main.bib

@article{psw,
  ISSN = {0025-5718},
  abstract = {The odd composite $n \leqslant 25 \cdot 10^9$ such that $2^{n-1} \equiv 1 (\operatorname{mod n})$ have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Carmichael numbers. The theoretical upper bound and the heuristic lower bound due to Erdos for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences.},
  author = {Pomerance, Carl and Selfridge, J. L. and Wagstaff, Jr., Samuel S.},
  copyright = {Copyright 1980 American Mathematical Society},
  journal = {Mathematics of Computation},
  jstor_articletype = {Full Length Article},
  jstor_date = {198007},
  jstor_formatteddate = {Jul., 1980},
  keywords = {10A15, 10-04, Pseudoprime, strong pseudoprime, Euler pseudoprime, Carmichael number, primality testing, Lucas sequence},
  month = {jul},
  number = {151},
  pages = {1003--1026},
  publisher = {American Mathematical Society},
  title = {The Pseudoprimes to $25 \cdot 10^9$},
  url = {http://links.jstor.org/sici?sici=0025-5718%28198007%2935%3A151%3C1003%3ATPT%3E2.0.CO%3B2-D},
  volume = {35},
  year = {1980},
}

@Article{Kal95,
  author = 	 {Erich Kaltofen},
  title = 	 {Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems},
  journal = 	 {Mathematics of Computation},
  year = 	 {1995},
  OPTkey = 	 {},
  volume =	 {64},
  number =	 {210},
  pages =	 {777-806},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{KalSau91,
  author = 	 {E. Kaltofen and B. Saunders},
  title = 	 {On Wiedemann's method of solving sparse linear systems},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proc. AAECC-9, LNCS 539},
  pages =	 {29-38},
  year =	 {1991},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Springer-Verlag},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{Dum02,
  author = 	 {J. Dumas and others},
  title = 	 {LinBox: a generic library for exact linear algebra},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ICMS'02},
  pages =	 {40-50},
  year =	 {2002},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Beijing, China},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {World Scientific},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Cop94,
  author = 	 {D. Coppersmith},
  title = 	 {Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm},
  journal = 	 {Math. Comp.},
  year = 	 {1994},
  OPTkey = 	 {},
  volume =	 {62},
  number =	 {205},
  pages =	 {333-350},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{EbeKal97,
  author = 	 {W. Eberly and E. Kaltofen},
  title = 	 {On randomized Lanczos algorithms},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'97},
  pages =	 {176-183},
  year =	 {1997},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Maui, USA},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{lehmer1930,
  ISSN = {0003-486X},
  author = {D. H. Lehmer},
  copyright = {Copyright 1930 Annals of Mathematics},
  journal = {The Annals of Mathematics},
  jstor_articletype = {Full Length Article},
  jstor_date = {193007},
  jstor_formatteddate = {Jul., 1930},
  language = {EN},
  month = {jul},
  number = {3},
  pages = {419--448},
  publisher = {Annals of Mathematics},
  series = {2},
  title = {An Extended Theory of Lucas' Functions},
  url = {http://links.jstor.org/sici?sici=0003-486X%28193007%292%3A31%3A3%3C419%3AAETOLF%3E2.0.CO%3B2-E},
  volume = {31},
  year = {1930},
}

@Article{bruce,
  ISSN = {0002-9890},
  author = {Bruce, J. W.},
  copyright = {Copyright 1993 Mathematical Association of America},
  journal = {The American Mathematical Monthly},
  jstor_articletype = {Full Length Article},
  jstor_date = {199304},
  jstor_formatteddate = {Apr., 1993},
  month = {apr},
  number = {4},
  pages = {370--371},
  publisher = {Mathematical Association of America},
  title = {A Really Trivial Proof of the Lucas-Lehmer Test},
  url = {http://links.jstor.org/sici?sici=0002-9890%28199304%29100%3A4%3C370%3AARTPOT%3E2.0.CO%3B2-D},
  volume = {100},
  year = {1993},
}

@Article{apr,
  ISSN = {0003-486X},
  author = {Adleman, Leonard M. and Pomerance, Carl and Rumely, Robert S.},
  copyright = {Copyright 1983 Annals of Mathematics},
  journal = {The Annals of Mathematics},
  jstor_articletype = {Full Length Article},
  jstor_date = {198301},
  jstor_formatteddate = {Jan., 1983},
  month = {jan},
  number = {1},
  pages = {173--206},
  publisher = {Annals of Mathematics},
  series = {2},
  title = {On Distinguishing Prime Numbers from Composite Numbers},
  url = {http://links.jstor.org/sici?sici=0003-486X%28198301%292%3A117%3A1%3C173%3AODPNFC%3E2.0.CO%3B2-I},
  volume = {117},
  year = {1983},
}

@Article{cl,
  ISSN = {0025-5718},
  abstract = {We present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by Adleman and Rumely. The new algorithm performs well in practice. It is the first primality test in existence that can routinely handle numbers of hundreds of decimal digits.},
  author = {Cohen, H. and Lenstra, Jr., H. W. },
  copyright = {Copyright 1984 American Mathematical Society},
  journal = {Mathematics of Computation},
  jstor_articletype = {Full Length Article},
  jstor_date = {198401},
  jstor_formatteddate = {Jan., 1984},
  keywords = {10A25, 10-04, 12C20, Primality testing, Jacobi sum},
  month = {jan},
  number = {165},
  pages = {297--330},
  publisher = {American Mathematical Society},
  title = {Primality Testing and Jacobi Sums},
  url = {http://links.jstor.org/sici?sici=0025-5718%28198401%2942%3A165%3C297%3APTAJS%3E2.0.CO%3B2-M},
  volume = {42},
  year = {1984},
}

@Article{solovay,
author = {R. Solovay and V. Strassen},
collaboration = {},
title = {A Fast Monte-Carlo Test for Primality},
publisher = {SIAM},
year = {1977},
journal = {SIAM Journal on Computing},
volume = {6},
number = {1},
pages = {84-85},
keywords = {Monte-Carlo tests; primality},
url = {http://link.aip.org/link/?SMJ/6/84/1},
doi = {10.1137/0206006}
}

@Article{rabin,
  author = 	 {Michael O. Rabin},
  title = 	 {Probabilistic algorithm for testing primality},
  journal = 	 {Journal of Number Theory},
  year = 	 {1980},
  OPTkey = 	 {},
  volume =	 {12},
  number =	 {1},
  pages =	 {128-138},
  month =	 {February},
  OPTnote = 	 {},
  OPTannote = 	 {},
  url={http://www.sciencedirect.com/science/article/B6WKD-4CRP54N-KX/2/3f3a40d83c5897abfaae6b8e146e0616}
}

@Book{taocp2,
 author = {Donald E. Knuth},
 title = {The art of computer programming, volume 2: seminumerical algorithms},
 year = {1997},
 edition = {3},
 isbn = {0-201-89684-2},
 publisher = {Addison-Wesley Longman Publishing Co., Inc.},
 address = {Boston, MA, USA},
 }

