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Created January 8, 2022 19:49
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Numerical Mathematics Study Guide
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NumMath_TUHH_2015_pt1.md

Numerical Mathematics Study Guide

Part 1 of the TUHH Engineering Math Notes by by 0xdawg

Usage

The subdivisions or "Chapters" are clusters of knowledge around Numerical Mathematics, more or less strictly ordered by increasing specificity or involvedness. Use this to guide your learning journey around anything that involves calculations on computers with limited precision (the vast majority of cases). This is relevant for instance for complex computations involving a very large number of multiplication or division operations, especially when the sought values are in the [0..1] range.

Practical relevance for this lies in the training of neural networks and their deployment on less capable infrastructure, for instance. Other applications revolve around eigenvalue, singular value computations etc. as part of numerical simulations aka. interpolation, approximation, FEMs and other methods.

One

  • What does Numerical Mathematics concern itself with?

  • What is the main goal in studying Numerical Mathematics?

  • What is the Forward Error?

  • [] In which calculation steps is it relevant?

  • [o] What is the Backward Error and what is its relevance?

  • Is there a interrelation between these two error types?

  • What holds in IEEE arithmetic?

  • What does machine precision actually describe? What computations does it have relevance on? What is its symbol?

  • [] What computations can be analysed?

  • [] What is the forward error of a scalar product? What is the backward error in this case?

  • [] What is the condition K of a matrix?

  • [] Why is the concept of matrix condition relevant?

  • [] What does the condition of a matrix actually describe or state?

  • [] What is Perturbation Theory?

  • [] What does Annihilation mean in this context?

  • [] What is \Gamma_n ?

  • [] What is the condition of a differentiable scalar function evaluation?

  • [] When do we refer to an algorithm as stable?

  • [] What is mixed stability?

  • [] What other forms of stability do we know?

  • [] When is a problem ill-posed?

  • [] When is a problem badly conditioned?

  • [] What can we conclude from a condition of 10^8 with data precision at only three decimals?

  • [] Can you name an sketch a stable algorithm?

  • [] How would you evaluate the zeroes of an arbitrary polynomial numerically?

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