# 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
• 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”