@Book{riesel,
  author =	 {Hans Riesel},
  ALTeditor = 	 {},
  title = 	 {Prime Numbers and Computer Methods for Factorization},
  publisher = 	 {Boston; Basel; Stuttgart: Birkh\"auser},
  year = 	 {1985},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {PM 57},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{cohen,
  author =	 {Henri Cohen},
  ALTeditor = 	 {},
  title = 	 {A Course in Computational Algebraic Number Theory},
  publisher = 	 {Springer Verlag},
  year = 	 {1993},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {GTM 138},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{mca,
  author =	 {Joachim von zur Gathen and J\"urgen Gerhard},
  ALTeditor = 	 {},
  title = 	 {Modern Computer Algebra},
  publisher = 	 {Cambridge University Press},
  year = 	 {2002},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  edition =	 {2nd},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{wangdongming,
  author =	 {王东明，夏壁灿，李子明},
  ALTeditor = 	 {},
  title = 	 {计算机代数（第2版）},
  publisher = 	 {清华大学出版社},
  year = 	 {2007},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{zsg05,
  author = 	 {张树功，雷娜，刘停战},
  ALTeditor = 	 {},
  title = 	 {计算机代数基础---代数与符号计算的基本原理},
  publisher = 	 {科学出版社},
  year = 	 {2005},
  address =      {北京},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{rz79,
  author = 	 {Richard Zippel},
  title = 	 {Probabilistic algorithms for sparse polynomials},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =    {Lec. Notes Comp. Sci.},
  pages = 	 {216-226},
  year = 	 {1979},
  OPTeditor = 	 {},
  OPTvolume = 	 {72},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {London, UK},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =    {Springer-Verlag},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{brent1980,
  author = 	 {Richard P. Brent},
  title = 	 {An improved Monte Carlo factorization algorithm},
  journal = 	 {BIT Numerical Mathematics},
  year = 	 {1980},
  OPTkey = 	 {},
  volume =	 {20},
  number =	 {2},
  pages = 	 {176--194},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{gower08,
  author = 	 {Jason E. Gower and Wagstaff, Jr., Samuel S.},
  title = 	 {Square form factorization},
  journal = 	 {Mathematics of computation},
  year = 	 {2008},
  OPTkey = 	 {},
  volume =	 {77},
  number =	 {261},
  pages =	 {551-588},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{williams,
  author = 	 {H. C. Williams},
  title = 	 {A $p+1$ Method of Factoring},
  journal = 	 {Mathematics of Computation},
  year = 	 {1982},
  OPTkey = 	 {},
  volume =	 {39},
  number =	 {159},
  pages =	 {225-234},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{dixon,
  author = 	 {John D. Dixon},
  title = 	 {Asymptotically Fast Factorization of Integers},
  journal = 	 {Mathematics of Computation},
  year = 	 {1981},
  OPTkey = 	 {},
  volume =	 {36},
  number =	 {153},
  pages =	 {255-260},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{mon85,
  author = 	 {Peter L. Montgomery},
  title = 	 {Modular Multiplication Without Trial Division},
  journal = 	 {Mathematics of Computation},
  year = 	 {1985},
  OPTkey = 	 {},
  volume =	 {44},
  number =	 {170},
  pages =	 {519-521},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jt70,
  author = 	 {M. A. Jenkins and J. F. Traub},
  title = 	 {A three-stage variable-shift iteration for polynomial zeros in relation to generalized Rayleigh iteration},
  journal = 	 {Numer. Math.},
  year = 	 {1970},
  OPTkey = 	 {},
  volume = 	 {14},
  OPTnumber = 	 {},
  pages = 	 {252-263},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@MastersThesis{wrm01,
  author = 	 {Wankere R. Mekwi},
  title = 	 {Iterative Methods for Roots of Polynomials},
  school = 	 {Exeter College, University of Oxford},
  year = 	 {2001},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{wwapl94,
  author = 	 {William W. Adams and Philippe Loustaunau},
  ALTeditor = 	 {},
  title = 	 {An Introduction to Gr{\"o}bner Bases},
  publisher = 	 {American Mathematical Society},
  year = 	 {1994},
  OPTkey = 	 {},
  volume = 	 {3},
  OPTnumber = 	 {},
  series = 	 {Graduate Studies in Mathematics},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{hun74,
  author = 	 {T. W. Hungerford},
  ALTeditor = 	 {},
  title = 	 {Algebra},
  publisher = 	 {Springer Verlag},
  year = 	 {1974},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {Berlin and New York},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{fkq85,
  author = 	 {冯克勤译，聂灵沼校},
  ALTeditor = 	 {},
  title = 	 {代数学},
  publisher = 	 {湖南教育出版社},
  year = 	 {1985},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {长沙},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{zm02,
  author = 	 {周梦},
  ALTeditor = 	 {},
  title = 	 {计算代数与应用},
  publisher = 	 {武汉大学出版社},
  year = 	 {2002},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {武汉},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{rjbjhd89,
  author = 	 {R. J. Bradford and J. H. Davenport},
  title = 	 {Effective tests for cyclotomic polynomials},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle = {Symbolic and Algebraic Computation},
  pages = 	 {244-251},
  year = 	 {1989},
  OPTeditor = 	 {},
  volume = 	 {358},
  OPTnumber = 	 {},
  series = 	 {Lecture Notes in Computer Science},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =    {Springer Berlin/ Heidelberg},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jvzg90,
  author = 	 {J. von zur Gathen},
  title = 	 {Functional decomposition of polynomials: the tame case},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1990},
  OPTkey = 	 {},
  volume = 	 {9},
  OPTnumber = 	 {},
  pages = 	 {281-299},
  month = 	 {March},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{fkq03,
  author = 	 {冯克勤},
  ALTeditor = 	 {},
  title = 	 {初等数论及应用},
  publisher = 	 {北京师范大学出版社},
  year = 	 {2003},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{BRPKHT78,
  author = 	 {R. P. Brent and H. T. Kung},
  title = 	 {Fast algorithms for manipulating formal power series},
  journal = 	 {J. Assoc. Comput. Mach.},
  year = 	 {1978},
  OPTkey = 	 {},
  volume = 	 {25},
  number = 	 {4},
  pages = 	 {581-595},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{KHT74,
  author = 	 {H. T. Kung},
  title = 	 {On Computing Reciprocals of Power Series},
  journal = 	 {Numer. Math.},
  year = 	 {1974},
  OPTkey = 	 {},
  volume = 	 {22},
  OPTnumber = 	 {},
  pages = 	 {341-348},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{nlzdss00,
  author = 	 {聂灵沼，丁石孙},
  ALTeditor = 	 {},
  title = 	 {代数学引论（第2版）},
  publisher = 	 {高等教育出版社},
  year = 	 {2000},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {北京},
  edition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{hp98,
  author = 	 {Xiaohan HUANG and Victor Y. Pan},
  title = 	 {Fast Rectangular Matrix Multiplication and Applications},
  journal = 	 {Journal of Complexity},
  year = 	 {1998},
  OPTkey = 	 {},
  volume = 	 {14},
  OPTnumber = 	 {},
  pages = 	 {257-299},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{ks98,
  author = 	 {Erich Kaltofen and Victor Shoup},
  title = 	 {Subquadratic-Time Factoring of Polynomials over Finite Fields},
  journal = 	 {Mathematics of Computation},
  year = 	 {1998},
  OPTkey = 	 {},
  volume = 	 {67},
  OPTnumber = 	 {},
  pages = 	 {1179-1197},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Win68,
  author = 	 {S. Winograd},
  title = 	 {A New Algorithm for Inner Product},
  journal = 	 {IEEE Trans. Comp.},
  year = 	 {1968},
  OPTkey = 	 {},
  volume =	 {17},
  OPTnumber = 	 {},
  pages =	 {693-694},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Article{Hig89,
  author = 	 {N. J. Higham},
  title = 	 {The Accuracy of Solutions of Triangular Systems},
  journal = 	 {SIAM J. Numer. Anal.},
  year = 	 {1989},
  OPTkey = 	 {},
  volume =	 {26},
  OPTnumber = 	 {},
  pages =	 {1252-1265},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Book{GolVan01,
  author = 	 {G. H. Golub and C. F. van Loan著，袁亚湘等译},
  OPTeditor =	 {},
  title = 	 {矩阵计算},
  publisher = 	 {科学出版社},
  year = 	 {2001},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Str68,
  author = 	 {V. Strassen},
  title = 	 {Gaussian Elimination is not Optimal},
  journal = 	 {Numer. Math.},
  year = 	 {1969},
  OPTkey = 	 {},
  volume =	 {13},
  OPTnumber =	 {},
  pages =	 {354-356},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Book{ZhaXu04,
  author = 	 {张贤科，许甫华},
  OPTALTeditor = {},
  title = 	 {高等代数学（第2版）},
  publisher = 	 {清华大学出版社},
  year = 	 {2004},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTtype = 	 {},
  address =	 {北京},
  month =	 {7},
  pages =	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Article{Pan84,
  author = 	 {Victor Pan},
  title = 	 {How Can We Speed Up Matrix Multiplication?},
  journal = 	 {SIAM Review},
  year = 	 {1984},
  OPTkey = 	 {},
  volume =	 {26},
  number =	 {3},
  OPTpages = 	 {},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Book{Van92,
  author =	 {C. F. Van Loan},
  editor =	 {},
  title = 	 {Computational Frameworks for the Fast Fourier Transform},
  publisher = 	 {SIAM Publications},
  year = 	 {1992},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Philadelphia, PA},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}
 
