现代计算机代数(英文版)
2001-4
世界图书出版公司
J.Von zur Gathen
753
Modern Computer Algebra Computer algebra systems are gaining more and more importance in all areasof science and engineering. This textbook gives a thorough introduction to thealgorithmic basis of the mathematical engine in computer algebra systems. It is designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics. Its comprehensiveness and authority make it also an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). Some of this material has never appeared before in book form. Finally, a great deal of historical information and illustration enlivens the text. Joachim yon zur Gathen has a PhD from Universitat Ztirich, taught at University of Toronto from 1981 to 1994, and is now at Universitat Paderbom. Jtirgen Gerhard is completing his PhD and is wissenschaftlicher Mitarbeiter at Universitat Paderborn.
Introduction 1 Cyclohexane， cryptography， codes， and computer algebra 1.1 Cyclohexane conformations 1.2 The RSA cryptosystem 1.3 Distributed data structures 1.4 Computer algebra systems I Euclid 2 Fundamental algorithms 2.1 Representation and addition of numbers 2.2 Representation and addition of polynomials 2.3 Multiplication 2.4 Division with remainder Notes Exercises 3 The Euclidean Algorithm 3.1 Euclidean domains 3.2 The Extended Euclidean Algorithm 3.3 Cost analysis for Z and F[x] Notes Exercises 4 Applications of the Euclidean Algorithm 4.1 Modular arithmetic 4.2 Modualr inverses via Euclid 4.3 Repeated squaring 4.4 Modular inverses via Fermat 4.5 Linear Diophantine equations …… 5 Modualr algorithms and interpolation 6 The resultant and gcd computaion 7 Application:Decoding BCH codesⅡ Newton 8 Fast multiplication 9 Newton iteration 10 Fast polynomial evalution and interpoation 11 Fast Euclidean Algorithm 12 Fast Linear algebra 13 Fourier Transform and image compressionⅢ GauB 14 Factoring polynomials over finite fields 15 Hensel lifting and factoring polynomials 16 Short vectors in lattices 17 Applications of basis reductionⅣ Fermat 18 Primality testing 19 Factoring integers 20 Application:Public key cryptographyⅤ Hibert 21 Grobner bases 22 Symbolic integration 23 Symbolic Summation 24 Applications Appendix 25 Fundamental conceptsSources of illustrationsSources of quotaionsList of algorithmsLsit of figureds and tablesReferencesList of notationIndex
