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
    • Basic quadrature rules:
      • 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
    • Adaptive step size control:
      • 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

requirements

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)

Link zum Mood­le-Kurs: https://moodle.ruhr-uni-bochum.de/m/course/view.php?id=26453