## 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: Thursdays, 3. DS (11:10-12:40) ONLINE / **FIRST TUTORIAL: OCT 29, 2020**

**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. Recurrent links (i.e., the same every week) are announced here below. In order to keep this as close as possible to a real lecture experience, the webcasts are not recorded.**

Webcast link for the lectures (recurrent, same every week): https://us02web.zoom.us/j/88336989253?pwd=WVQ1TkxyM2JKempOZ2w1dXlPeHRUUT09

Meeting ID: 883 3698 9253

Password: 164596

Webcast link for the exercise tutorials (recurrent, same every week): https://us02web.zoom.us/j/84364851860?pwd=d09OcURPeU5hUzc0ZnBWeU9yempmUT09

Meeting ID: 843 6485 1860

Password: 667585

**Format**

2 SWS lecture, 2 SWS exercise, self-study

5 credits

**Exam**

**Online exam in OPAL (exceptionally due to Covid-19 situation and special decision by university Senate and Rectorate) of 90 minutes duration. Time and date to be announced.**

Proper registration to the exam in SELMA is mandatory. Only registered students will receive a link and code to participate in the online exam. This is an open-book exam. All sources and means of help are allowed to be used, but the exam must be completed alone and by yourself without the help of any third person.

**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 2020/21**

Exam Review dates and details are going to be announced after completion of the exam.

**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 (Blackboard 01 PDF, Exercise 01 PDF, Solution 01 PDF)
- Lecture 2 - linear systems of equations, LU decomposition of matrices, Gaussian elimination, iterative linear solvers, Jacobi method (Blackboard 02 PDF, Exercise 02 PDF, Solution 02 PDF)
- Lecture 3 - Gauss-Seidel method, SOR method, Conjugate gradient methods, preconditioning schemes (Blackboard 03 PDF, Exercise 03 PDF, Solution 03 PDF)
- Lecture 4 - Least-Squares methods, QR decomposition, singular value decomposition (Blackboard 04 PDF, Exercise 04 PDF, Solution 04 PDF)
- Lecture 5 - Non-linear least squares, non-linear equations, Newton method, bisection method, secant method (Blackboard 05 PDF, Exercise 05 PDF, Solution 05 PDF)
- Lecture 6 - Non-linear systems of equations, quasi-Newton method, rank-1 update, Broyden algorithm, Lagrange interpolation, barycentric interpolation (Blackboard 06 PDF, Exercise 06 PDF, Solution 06 PDF)
- Lecture 7 - Interpolation algorithms: Aitken-Neville algorithm, Hermite and Spline interpolation (Blackboard 07 PDF, Exercise 07 PDF, Solution 07 PDF)
- Optional: Lecture 8 - Trigonometric interpolation: Discrete Fourier transform and fast Fourier transform algorithms (skipped due to Covid-19 related shortening of winter term by the university)
- Lecture 9 - Numerical integration (quadrature): trapezoidal rule, Simpson rule, Romberg extrapolation, Gauss quadrature (Blackboard 08 PDF, Exercise 08 PDF, Solution 08 PDF)
- Lecture 10 - Numerical differentiation: finite difference methods, Romberg extrapolation, Initial value problems of ordinary differential equations, the explicit Euler scheme (Blackboard 09 PDF, Exercise 09 PDF, Solution 09 PDF)
- Lecture 11 - second-order methods, Heun's method, Runge-Kutta methods, variable step size control, embedded Runge-Kutta, Richardson extrapolation (Blackboard 10 PDF, Exercise 10 PDF, Solution 10 PDF, Butch Table Solver PDF, Butch Table Solver Jupyter)
- Lecture 12 - implicit methods, multistep methods, Systems of ODEs, higher-order ODEs (Blackboard 11 PDF, Exercise 11 PDF, Solution 11 PDF)
- Lecture 13 - numerical stability, stiff problems, partial differential equations introduction (Blackboard 12 PDF, Exercise 12 PDF)
- Lecture 14 - Partial differential equations 1: parabolic problems, elliptic problems
- Optional: Lecture 15 - Partial differential equations 2: hyperbolic problems, Courant-Friedrichs-Lewy condition