Backward Errors and Condition Numbers of Regular and Singular Points on Algebraic Curves

Abstract

Expressions for the componentwise and normwise backward errors at an arbitrary point on an algebraic curve are derived and the formulae that relate them to the condition numbers are established. The expressions for the condition numbers at a regular point that is near a singular point are examined and it is shown that restrictions on their validity arise in this circumstance. In particular, the lowest order approximation that is used when condition numbers are derived places tight bounds on the maximum allowable perturbation on this class of point. An example that illustrates this limitation is given.