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Excerpt from On the Conditioning of the Nonsymmetric Eigenproblem: Theory and Software <p>The condition number of a problem measures the sensitivity of the solution to small changes in the input. We call the problem ill-conditioned if its condition number is large, and ill-posed if its condition number is infinite. We may use condition numbers to bound errors in computed solutions of numerical problems. <p>We illustrate this with a simple example. It is well known that the condition number for solving a system of linear equations is ic(a) E A where [i II is any matrix operator norm (we will be more specific about norms later). Suppose that linear system Ar. B is solved via Gaussian elimination with partial pivoting, or some other stable scheme. Let f be the computed solution. Then one may bound the error by. <p>About the Publisher <p>Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com <p>This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.