@Article{Pai79,
  author = 	 {C. C. Paige},
  title = 	 {Computer Solution and Perturbation Analysis of Generalized Least Squares Problems},
  journal = 	 {Math. Comp.},
  year = 	 {1979},
  OPTkey = 	 {},
  volume =	 {33},
  OPTnumber = 	 {},
  pages =	 {171-184},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@Article{Fra61,
  author = 	 {J. G. F. Francis},
  title = 	 {The QR Transformation: A Unitary Analogue to the LR Transformation, Parts I and II},
  journal = 	 {Comp. J.},
  year = 	 {1961},
  OPTkey = 	 {},
  volume =	 {4},
  OPTnumber = 	 {},
  pages =	 {265-272, 332-345},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@Article{GolKah65,
  author = 	 {G. H. Golub and W. Kahan},
  title = 	 {Calculating the Singular Values and Pseudo-Inverse of a Matrix},
  journal = 	 {SIAM J. Num. Anal. Ser. B},
  year = 	 {1965},
  OPTkey = 	 {},
  volume =	 {2},
  OPTnumber = 	 {},
  pages =	 {205-224},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Cha82,
  author = 	 {T. Chan},
  title = 	 {An Improved Algorithm for Computing the Singular Value Decomposition},
  journal = 	 {ACM Trans. Math. Soft.},
  year = 	 {1982},
  OPTkey = 	 {},
  volume =	 {8},
  number =	 {1},
  pages =	 {72-83},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@Article{Wil68,
  author = 	 {J. H. Wilkinson},
  title = 	 {Global Convergence of Tridiagonal QR Algorithm With Origin Shifts},
  journal = 	 {Lin. Alg. and Its Applic.},
  year = 	 {1968},
  OPTkey = 	 {},
  volume =	 {I},
  OPTnumber = 	 {},
  pages =	 {409-420},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@Article{Ruh70,
  author = 	 {A. Ruhe},
  title = 	 {An Algorithm for Numerical Determination of the Structure of a General Matrix},
  journal = 	 {BIT},
  year = 	 {1976},
  OPTkey = 	 {},
  volume =	 {10},
  OPTnumber = 	 {},
  pages =	 {196-216},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Ste71,
  author = 	 {G. W. Stewart},
  title = 	 {Algorithm 406: HQR3 and EXCHNG: Fortran Subroutines for Calculating and Ordering the Eigenvalues of a Real Upper Hessenberg Matrix},
  journal = 	 {ACM Trans. Math. Soft.},
  year = 	 {1971},
  OPTkey = 	 {},
  volume =	 {2},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  month =	 {275-280},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Wil65,
  author =	 {J. H. Wilkinson},
  ALTeditor = 	 {},
  title = 	 {The Algebraic Eigenvalue Problem},
  publisher = 	 {Clarendon Press, Oxford, UK},
  year = 	 {1965},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Par74,
  author = 	 {B. N. Parlett},
  title = 	 {A Recurrence Among the Elements of Functions of Triangular Matrices},
  journal = 	 {Lin. Alg. and Its Applic.},
  year = 	 {1976},
  OPTkey = 	 {},
  volume =	 {14},
  OPTnumber = 	 {},
  pages =	 {117-121},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{TreBau06,
  author =	 {L. N. Trefethen and D. Bau著，{陆金甫}，{关治}译},
  ALTeditor = 	 {},
  title = 	 {数值线性代数},
  publisher = 	 {人民邮电出版社},
  year = 	 {2006},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{BarSte72,
  author = 	 {R. H. Bartels and G. W. Stewart},
  title = 	 {Solution of the Equation $AX + XB =C$},
  journal = 	 {Comm. ACM},
  year = 	 {1972},
  OPTkey = 	 {},
  volume =	 {15},
  OPTnumber = 	 {},
  pages =	 {820-826},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Ste72b,
  author = 	 {G. W. Stewart},
  title = 	 {On the Sensitivity of the Eigenvalue Problem $Ax=\lambda Bx$},
  journal = 	 {SIAM J. Num. Anal.},
  year = 	 {1972},
  OPTkey = 	 {},
  volume =	 {9},
  OPTnumber = 	 {},
  pages =	 {669-686},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{CliPle76,
  author = 	 {R. E. Cline and R. J. Plemmons},
  title = 	 {L2-Solution to Underdetermined Linear Systems},
  journal = 	 {SIAM Review},
  year = 	 {1976},
  OPTkey = 	 {},
  volume =	 {18},
  OPTnumber = 	 {},
  pages =	 {92-106},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{web95,
  author = 	 {Kenneth Weber},
  title = 	 {The accelerated integer GCD algorithm},
  journal = 	 {ACM Transactions on Mathematical Software},
  year = 	 {1995},
  OPTkey = 	 {},
  volume =	 {21},
  OPTnumber = 	 {1},
  pages =	 {111-122},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jeb03,
  author = 	 {T. Jebelean},
  title = 	 {A generalization of the binary GCD algorithm},
  journal = 	 {Proceedings of the 1993 international symposium on Symbolic and algebraic computation },
  year = 	 {1993},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {111-116},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sor94,
  author = 	 {Jonathan Sorenson},
  title = 	 {Two fast GCD algorithms},
  journal = 	 {Journal of Algorithms},
  year = 	 {1994},
  OPTkey = 	 {},
  volume =	 {16},
  number =	 {1},
  pages =	 {110-144},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sed07,
  author = 	 {Sidi Mohamed Sedjelmaci},
  title = 	 {Jebelean--Weber's algorithm without spurious factors},
  journal = 	 {Information Processing Letters},
  year = 	 {2007},
  OPTkey = 	 {},
  volume =	 {102},
  number =	 {6},
  pages =	 {247-252},
  month =	 {6},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{yao07,
  author =	 {{M.  Mend\`{e}s France} and  G. Tenenbaum著，姚家燕译},
  ALTeditor = 	 {},
  title = 	 {素数论},
  publisher = 	 {清华大学出版社},
  year = 	 {2007},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {研究生数学丛书},
  address =	 {北京},
  OPTedition = 	 {},
  month =	 {10},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{fra63,
  author = 	 {Evelyn Frank},
  title = 	 {On continued fraction expansions for binomial quadratic surds. III},
  journal = 	 {Numerische Mathematik},
  year = 	 {1963},
  OPTkey = 	 {},
  volume =	 {5},
  number =	 {1},
  pages =	 {113-117},
  month =	 {12},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{lag85,
  author = 	 {J.C. Lagaria and V.S. Miller and A.M. Odlyzko},
  title = 	 {Computing $\pi(x)$: The Meissel-Lehmer Method},
  journal = 	 {Mathematics of Computation},
  year = 	 {1985},
  OPTkey = 	 {},
  volume =	 {44},
  number =	 {170},
  pages =	 {537-560},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{del96,
  author = 	 {M. Del\'eglise and J. Rivat},
  title = 	 {Computing $\pi(x)$: The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method},
  journal = 	 {Mathematics of Computation},
  year = 	 {1996},
  OPTkey = 	 {},
  volume =	 {65},
  number =	 {213},
  pages =	 {235-245},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{lag87,
  author = 	 {J.C. Lagarias and A.M. Odlyzko},
  title = 	 {Computing $\pi(x)$: An Analytic Method},
  journal = 	 {Journal of Algorithms},
  year = 	 {1987},
  OPTkey = 	 {},
  volume =	 {8},
  number =	 {2},
  pages =	 {173-191},
  month =	 {6},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{del96m,
  author = 	 {M. Del\'eglise and J. Rivat},
  title = 	 {Computing the summation of the M\"obius function},
  journal = 	 {Experimental Mathematics},
  year = 	 {1996},
  OPTkey = 	 {},
  volume =	 {5},
  number =	 {4},
  pages =	 {291-295},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{gou01,
  author = 	 {X. Gourdon},
  title = 	 {Computation of $\ pi(x)$ : improvements to the Meissel, Lehmer, Lagarias, Miller, Odlyzko, Deleglise and Rivat method},
  note = 	 {\url{http://numbers.computation.free.fr/Constants/Primes/Pix/piNalgorithm.ps}},
  year = 	 {2001},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  month =	 {2},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{leh32,
  author = 	 {D. H. Lehmer},
  title = 	 {An inversive algorithm},
  journal = 	 {Bulletin of the American Mathematical Society},
  year = 	 {1932},
  OPTkey = 	 {},
  volume =	 {38},
  number =	 {10},
  pages =	 {693-694},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{dav88,
  author =	 {J. H. Davenport and Y. Siret and E. Tournier},
  ALTeditor = 	 {},
  title = 	 {Computer algebra: systems and algorithms for algebraic computation},
  publisher = 	 {Academic Press},
  year = 	 {1988},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {London, UK},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{bro05,
  author =	 {Manuel Bronstein},
  ALTeditor = 	 {},
  title = 	 {Symbolic Integration I: Transcendental Functions(2nd ed.)},
  publisher = 	 {Springer Verlag},
  year = 	 {2005},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{aca,
  author =	 {K.O. Geddes and S.R. Czapor and G. Labahn},
  ALTeditor = 	 {},
  title = 	 {Algorithms for Computer Algebra},
