course: Mathematics 4

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

dates in summer term

  • start: Tuesday the 02.04.2019
  • lecture Tuesdays: from 08:15 to 10.00 o'clock in HZO 80
  • tutorial (alternativ) Thursdays: from 12:00 to 14.00 o'clock in IA 1/109
  • tutorial (alternativ) Thursdays: from 12:00 to 14.00 o'clock in IA 1/53
  • tutorial (alternativ) Thursdays: from 12:00 to 14.00 o'clock in IA 1/53
  • tutorial (alternativ) Fridays: from 08:00 to 10.00 o'clock in IA 1/109

Exam

Form of exam:written
Registration for exam:FlexNow
Date:12.08.2019
Begin:08:30
Duration:120min
Rooms : HIC ,  HID
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”