Alan Turing introduced the sensitivity of the solution of a numerical problem to changes in its data as a way to measure the difficulty of solving the problem accurately. Condition numbers are now considered fundamental to sensitivity analysis. They have been used to measure the mathematical difficulty of many linear algebra problems, including linear systems, linear least-squares, and eigenvalue problems. By definition, unless exact arithmetic is used, it is expected to be difficultto accurately solve an ill-conditioned problem.
In this talk we focus on least-squares problems. After a historical overview of condition number for least-squares, we introduce two related condition numbers. The first is the partial condition number, which measures the sensitivity of a linear combination of the components of the solution. The second is related the truncated SVD solution of the problem, which is often used when the matrix is nearly rank deficient.
Throughout the talk we are interested in three types of results :closed formulas for condition numbers, sharp bounds and statistical estimates.