# course: Mathematics 4

number:
150116
teaching methods:
lecture with tutorials
media:
black board and chalk
responsible person:
Dr. rer. nat. Mario Lipinski
lecturer:
Dr. rer. nat. Mario Lipinski (Mathematik)
language:
german
HWS:
4
CP:
5
offered in:
summer term

## dates in summer term

• start: Monday the 20.04.2020
• lecture: siehe "Sonstiges"

## Exam

##### All statements pertaining to examination modalities (for the summer/winter term of 2020) are given with reservations. Changes due to new requirements from the university will be announced as soon as possible.
 Form of exam: written Registration for exam: FlexNow Date: 17.08.2020 Begin: 13:30 Duration: 120min Rooms : HIA ,  HIB Individual appointments of students to each exam location will be issued by the responsible chair.

## goals

Proficiency in applying the following mathematical methods to engineering problems:

• Basic algorithms of numerical mathematics

## content

1. Systems of linear equations
• Direct solvers: LU decomposition, Cholesky decomposition, matrix norms, error
• Iterative solvers:
• fixed point problems, Jacobi method, Gauss-Seidel method, SOR method
2. Systems of non-linear equations
• Newton’s method:
• geometric idea, convergence, error, relaxation
3. Interpolation
• Lagrange interpolation:
• problem, divided differences, error
• Hermite interpolation:
• problem, modified divided differences, error
• Cubic spline interpolation:
• problem, calculation of interpolating cubic spline functions
4. Numerical integration
• midpoint rule, trapezoidal rule, Simpson’s rule, order, error
• Gauss formulas:
• definition, Legendre polynomials
• Composite rules:
• definition, error
• Romberg’s method:
• motivation, Romberg tableau
5. Ordinary differential equations
• Basic numerical solvers:
• forward/backward Euler, trapezoidal rule, error, order
• Runge Kutta methods:
• definition, Butcher tableau
• Stability:
• model problem, domain of absolute stability, A-stability
• comparison of step sizes, comparison of orders
• Outlook:
• multistep methods, BDF formulas
6. Eigenvalues and eigenvectors
• Power iteration:
• definition, inverse power iteration
• Rayleigh quotient iteration:
• definition, inverse Rayleigh quotient iteration
• QR iteration:
• QR decomposition, QR iteration, Hessenberg matrices

None

## recommended knowledge

Contents of lectures “Mathematik 1-3”

## miscellaneous

Im Som­mer­se­mes­ter 2020 wird die­ser Kurs als reiner Online-Kurs durchgeführt.

Name des Mood­le-Kur­ses: Mathematik 4 für ET / IT (Numerik) (150116-SoSe20)