  publisher = 	 {Kluwer Academic Publishers},
  year = 	 {1992},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{bro90,
  author = 	 {Manuel Bronstein},
  title = 	 {Integration of elementary functions},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1990},
  OPTkey = 	 {},
  volume =	 {9},
  number =	 {2},
  pages =	 {117 - 173},
  month =	 {2},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{ris69,
  author = 	 {Robert H. Risch},
  title = 	 {The Problem of Integration in Finite Terms},
  journal = 	 {Transactions of the American Mathematical Society},
  year = 	 {1969},
  OPTkey = 	 {},
  volume =	 {139},
  OPTnumber = 	 {},
  pages =	 {167-189},
  month =	 {5},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{ris70,
  author = 	 {Robert H. Risch},
  title = 	 {The solution of the problem of integration in finite terms},
  journal = 	 {Bulletin of the American Mathematical Society},
  year = 	 {1970},
  OPTkey = 	 {},
  volume =	 {76},
  number =	 {3},
  pages =	 {605-608},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{ris79,
  author = 	 {Robert H. Risch},
  title = 	 {Algebraic Properties of the Elementary Functions of Analysis},
  journal = 	 {Robert H. Risch},
  year = 	 {1979},
  OPTkey = 	 {},
  volume =	 {101},
  number =	 {4},
  pages =	 {743-759},
  month =	 {8},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{dav84,
  author = 	 {J. H. Davenport},
  title = 	 {Int\'egration algorithmique des fonctions \'el\'ementairement transcendantes sur une courbe alg\'ebrique},
  journal = 	 {Annales de l'institut Fourier},
  year = 	 {1984},
  OPTkey = 	 {},
  volume =	 {34},
  number =	 {2},
  pages =	 {271-276},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{dav86,
  author = 	 {J. H. Davenport},
  title = 	 {The Risch differential equation problem},
  journal = 	 {SIAM Journal on Computing},
  year = 	 {1986},
  OPTkey = 	 {},
  volume =	 {15},
  number =	 {4},
  pages =	 {903 - 918},
  month =	 {11},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{rot77,
  author = 	 {M. Rothstein},
  title = 	 {A new algorithm for the integration of Exponential and Logarithmic Functions},
  journal = 	 {Proceedings of the 1977 MACSYMA Users Conference},
  year = 	 {1977},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {263-274},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{tra84,
  author = 	 {Trager B},
  title = 	 {Integration of Algebraic Functions},
  journal = 	 {Ph.D thesis, Dpt. of EECS, Massachusetts Institute of Technology},
  year = 	 {1984},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{put07,
  author =	 {Marius van der Put and Michael F. Singer},
  ALTeditor = 	 {},
  title = 	 {Galois theory of linear differential equations},
  publisher = 	 {科学出版社},
  year = 	 {2007},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{kap57,
  author =	 {Irving Kaplansky},
  ALTeditor = 	 {},
  title = 	 {An introduction to differential algebra},
  publisher = 	 {Hermann},
  year = 	 {1957},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Paris},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{kov05,
  author = 	 {Jerald J. Kovacic},
  title = 	 {An algorithm for solving second order linear homogeneous differential equations},
  journal = 	 {KSDA Lecture notes},
  year = 	 {2005},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  month =	 {9},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{kov01,
  author = 	 {Jerald J. Kovacic},
  title = 	 {An algorithm for solving second order linear homogeneous differential equations},
  journal = 	 {CCNY Colloquium lecture},
  year = 	 {2001},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  month =	 {9},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{ding04,
  author =	 {丁同仁，李承治},
  ALTeditor = 	 {},
  title = 	 {常微分方程教程},
  publisher = 	 {高等教育出版社},
  year = 	 {2004},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTedition = 	 {},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{kov86,
  author = 	 {Jerald J. Kovacic},
  title = 	 {An algorithm for solving second order linear homogeneous differential equations},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1986},
  OPTkey = 	 {},
  volume =	 {2},
  number =	 {1},
  pages =	 {3 - 43},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sau81,
  author = 	 {B. David Saunders},
  title = 	 {An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations},
  journal = 	 {Proceedings of the fourth ACM symposium on Symbolic and algebraic computation},
  year = 	 {1981},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {105-108},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{bro02,
  author = 	 {Manuel Bronstein},
  title = 	 {Solutions of linear ordinary differential equations in terms of special functions},
  journal = 	 {Proceedings of the 2002 international symposium on Symbolic and algebraic computation},
  year = 	 {2002},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {23-28},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{abr91,
  author = 	 {S. A. Abramov and K. Yu. Kvansenko},
  title = 	 {Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients},
  journal = 	 {Proceedings of the 1991 international symposium on Symbolic and algebraic computation},
  year = 	 {1991},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {267 - 270},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{bro92coe,
  author = 	 {Manuel Bronstein},
  title = 	 {On solutions of linear ordinary differential equations in their coefficient field},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1992},
  OPTkey = 	 {},
  volume =	 {13},
  number =	 {4},
  pages =	 {413 - 439},
  month =	 {4},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{bro92bar,
  author = 	 {Manuel Bronstein},
  title = 	 {Linear ordinary differential equations: breaking through the order 2 barrier},
  journal = 	 {Papers from the international symposium on Symbolic and algebraic computation},
  year = 	 {1992},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {42 - 48},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sin81,
  author = 	 {Michael F. Singer},
  title = 	 {Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations},
  journal = 	 {American Journal of Mathematics},
  year = 	 {1981},
  OPTkey = 	 {},
  volume =	 {103},
  number =	 {4},
  pages =	 {661-682},
  month =	 {8},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sin91,
  author = 	 {Michael F. Singer},
  title = 	 {Liouvillian solutions of linear differential equations with Liouvillian coefficients},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1991},
  OPTkey = 	 {},
  volume =	 {11},
  number =	 {3},
  pages =	 {251 - 273},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{feng00,
  author =	 {冯克勤},
  ALTeditor = 	 {},
  title = 	 {代数数论},
  publisher = 	 {科学出版社},
  year = 	 {2000},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTedition = 	 {},
  month =	 {6},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sel64,
  author = 	 {J. L. Selfridge and A. Hurwitz},
  title = 	 {Fermat numbers and Mersenne numbers},
  journal = 	 {Mathematics of Computation},
  year = 	 {1964},
  OPTkey = 	 {},
  volume =	 {18},
  OPTnumber = 	 {},
  pages =	 {146-148},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{mor75,
  author = 	 {Michael A. Morrison and John Brillhart},
  title = 	 {A Method of Factoring and the Factorization of $F_7$},
  journal = 	 {Mathematics of Computation},
  year = 	 {1975},
  OPTkey = 	 {},
  volume =	 {29},
  number =	 {129},
  pages =	 {183-205},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{shuzhifenxi,
  author = 	 {李庆扬，王能超，易大义},
  ALTeditor = 	 {},
  title = 	 {数值分析},
  publisher = 	 {清华大学出版社},
  year = 	 {2001},
  address =      {北京},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{ita,
  author =	 {H. Corman著，潘金贵等译},
  ALTeditor = 	 {},
  title = 	 {算法导论(原书第2版)},
  publisher = 	 {机械工业出版社},
  year = 	 {2006},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{BBP,
  author = 	 {D. H. Bailey and P. B. Borwein and S. Plouffe},
  title = 	 {On the computation of the n'th decimal digit of various transcendental numbers},
  journal = 	 {Mathematics of Computation},
  year = 	 {1996},
  OPTkey = 	 {},
  volume =	 {},
  number =	 {},
  pages =	 {1-2},
  month =	 {November},
  OPTnote = 	 {},
  OPTannote = 	 {},
  url={http://lacim.uqam.ca/~plouffe/Simon/articlepi.html}
}

