# 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

- 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

- Systems of non-linear equations
- Newton’s method:
- geometric idea, convergence, error, relaxation

- Newton’s method:
- Interpolation
- Lagrange interpolation:
- problem, divided differences, error

- Hermite interpolation:
- problem, modified divided differences, error

- Cubic spline interpolation:
- problem, calculation of interpolating cubic spline functions

- Lagrange interpolation:
- 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

- Basic quadrature rules:
- 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

- Basic numerical solvers:
- 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

- Power iteration:

## requirements

None

## recommended knowledge

Contents of lectures “Mathematik 1-3”