Elementary Functions: Algorithms and Implementation

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Springer Science & Business Media, 2006 - Computers - 265 pages
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"An important topic, which is on the boundary between numerical analysis and computer science.... I found the book well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find. Moreover, there are very few books on these topics and they are not recent."

–Numerical Algorithms (review of the first edition)

This unique book provides concepts and background necessary to understand and build algorithms for computing the elementary functions—sine, cosine, tangent, exponentials, and logarithms. The author presents and structures the algorithms, hardware-oriented as well as software-oriented, and also discusses issues related to accurate floating-point implementation. The purpose is not to give "cookbook recipes" that allow one to implement a given function, but rather to provide the reader with tools necessary to build or adapt algorithms for their specific computing environment.

This expanded second edition contains a number of revisions and additions, which incorporate numerous new results obtained during the last few years. New algorithms invented since 1997—such as Matula’s bipartite method, another table-based method due to Ercegovac, Lang, Tisserand, and Muller—as well as new chapters on multiple-precision arithmetic and examples of implementation have been added. In addition, the section on correct rounding of elementary functions has been fully reworked, also in the context of new results. Finally, the introductory presentation of floating-point arithmetic has been expanded, with more emphasis given to the use of the fused multiply-accumulate instruction.

The book is an up-to-date presentation of information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find the book a useful reference and resource.

  

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Contents

Introduction
1
Some Basic Things About Computer Arithmetic
9
Algorithms Based on Polynomial Approximation andor Table Lookup MultiplePrecision Evaluation of Functions
25
Polynomial or Rational Approximations
27
MultiplePrecision Evaluation of Functions
89
ShiftandAdd Algorithms
101
Introduction to ShiftandAdd Algorithms
103
The CORDIC Algorithm
133
Some Other ShiftandAdd Algorithms
157
Range Reduction Final Rounding and Exceptions
171
Range Reduction
173
Final Rounding
193
Miscellaneous
217
Examples of Implementation
225
Copyright

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Page 244 - Redundant and online CORDIC: Application to matrix triangularization and SVD,
Page 257 - DA Sunderland, RA Strauch, SS Wharfield, HT Peterson, and CR Cole, "CMOS/SOS Frequency Synthesizer LSI Circuit for Spread Spectrum Communications," IEEE Journal of Solid State Circuits, August 1984, pp.
Page 240 - Reprinted in EE Swartzlander, Computer Arithmetic, Vol. 1, IEEE Computer Society Press Tutorial, Los Alamitos, CA, 1990.
Page 257 - N. Takagi, T. Asada and S. Yajima, A hardware algorithm for computing sine and cosine using redundant binary representation, Trans.
Page 258 - D. Timmermann, H. Hahn, BJ Hosticka, and G. Schmidt, "A programmable CORDIC chip for digital signal processing applications," IEEE Journal of Solid-State Circuits, Vol.
Page 259 - A. Ziv. Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded last Bit.
Page 258 - Walther. A unified algorithm for elementary functions. In Joint Computer Conference Proceedings, 1971. Reprinted in EE Swartzlander, Computer Arithmetic...

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