@Article{Bellard-BBP,
  author = 	 {F. Bellard},
  title = 	 {A new formula to compute the n'th binary digit of pi},
  journal = 	 {Bellard's Website},
  year = 	 {1997},
  OPTkey = 	 {},
  volume =	 {},
  number =	 {},
  pages =	 {1-2},
  month =	 {January},
  OPTnote = 	 {},
  OPTannote = 	 {},
  url={http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html}
}

@Misc{pi-record,
  OPTkey = 	 {},
  author = 	 {Yasumasa Kanada},
  title =	 {Current published world record of pi calculation},
  OPThowpublished = {},
  month =	 {October},
  year =	 {2005},
  note =	 {\url{http://www.super-computing.org/pi_current.html}},
  OPTannote =	 {}
}

@Article{bs,
  author = 	 {David V. Chudnovsky and Gregory V. Chudnovsky},
  title = 	 {Computer algebra in the service of mathematical physics and number theory},
  journal = 	 {Computers and Mathematics},
  year = 	 {1990},
  OPTkey = 	 {},
  volume =	 {09},
  number =	 {},
  pages =	 {232},
  month =	 {},
  OPTnote = 	 {},
  OPTannote = 	 {},
}

@Article{Brent-agm,
  author = 	 {R. P. Brent},
  title = 	 {Multiple-precision zero-finding methods and the complexity of elementary function evaluation},
  journal = 	 {Analytic Computational Complexity},
  year = 	 {1975},
  OPTkey = 	 {},
  volume =	 {},
  number =	 {},
  pages =	 {151-176},
  month =	 {},
  OPTnote = 	 {},
  OPTannote = 	 {},
  url={http://wwwmaths.anu.edu.au/~brent/pub/pub028.html}
}

