Abstract
It is folklore particularly in numerical and computer sciences that, instead of solving some general problem $f:A\to B$, additional structural information about the input $x\in A$ (that is any kind of promise that $x$ belongs to a certain subset $A'\subseteq A$) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problem show much advice is necessary and sufficient to render them computable.
Specifically, finding a nontrivial solution to a homogeneous linear equation $A\cdot\vec x=0$ for a given singular real $n\times n$matrix $A$ is possible when knowing $\rank(A)\in\{0,1,\ldots,n1\}$; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric $n\times n$matrix $A$ is possible when knowing the number of distinct eigenvalues: an integer between $1$ and $n$ (the latter corresponding to the nondegenerate case). And again we show that $n$fold (i.e. roughly $\log n$ bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding \emph{some single} eigenvector of $A$ requires and suffices with $\Theta(\log n)$fold advice.
BibTeX  Entry
@InProceedings{ziegler:OASIcs:2009:2277,
author = {Martin Ziegler},
title = {{Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability}},
booktitle = {6th International Conference on Computability and Complexity in Analysis (CCA'09)},
series = {OpenAccess Series in Informatics (OASIcs)},
ISBN = {9783939897125},
ISSN = {21906807},
year = {2009},
volume = {11},
editor = {Andrej Bauer and Peter Hertling and KerI Ko},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/2277},
URN = {urn:nbn:de:0030drops22770},
doi = {10.4230/OASIcs.CCA.2009.2277},
annote = {Keywords: Nonuniform computability, recursive analysis, topological complexity, linear algebra}
}
Keywords: 

Nonuniform computability, recursive analysis, topological complexity, linear algebra 
Collection: 

6th International Conference on Computability and Complexity in Analysis (CCA'09) 
Issue Date: 

2009 
Date of publication: 

25.11.2009 