Excerpt from The Probability That a Numerical, Analysis Problem Is Difficult
Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zero finding, share the following property: the difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigenproblems, and polynomial zero finding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a "random" problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic.
To investigate the probability that a numerical analysis problem is difficult, we need to do three things:
1) Choose a measure of difficulty,
2) Choose a probability distribution on the set of problems,
3) Compute the distribution of the measure of difficulty induced by the distribution on the set of problems.
The measure of difficulty we shall use in this paper is the condition number, which measures the sensitivity of the solution to small changes in the problem. For the problems we consider in this paper (matrix inversion, polynomial zero finding and eigenvalue calculation), there are well known condition numbers in the literature of which we shall use slightly modified versions to be discussed more fully later. The condition number is an appropriate measure of difficulty because it can be used to measure the expected loss of accuracy in the computed solution, or even the number of iterations required for an iterative algorithm to converge to a solution.
The probability distribution on the set of problems for which we will attain most of our results will be the "uniform distribution" which we define as follows.
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