@Article{gor98,
  author = 	 {Daniel M. Gordon},
  title = 	 {A survey of fast exponentiation methods},
  journal = 	 {Journal of Algorithms},
  year = 	 {1998},
  OPTkey = 	 {},
  volume =	 {27},
  number =	 {1},
  pages =	 {129 - 146},
  month =	 {3},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{hong96,
  author = 	 {Seong Min Hong and Sang Yeop Oh and Hyun Soo Yoon},
  title = 	 {New modular multiplication algorithms for fast modular exponentiation},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Advances in Cryptology—Proceedings of Eurocrypt '96},
  pages =	 {166 - 177},
  year =	 {1996},
  OPTeditor = 	 {},
  volume =	 {1070},
  OPTnumber = 	 {},
  series =	 {Lecture Notes in Computer Science},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Springer Berlin / Heidelberg},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@InProceedings{bos90,
  author = 	 {Jurjen Bos and Matthijs Coster},
  title = 	 {Addition chain heuristics},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Advances in Cryptology - Proceedings of Crypto '89},
  pages =	 {400 - 407},
  year =	 {1990},
  OPTeditor = 	 {},
  volume =	 {435},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Springer-Verlag},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{aks04,
  author = 	 {Manindra Agrawal and Neeraj Kayal and Nitin Saxena},
  title = 	 {PRIMES is in P},
  journal = 	 {Annals of Mathematics},
  year = 	 {2004},
  OPTkey = 	 {},
  volume =	 {160},
  number =	 {2},
  pages =	 {781-793},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{gk86,
  author = 	 {S. Goldwasser and J. Kilian},
  title = 	 {Almost all primes can be quickly certified},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proceedings of the eighteenth annual ACM symposium on Theory of computing},
  pages =	 {316 - 329},
  year =	 {1986},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{atk93,
  author = 	 {A. O. L. Atkin and F. Morain},
  title = 	 {Elliptic curves and primality proving},
  journal = 	 {Mathematics of Computation},
  year = 	 {1993},
  OPTkey = 	 {},
  volume =	 {61},
  OPTnumber = 	 {},
  pages =	 {29 - 68},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{bai80,
  author = 	 {Robert Baillie and Wagstaff, Jr., Samuel S.},
  title = 	 {Lucas Pseudoprimes},
  journal = 	 {Mathematics of Computation},
  year = 	 {1980},
  OPTkey = 	 {},
  volume =	 {35},
  number =	 {152},
  pages =	 {1391 - 1417},
  month =	 {10},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{pom84,
  author = 	 {Carl Pomerance},
  title = 	 {Are There Counterexamples to the Baillie-PSW Primality Test?},
  note = 	 {\url{http://www.pseudoprime.com/dopo.pdf}},
  OPTkey = 	 {},
  OPTmonth = 	 {},
  year =	 {1984},
  OPTannote = 	 {}
}

@Misc{mar04,
  author = 	 {Martin, M.},
  title = 	 {Re: Baillie-PSW - Which variant is correct?},
  note = 	 {\url{http://groups.google.com/group/sci.crypt/msg/48ec324b9fc9f866}},
  OPTkey = 	 {},
  OPTmonth = 	 {},
  year =	 {2004},
  OPTannote = 	 {}
}

@Book{wwj84,
  author = 	 {吴文俊},
  ALTeditor = 	 {},
  title = 	 {几何定理机器证明的基本原理（初等几何部分）},
  publisher = 	 {科学出版社},
  year = 	 {1984},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{piandtheagm,
  author =	 {J. M. Borwein and P. B. Borwein},
  ALTeditor = 	 {},
  title = 	 {Pi and the AGM},
  publisher = 	 {Addison-Wesley Longman Publishing Co., Inc.},
  year = 	 {1987},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  edition =	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{LLL82,
  author = 	 {A. K. Lenstra and Lenstra, Jr., H. W.  and L. Lov\'asz},
  title = 	 {Factoring Polynomials with Rational Coefficients},
  journal = 	 {Mathematicsche Annaln},
  year = 	 {1982},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  number = 	 {261},
  pages = 	 {515-534},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{gmplib,
  OPTkey = 	 {},
  author = 	 {{The GMP team}},
  title =	 {The GNU MP Bignum Library},
  OPThowpublished = {},
  OPTmonth = 	 {},
  OPTyear = 	 {},
  note =	 {\url{http://gmplib.org/}},
  OPTannote = 	 {}
}

@Misc{qpi,
  key =		 {},
  author =	 {S. Pagliarulo},
  title =	 {Stu's pi page},
  OPThowpublished = {},
  OPTmonth = 	 {},
  OPTyear = 	 {},
  note =	 {\url{http://home.istar.ca/~lyster/otherconstants.html}},
  OPTannote = 	 {}
}

@Misc{Weisstein-machin,
   author = 	 {E. W. Weisstein},
   title = 	 {From MathWorld--A Wolfram Web Resource: Machin-Like Formulas},
   journal = 	 {},
   year = 	 {},
   OPTkey = 	 {},
   volume =	 {},
   number =	 {},
   pages =	 {},
   month =	 {},
   note = 	 {\url{http://mathworld.wolfram.com/Machin-LikeFormulas.html}},
   OPTannote = 	 {}
 }

@Misc{Weisstein-horner,
   author = 	 {E. W. Weisstein},
   title = 	 {From MathWorld--A Wolfram Web Resource: Horner's Rule},
   journal = 	 {},
   year = 	 {},
   OPTkey = 	 {},
   volume =	 {},
   number =	 {},
   pages =	 {},
   month =	 {},
   note = 	 {\url{http://mathworld.wolfram.com/HornersRule.html}},
   OPTannote = 	 {}
 }

 @Article{Lehmer-machin,
   author = 	 {D. H. Lehmer},
   title = 	 {On Arccotangent Relations for $\pi$},
   journal = 	 {Amer. Math. Monthly},
   year = 	 {1938},
   OPTkey = 	 {},
   volume =	 {45},
   number =	 {},
   pages =	 {657-664},
   month =	 {},
   OPTnote = 	 {},
   OPTannote = 	 {},
   url={}
 }

 @Manual{gmpman,
  title = 	 {The GNU Multiple Precision Arithmetic Library},
  OPTkey = 	 {},
  author = 	 {{The GMP team}},
  OPTorganization = {},
  OPTaddress = 	 {},
  edition = 	 {4.2.3},
  month = 	 {July},
  year = 	 {2008},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Hwang-machin,
   author = 	 {C. L. Hwang},
   title = 	 {More Machin-Type Identities.},
   journal = 	 {Math. Gaz.},
   year = 	 {1997},
   OPTkey = 	 {},
   volume =	 {81},
   number =	 {},
   pages =	 {120-121},
   month =	 {},
   OPTnote = 	 {},
   OPTannote = 	 {},
   url={}
 }

