A warm welcome to everybody to this week's seminar at the Chair of Dynamics, Control

and Numerics at FAU Erlangen.

We are very happy to have Professor Jescher Respondeg as today's speaker.

He's a professor at Silesian University of Technology in Gliwice and an expert in the

intersection of numerical methods and mathematical control theory.

Professor Respondeg is best known for his work on special matrices and their application

in control theory.

In 2016, he was awarded the honorary title Doctor of Science from the prestigious Poznan

University of Technology for his scientific achievements.

And today he will give a talk on numerical algorithms as a tool that unites scientific

disciplines.

Please, Professor Respondeg, the screen is yours.

Thanks for giving me a voice. For the introduction, I shall divide my presentation into three

parts.

The first part will be general, as I said, general some general ideas about the numerical

problems.

The second part will be about the first matrix algorithm.

The last part will be about the applications of the first algorithm matrices just to the

control theory, the results which are also personally by me obtained.

The idea of computer simulation, the idea of computer simulation is writing the basic

laws of physics in the form of mathematical models to predict in the computer the development

of various physical phenomena, behaviour of technical constructions.

But what I want to emphasize here is that mathematical models of the physical phenomena

have been even from centuries, for example, Newton's law of dynamics.

However, the real possibilities of implementing the physical laws, for example, Newton's laws,

was very narrow because just to the lack of numerical algorithms and computers to be performated.

I will support now that idea by a few examples, notes from the school.

Many of us, every one of us meet in the school with a formula of the range of missiles.

The formula to show the range of the missile is expressed just by this algebraic, quite

simple formula.

But unfortunately, this formula obeys the air resistance.

That's why it's rather very inaccurate.

But it is, that's why we need to improve.

But this formula can be calculated even before the age of the computer by the mathematical

tables.

But it is possible by the same law, it means Newton's law, to construct formula for, now

let's go to the next slide.

It is possible to, by the same Newton's law, it is possible to calculate the range of the

missile with taking the air resistance into the account.

But now the model complicates.

You can see that we are now, now comes, besides the gravity force, also comes the air force

with some exponent.

The first approximation of the air resistance is for exponent equal to one.

And this equation can still be solved without the help of computer.

I was also interested in physics, so I am fully aware how the formula looks now as a

solution.

But the most close to reality is the model when the exponent of the speed is the second

power.

Unfortunately, this differential equation cannot be expressed like the previous by the

explicit formula, it means the range of the missile L is equal to something finite, it's