## Numerical Mathematics 2

 Faculteit Science and Engineering Jaar 2022/23 Vakcode WBMA023-05 Vaknaam Numerical Mathematics 2 Niveau(s) bachelor Voertaal Engels Periode semester I b ECTS 5 Rooster rooster.rug.nl

Uitgebreide vaknaam Numerical Mathematics 2
Leerdoelen The student is able to:
1. describe the methods mentioned in the course description,
2. reproduce basic concepts and theorems from Numerical Analysis related to the subjects mentioned in the description,
3. apply techniques from vector analysis and linear algebra to analyze the treated numerical methods,
4. create an algorithm from a given numerical method and to implement it in MATLAB,
5. assess some of the difficulties that may come up when solving a problem on a computer by a specific algorithm, specifically, the difficulties arising from finite precision computations (propagation of round-off errors), limited memory availability, and finite computation speed (complexity of the algorithm),
6. make a judicious choice from the discussed methods for a particular problem.
Omschrijving In this course, we will treat both the theory and algorithms of advanced numerical methods. Advanced methods are built on advanced theoretical considerations. Hence, the theory will play an important role in this course.
We will study tools for solving the following problems:
• Linear systems: Under which conditions can a linear system be solved safely, i.e. without rounding errors destroying the solution? What to do with under- and overdetermined system?
• (Generalized) eigenvalue problems: How can one determine all the eigenvalues of a system? How can one find the most interesting eigenvalues of a matrix? Among others, we will discuss the Lanczos method.
• Approximation, Interpolation, and Integration problems:
o A least-squares problem can be solved elegantly when the approximating polynomial is constructed with orthogonal polynomials, like Gauss Legendre and Chebyshev polynomials.
o As a spin-off, orthogonal polynomials learn us how to choose the interpolation points in interpolation problems.
o Moreover, since numerical integration methods are based on interpolation they lead to the very accurate Gauss and Chebyshev integration methods.
Uren per week
Onderwijsvorm Hoorcollege (LC), Werkcollege (T)
(The Lab Sessions are mandatory. Deadlines will be set; not meeting a deadline will result in zero points for the associated exercise.)
Toetsvorm Practisch werk (PR), Tussentoets (IT)
(The course will be subdivided into three units. Each unit will consist of a practical and a test, which both will be marked, say PRi and Ti for the i-th unit. Each PRi and Ti should be 5 or higher to pass. The final mark will be (2*PR1+3*PR2+3*PR3+3*T1+4*T2+5*T3)/20, and should before rounding be 5.5 or higher to pass. Each no-pass can be repaired. For the practical, this is during the next unit and for the test, this is in the re-examination. A pass mark for a practical cannot be upgraded by a repair. In case the final mark is 5.5 or higher, a pass mark for a test cannot be upgraded by repair)
Vaksoort bachelor
Coördinator dr. ir. F.W. Wubs
Docent(en) dr. ir. G. Tiesinga ,dr. ir. F.W. Wubs
Verplichte literatuur
Titel Auteur ISBN Prijs
Numerical Mathematics, Text in Applied Mathematics 37, Springer, second edition, 2006. Quarteroni, Sacco and Saleri 978-3-540-34658
Entreevoorwaarden Prior knowledge: Knowledge and skill in handling basic numerical techniques and programming in MATLAB as taught in the course Numerical Mathematics I and good knowledge of linear algebra and vector analysis is necessary.
Opmerkingen This course prepares for advanced courses on Numerical Mathematics
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