@Book{Berndt-rama,
  author = 	 {B. C. Berndt},
  ALTeditor = 	 {},
  title = 	 {Ramanujan's Notebooks, Part IV},
  publisher = 	 {Springer-Verlag},
  year = 	 {1994},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {New York},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{Chudnovsky-rama,
  author = 	 {David V. Chudnovsky and Gregory V. Chudnovsky},
  title = 	 {Approximations and Complex Multiplication According to Ramanujan},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987},
  pages =	 {375-472},
  year =	 {1987},
  editor =	 {G. E. Andrews and B. C. Berndt and R. A. Rankin},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Boston, MA},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Academic Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{Borwein-rama,
  author = 	 {J. M. Borwein and P. B. Borwein},
  title = 	 {More Ramanujan-Type Series for 1/PI},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987},
  pages =	 {359-374},
  year =	 {1988},
  editor =	 {G. E. Andrews and B. C. Berndt and R. A. Rankin},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {New York},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Academic Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{Weisstein-form,
   author = 	 {E. W. Weisstein},
   title = 	 {From MathWorld--A Wolfram Web Resource: Pi Formulas},
   journal = 	 {},
   year = 	 {},
   OPTkey = 	 {},
   volume =	 {},
   number =	 {},
   pages =	 {},
   month =	 {},
   note = 	 {\url{http://mathworld.wolfram.com/PiFormulas.html}},
   OPTannote = 	 {}
 }

@Article{Borwein-iter,
  author = 	 {J. M. Borwein and P. B. Borwein and D. H. Bailey},
  title = 	 {Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi},
  journal = 	 {Amer. Math. Monthly 96},
  year = 	 {1989},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages = 	 {201-219},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Beeler-iter,
  author = 	 {M. Beeler and R. W. Gosper and R. Schroeppel},
  ALTeditor = 	 {},
  title = 	 {HAKMEM},
  publisher = 	 {MIT Artificial Intelligence Laboratory, Memo AIM-239},
  year = 	 {1972},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {69},
  OPTseries = 	 {},
  address =	 {Cambridge, MA},
  OPTedition = 	 {},
  OPTmonth = 	 {Feb},
  note = 	 {\url{http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item140}},
  OPTannote = 	 {}
}

@Misc{mjhdrs06,
  OPTkey = 	 {},
  author =	 {M. Jason Hinek and Douglas R. Stinson},
  title =	 {An Inequality About Factors of Multivariate Polynomials},
  OPThowpublished = {},
  OPTmonth = 	 {},
  year =	 {2006},
  note =	 {\url{http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-15.pdf}},
  OPTannote = 	 {}
}


@InProceedings{jsc04,
  author = 	 {J. S. Coron},
  title = 	 {Finding small roots of bivariate integer polynomial equations revisited},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =    {Lecture Notes in Computer Science},
  pages = 	 {492-505},
  year = 	 {2004},
  OPTeditor = 	 {},
  volume = 	 {3027},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{rz93,
  author = 	 {Richard Zippel},
  ALTeditor = 	 {},
  title = 	 {Effecive Polynmial Computation},
  publisher = 	 {Kluwer Academic Publishers},
  year = 	 {1993},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address = 	 {Boston/Dordrecht/London},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jmdyyy73,
  author = 	 {Joel Moses and David Y. Y. Yun},
  title = 	 {The {EZ GCD} algorithm},
  journal = 	 {Proc. ACM73},
  year = 	 {1973},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages = 	 {159-166},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{wang75,
  author = 	 {P. S. Wang and L. P. Rothschild},
  title = 	 {Factoring multivariate polynomials over the integers},
  journal = 	 {Math. Comp.},
  year = 	 {1975},
  OPTkey = 	 {},
  volume =	 {29},
  OPTnumber = 	 {},
  pages =	 {935-950},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{wang78,
  author = 	 {P. S. Wang},
  title = 	 {An Improved Multivariate Polynomial Factoring Algorithm},
  journal = 	 {Math. Comp.},
  year = 	 {1978},
  OPTkey = 	 {},
  volume =	 {32},
  OPTnumber = 	 {},
  pages =	 {1215-1231},
  month =	 {October},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Mcc73,
  author = 	 {M. McClellan},
  title = 	 {The exact solution of systems of linear equations with polynomial coefficients},
  journal = 	 {Journal of the Association for Computing Machinery},
  year = 	 {1973},
  OPTkey = 	 {},
  volume =	 {20},
  number = 	 {4},
  pages =	 {563-588}
}
  
@Article{Dix82,
  author = 	 {John D. Dixon},
  title = 	 {Exact solution of linear equations using p-adic expansions},
  journal = 	 {Numer. Math.},
  year = 	 {1982},
  OPTkey = 	 {},
  volume =	 {40},
  OPTnumber = 	 {},
  pages =	 {137-141},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{RocSzu92,
  author =	 {A. Rockett and P. Szusz},
  OPTeditor =	 {},
  title = 	 {Continued fractions},
  publisher = 	 {World Scientific},
  year = 	 {1992},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Unpublished{CheSto04,
  author = 	 {Zhuliang Chen and Arne Storjohann},
  title = 	 {An implementation of certified linear system solving for integer matrices},
  note =	 {Poster at ISSAC},
  OPTkey = 	 {},
  OPTmonth =	 {},
  year =	 {2004},
  OPTannote = 	 {}
}

@Unpublished{CheSto04a,
  author = 	 {Zhuliang Chen and Arne Storjohann},
  title = 	 {A fast implementation of system solving for integer matrices},
  note = 	 {Poster at ECCAD'2004},
  OPTkey = 	 {},
  OPTmonth = 	 {},
  year =	 {2004},
  OPTannote = 	 {}
}

@Article{Sto05,
  author = 	 {Arne Storjohann},
  title = 	 {The shifted number system for fast linear algebra on integer matrices},
  journal = 	 {Journal of complexity},
  year = 	 {2005},
  OPTkey = 	 {},
  volume =	 {21},
  OPTnumber = 	 {},
  pages =	 {609-650},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Sto03,
  author = 	 {Arne Storjohann},
  title = 	 {High-order lifting and integrality certification},
  journal = 	 {Journal of symbolic computation},
  year = 	 {2003},
  OPTkey = 	 {},
  volume =	 {36},
  OPTnumber = 	 {},
  pages =	 {613-648},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{DumGio04,
  author = 	 {Jean-Guillaume Dumas and Pascal Giogi and Cl\'ement Pernet},
  title = 	 {FFPACK: finite field linear algebra package},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'04},
  OPTpages = 	 {},
  year =	 {2004},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Santander, Spain},
  month =	 {July, 4-7},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{WanPan03,
  author = 	 {X. Wang and V. Pan},
  title = 	 {Acceleration of euclidean algorithm and rational number reconstruction},
  journal = 	 {SIAM J. Comput.},
  year = 	 {2003},
  OPTkey = 	 {},
  volume =	 {32},
  number =	 {2},
  pages =	 {548-556},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Pan87,
  author = 	 {Victor Pan},
  title = 	 {Complexity of parallel matrix computations},
  journal = 	 {Theor. Comp. Sc.},
  year = 	 {1987},
  OPTkey = 	 {},
  volume =	 {54},
  OPTnumber = 	 {},
  pages =	 {65-85},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{Gie97,
  author = 	 {Mark Giesbrecht},
  title = 	 {Effient parallel solution of sparse systems of linear diophantine equations},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {PASCO'97},
  pages =	 {1-10},
  year =	 {1997},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Maui, USA},
  month =	 {July},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{Vil07,
  author = 	 {Gilles Villard},
  title = 	 {Some recent progress in exact linear algebra and related questions},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'07},
  OPTpages = 	 {},
  year =	 {2007},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Waterloo, Ontario, Canada},
  month =	 {July 29-August 1},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@Unpublished{Wan05,
  author = 	 {Zhendong Wan},
  title = 	 {An algorithm to solve linear systems exactly using numerical methods},
  note = 	 {preprint submitted to elsevier science},
  OPTkey = 	 {},
  month =	 {12},
  year =	 {2005},
  OPTannote = 	 {}
}



