenvectors
1.5. Norms and inner products in vector spaces
1.6. Matrix norms

2.     Fundamentals of Numerical Mathematics
2.1. Well-posedness and conditioning
2.2. Stability of numerical methods
2.3. A priori and a posteriori error analysis
2.4. Sources of error in computational models

3.     Numerical Linear Algebra
3.1. Direct methods
3.1.1. Gaussian elimination
3.1.2. LU and Cholesky factorization
3.2. Least squares problems
3.2.1. Regression problems
3.2.2. Normal equations
3.2.3. QR factorization
3.3. Singular value decomposition
3.3.1. Pseudoinverse and underdetermined least squares problems
3.3.2. Fundamental theorem of linear algebra
3.4. Iterative methods for solving linear systems
3.4.1. Stationary methods: Richardson, Jacobi, Gauss-Seidel
3.4.2. Convergence of stationary methods
3.4.3. Preconditioning
3.4.4. Gradient method and conjugate gradient
3.5. Approximation of eigenvalues and eigenvectors
3.5.1. Power method
3.5.2. QR iteration methods

4.     Roots of Nonlinear Equations / Solving Nonlinear Systems
4.1. Nonlinear equations in one dimension
4.1.1. Geometric methods: bisection, false position (chord), secant
4.1.2. Fixed-point iteration
4.1.3. Newton's method
4.2. Systems of nonlinear equations
4.2.1. Fixed-point iterations
4.2.2. Newton's method
4.2.3. Nonlinear least squares problems, Gauss-Newton method

5.     Interpolation of Functions
5.1. Polynomial interpolation
5.1.1. Lagrange polynomial interpolation
5.1.2. Stability of polynomial interpolation, Runge's phenomenon
5.2. Piecewise interpolation and splines
5.3. Methods based on orthogonal basis functions

6.     Numerical Integration
6.1. Interpolation methods: midpoint, trapezoidal, Simpson’s rule
6.2. Newton-Coates methods
6.3. Error analysis and degree of accuracy
6.4. Methods based on orthogonal basis functions

7.     Numerical Solution of Ordinary Differential Equations
7.1. Forward, backward, and centered difference formulas
7.2. The Cauchy problem
7.2.1. Explicit and implicit Euler methods
7.2.2. Stability
7.2.3. Heun’s method and Crank-Nicolson method
7.3. Analysis of single-step methods
7.4. Multistep methods: Adams and BDF

8.     Data-driven Methods
8.1. Applied Koopman theory and dynamic mode decomposition
8.2. Model order reduction and proper orthogonal decomposition
8.3. Kalman filtering and data assimilation

Knowledge and Understanding. Acquire programming techniques for numerical methods in linear algebra and functional analysis. Upon completion of the course, students will have acquired a solid understanding of the mathematical topics covered.

Applied Knowledge and Understanding. Gain the ability to implement numerical methods for linear algebra and functional analysis. Develop skills in programming, testing, and interpreting results. Acquire the capability to solve mathematical problems using MATLAB.

Independent Judgment. Acquire the ability to determine the most appropriate numerical method for solving linear algebra and differential problems based on the considerations and requirements of a specific problem, balancing factors including but not limited to accuracy, available computational resources, data precision, and computational time. Develop the skill to recognize potential sources of error in computational models and propose potential solutions.

Communication Skills. Acquire the ability to rigorously illustrate mathematical problems studied in the course and clearly present their associated numerical methods, highlighting the main properties of each.

Learning Skills. Develop the ability to study and solve problems similar, though not necessarily identical, to those addressed during the lectures.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Classroom lessons, Matlab excercises in the computer lab

Attendance

Not mandatory

Course books

Quarteroni et al. (2014). Numerical Mathematics, Springer.

Assessment

Optional Project: Students will have the opportunity to undertake a project individually or in groups of up to three people. The project will involve solving a problem arising from an application context. Both the code used to solve the problem and a written report (necessarily typed) clearly presenting the results, including all computational details, must be submitted.
The problem will be solvable exclusively by employing the methods covered in the course; however, these methods may be applied in contexts or with specific adaptations related to the particular problem, which may not have been explicitly discussed during lectures.
The project must be submitted at least 10 days prior to the exam date. Group projects must also include a self-certification detailing the specific contributions of each group member. A passing grade for the project is at least 18/30.

Oral Examination: It consists of two questions on the course topics. A minimum grade of 18/30 is required to pass.

For students who do not undertake the project, the oral examination will account for 100% of the final grade. For students who do undertake the project, the project will account for 50% of the final grade, with the oral examination comprising the remaining 50%. However, it will not be possible for students who undertake the project to discard it, even if their oral exam grade is higher.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

Same as for attending students.

Attendance

Same as for attending students.

Course books

Same as for attending students.

Assessment

Same as for attending students.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Students who have registered a disability certification or Specific Learning Disabilities (SLD/DSA) certification with the Office for Inclusion and Right to Study may request to use concept maps (with keywords) during the exam.

For this purpose, it is necessary to send the maps to the course instructor two weeks before the exam date. The instructor will verify their compliance with the university's guidelines and may request modifications if necessary.