Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation
CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE
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The author considers methods for finding high-precision approximations to simple zeros of smooth functions. As an application, the author gives fast methods for evaluating the elementary functions logx, expx , sinx etc. to high precision. For example, if x is a positive floating-point number with an n-bit fraction, then under rather weak assumptions an n-bit approximation to logx or expx may be computed in time asymptotically equal to 13Mn log of n to the base 2 as n approaches infinity, where Mn is the time required to multiply floating-point numbers with n-bit fractions. Similar results are given for the other elementary functions, and some analogies with operations on formal power series are mentioned.
- Numerical Mathematics