## Basic Numerical Methods

Upon completing this module, the students will acquire the basics of numerical mathematics and numerical simulation methods. This includes the theoretical understanding of how a computer calculates with finite floating-point numbers and what kind of errors and inaccuracies may arise from these, and how to reduce or control them same. They will be familiar with basic numerical methods for modelling and simulating statistical models, linear algebra models, and ordinary and partial differential equations. They will be able to estimate the approximation errors of the methods and determine the algorithmic intensity, and will be able to implement these methods themselves.

**Contents**

Floating point arithmetic, rounding errors, cancellation, numerical interpolation (Lagrange, Newton, Splines), Taylor developments, finite differences and their approximation errors, explicit and implicit time integrators, qudrature, direct and iterative algorithms for matrix inversion, matrix decomposition (LU), solution for the Poisson equation.

**Program / Module**

M.Sc. Computational Modeling and Simulation

Module: CMS-COR-NUM - Basic Numerical Methods

**Time/Place**

**Winter Term**

Lecture: Mondays, 5. DS (14:50-16:20) ONLINE / **FIRST LECTURE: OCT 26, 2020**

Exercises: t.b.a., ONLINE

**LECTURES AND EXERCISES WILL BE ENTIRELY ONLINE FOR THE WHOLE SEMESTER. They will be held as Zoom live screen-casts with the possibility to ask questions. Links for the LECTURES will be announced here below (a separate link for every week) a day prior to the lecture. In order to keep this as close as possible to a real lecture experience, the webcasts are not recorded.**

Webcast link for the exercises (recurrent, same every week): tba

Meeting ID: tba

Password: tba

**Format**

2 SWS lecture, 2 SWS exercise, self-study

5 credits

**Exam**

**In presence in March 2021. Date and place to be announced.**

**Since there are more than 10 registered students, the module examination consists of a written examination in presence with a duration of 90 minutes.**

At the exam, the following may be used:

- 4 A4 sheets (8 pages if you print duplex) of hand-written summary. We recommend writing the summary by hand, but it can also be machine-written. In the latter case, the font size must be 8 points or larger throughout.
- A standard pocket calculator (devices with network or bluetooth access, as well as devices capable of storing and displaying documents are not allowed)

**Grade scale:**

All exams are graded in absolute terms w.r.t. the following pre-defined grade scale that remains constant over the years:

- The top grade of 1.0 is reached with 80% of the maximum possible points
- Half of that, i.e., 40% of the maximum possible points, are required to pass
- Below 40%, or no-show, is a fail.

**Exam Review winter term 2019/20**

You can come and look at your exam, and ask questions about its correction and the answers given during the exam review times. Due to the current travel and contact restrictions, we offer three exam review dates: one in June, one during the summer break, and one around the beginning of the winter semester 2020/21:

**June 18, 2020, 2pm**, Outdoors at the pond between the BAR building and the Mensa (Maps Link). The review only takes place if there is no rain. In case of rain, a new date will be found. In order to participate, You MUST wear a face mask and you are only allowed to come forward one by one.**July 20, 2020, 2pm**, Outdoors in the seating area in front of the Center for Systems Biology Dresden, Pfotenhauerstr. 108 (Maps Link). The review only takes place if there is no rain. In case of rain, a new date will be found. In order to participate, You MUST wear a face mask and you are only allowed to come forward one by one.**October 19, 2020, 2pm**, Outdoors in the seating area in front of the Center for Systems Biology Dresden, Pfotenhauerstr. 108 (Maps Link). The review only takes place if there is no rain. In case of rain, a new date will be found. In order to participate, You MUST wear a face mask and you are only allowed to come forward one by one.

**IMPORTANT: All students attending an exam review must fill in and sign the exam review form they are going to receive during the review. Undocumented exam reviews are not permitted. You must participate in person.**

**Registration to the course**

For students of the Master program "Computational Modeling and Simulation: via CampusNet SELMA

For students of the Computer Science programs: via jExam

**Teachers**

Lecture: Prof. Ivo F. Sbalzarini

Exercises: Abhinav Singh

**Instruction language:** ENGLISH

Lecture notes are available as PDF here.

Below is the weekly syllabus and the exercise/solution handouts:

- Lecture 1 - finite-precision arithmetics, IEEE number representation, roundoff and extinction, error propagation, condition numbers, backward error analysis
- Lecture 2 - linear systems of equations, LU decomposition of matrices, Gaussian elimination, iterative linear solvers, Jacobi method
- Lecture 3 - Gauss-Seidel method, SOR method, Conjugate gradient methods, preconditioning schemes
- Lecture 4 - Least-Squares methods, QR decomposition, singular value decomposition
- Lecture 5 - Non-linear least squares, non-linear equations, Newton method, bisection method, secant method
- Lecture 6 - Non-linear systems of equations, quasi-Newton method, rank-1 update, Broyden algorithm, Lagrange interpolation, barycentric interpolation
- Lecture 7 - Interpolation algorithms: Aitken-Neville algorithm, Hermite and Spline interpolation
- Lecture 8 - Trigonometric interpolation: Discrete Fourier transform and fast Fourier transform algorithms
- Lecture 9 - Numerical integration (quadrature): trapezoidal rule, Simpson rule, Romberg extrapolation, Gauss quadrature
- Lecture 10 - Numerical differentiation: finite difference methods, Romberg extrapolation, Initial value problems of ordinary differential equations, the explicit Euler scheme
- Lecture 11 - second-order methods, Heun's method, Runge-Kutta methods, variable step size control, embedded Runge-Kutta, Richardson extrapolation
- Lecture 12 - implicit methods, multistep methods, Systems of ODEs, higher-order ODEs
- Lecture 13 - numerical stability, stiff problems, partial differential equations introduction
- Optional: Lecture 14 - Partial differential equations 1: parabolic problems, elliptic problems
- Optional: Lecture 15 - Partial differential equations 2: hyperbolic problems, Courant-Friedrichs-Lewy condition