Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability

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 $\rank(A)\in\{0,1,\ldots,n-1\}$; 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.