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2021 | OriginalPaper | Chapter

10. Conditioning and Well-Posedness

Author : Tobias Weinzierl

Published in: Principles of Parallel Scientific Computing

Publisher: Springer International Publishing

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Abstract

Hadamard’s definition of well-posedness makes us start to think if a particular problem is solvable on a computer at all. We introduce the terms well-conditioned and ill-conditioned, and finally define the condition number to quantify how sensitive an algorithm is to slight changes of its input. After a brief excursus what the condition number means for linear equation systems—it turns out to be a special case of our more generic definition—we discuss the interplay of the conditioning with forward stability and introduce backward stability as evaluation condition number.

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previous chapter Vectorisation of the Model Problem
next chapter Taylor Expansion
Footnotes
1
As always, there are people developing solutions to ill-posed problems. But that is out of our comfort zone here.
 
2
We revise Taylor expansion in Chap. 11. For the time being, it is sufficient to accept that \(f(x+\delta ) \approx f(x) + \delta \cdot f'(x)\) is a reasonable assumption here since \(\delta \) is small.
 
Metadata
Title
Conditioning and Well-Posedness
Author
Tobias Weinzierl
Copyright Year
2021
Book
Principles of Parallel Scientific Computing

Print ISBN: 978-3-030-76193-6

Electronic ISBN: 978-3-030-76194-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-76194-3_10

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