Abstract
A set of simple stopping criteria is presented, which improve the efficiency of iterative root finding by terminating the iterations immediately when no further improvement of the roots is possible. The criteria use only the function evaluations already needed by the root finding procedure to which they are applied. The improved efficiency is achieved by formulating the stopping criteria in terms of fractional significant digits. Test results show that the new stopping criteria reduce the iteration work load by about one-third compared with the most efficient stopping criteria currently available. This is achieved without compromising the accuracy of the extracted roots.
Highlights
Stopping criteria for root finding procedures for nonlinear functions fall into two categories: (1) those that rely on the user to specify a tolerance within which the roots are needed and (2) those that seek to terminate the iterations automatically when an iterate has been reached whose accuracy cannot be improved
Category (1) is easy to implement using stopping criteria such as |xi − xi−1| < e or |xi − xi−1|/|xi−1| < e, where xi and xi−1 are successive iterates and e is a user-supplied upper limit on the absolute or relative error. The drawback of such stopping criteria is that they shift the responsibility for providing accurate results from the program developer to the user
A category (2) stopping criterion for polynomial root finding was proposed by Adams [6]
Summary
A new set of stopping criteria for iterative root finding and eigenvalue extraction has been presented, which terminates the iterations immediately when no further improvement of the results is possible. The new criteria, called JLN, have been tested numerically against the existing stopping criteria of Igarashi [3,5], Grant & Hitchins [7] and Ward [9]. Igarashi occasionally failed to trigger, leading to multiple redundant iterations and uncertainty as to whether a root had been found. Both were rejected in their current formulations. (2) Ward and JLN were the only criteria that did not fail. When simplicity is more important than efficiency, Ward’s criterion is preferable. A numerical implementation of the JLN stopping criteria is available from the author on request
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