@InProceedings{WanPan02,
  author = 	 {X. Wang and V. Pan},
  title = 	 {Acceleration of euclidean algorithms and extensions},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'02},
  OPTpages = 	 {},
  year =	 {2002},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@Article{Wie86,
  author = 	 {D. Wiedemann},
  title = 	 {Solving sparse linear equations over finite fields},
  journal = 	 {IEEE transactions on infomation theory},
  year = 	 {1986},
  OPTkey = 	 {},
  volume =	 {32},
  number =	 {1},
  pages =	 {54-62},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@InProceedings{Mon95,
  author = 	 {P. Montegomery},
  title = 	 {A block lanczos algorithm for finding dependencis over gf(2)},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Advances in cryptology, EUROCRYPT'95, LNCS 921},
  pages =	 {106-121},
  year =	 {1995},
  OPTeditor = 	 {Guillou L., Quisquarter J.},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Springer-Verlag},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Lad76,
  author = 	 {Laderman},
  title = 	 {A noncommutative algorithm for multiplying $3\times 3$ matrices using 23 multiplications},
  journal = 	 {Bull. AMS},
  year = 	 {1976},
  OPTkey = 	 {},
  volume =	 {82},
  OPTnumber = 	 {},
  pages =	 {126-128},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Xu90,
  author =	 {徐献瑜},
  ALTeditor = 	 {},
  title = 	 {Pade逼近概论},
  publisher = 	 {上海科学技术出版社},
  year = 	 {1990},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {上海},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{CheEbe02,
  author = 	 {Li Chen and others},
  title = 	 {Efficient matrix preconditioners for black box linear algebra},
  journal = 	 {linear algebra and its applications},
  year = 	 {2002},
  OPTkey = 	 {},
  volume =	 {343-344},
  OPTnumber = 	 {},
  pages =	 {119-146},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@Article{Bareiss,
  author = 	 {E. H. Bareiss},
  title = 	 {Sylvester's identity and multistep integers preserving Gaussian elimination},
  journal = 	 {Mathematics of Computation},
  year = 	 {1968},
  OPTkey = 	 {},
  volume =	 {22},
  OPTnumber = 	 {},
  pages =	 {565-578},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Chee,
  author =	 {Chee-Keng Yap},
  ALTeditor = 	 {},
  title = 	 {Fundamental problems of algorithmic algebra},
  publisher = 	 {Oxford University Press},
  year = 	 {2000},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@TechReport{Par95,
  author = 	 {D. S. Parker},
  title = 	 {A randomizing butterfly transformation useful in block matrix computations},
  institution =  {Computer Science Department, University of California},
  year = 	 {1995},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTnumber = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{DumSau01,
  author = 	 {Jean-Guillaume Dumas and B. David Saunders and Gilles Villard},
  title = 	 {On efficient sparse integer matrix smith normal forms computations},
  journal = 	 {J. symbolic computation},
  year = 	 {2001},
  OPTkey = 	 {},
  volume =	 {32},
  OPTnumber = 	 {},
  pages =	 {71-99},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{CopWin87,
  author = 	 {Don Coppersmith and Shmuel Winograd},
  title = 	 {Matrix multiplication via arithmetic progressions},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {proceedings of the nineteenth annual acm symposium on theory of computing},
  OPTpages = 	 {},
  year =	 {1987},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{StoLab93,
  author = 	 {A. Storjohann and G. Labahn},
  title = 	 {A fast las vegas algorithm for computing the smith normal form of a polynomial matrix},
  journal = 	 {linear algebra and its applications},
  year = 	 {1993},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@PhdThesis{Sto00,
  author = 	 {Arne Storjohann},
  title = 	 {Algorithms for matrix canonical forms},
  school = 	 {Swiss federal institute of technology},
  year = 	 {2000},
  OPTkey = 	 {},
  OPTtype = 	 {},
  address =	 {Zurich},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{HafMcc91,
  author = 	 {J. Hafner and K. Maccurley},
  title = 	 {Asymptotically fast triangularization of matrices over rings},
  journal = 	 {SIAM J. Comput.},
  year = 	 {1991},
  OPTkey = 	 {},
  volume =	 {20},
  number =	 {6},
  pages =	 {1068-1083},
  month =	 {12},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@MastersThesis{Sto94,
  author = 	 {Arne Storjohann},
  title = 	 {Computation of hermite and smith normal forms of matrices},
  school = 	 {University of Waterloo},
  year = 	 {1994},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTaddress = 	 {Waterloo, Otario, Canada},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Sto98,
  author = 	 {Arne Storjohann},
  title = 	 {Computing hermite and smith normal forms of triangular integer matrices},
  journal = 	 {linear algebra and its applications},
  year = 	 {1998},
  OPTkey = 	 {},
  volume =	 {282},
  OPTnumber = 	 {},
  pages =	 {25-45},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{GieSto02,
  author = 	 {Mark Giesbrecht and Arne Storjohann},
  title = 	 {Computing rational forms of integer matrices},
  journal = 	 {J. symbolic computation},
  year = 	 {2002},
  OPTkey = 	 {},
  volume =	 {34},
  OPTnumber = 	 {},
  pages =	 {157-172},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Proceedings{BarMac90,
  title = 	 {Fast error-free algorithms for polynomial matrix computations},
  year = 	 {1990},
  OPTkey = 	 {},
  booktitle =	 {proceedings of the 29th conference on decision and control},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Honolulu, Hawaii},
  month =	 {12},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Ger01,
  author = 	 {J\"urgen Gerhard},
  title = 	 {Fast modular algorithms for squarefree factorization and hermite integration},
  journal = 	 {applicable algebra in engineering, communication and computing},
  year = 	 {2001},
  OPTkey = 	 {},
  volume =	 {11},
  OPTnumber = 	 {},
  pages =	 {203-226},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{DomKan87,
  author = 	 {P. Domich and R. Kannan and L. Trotter},
  title = 	 {Hermite normal form computation using modulo determinant arithmetic},
  journal = 	 {mathematics of operations research},
  year = 	 {1987},
  OPTkey = 	 {},
  volume =	 {12},
  number =	 {1},
  pages =	 {50-59},
  month =	 {2},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{HavMaj97,
  author = 	 {G. Havas and B. Majewski},
  title = 	 {Integer matrix diagonalization},
  journal = 	 {J. symbolic computation},
  year = 	 {1997},
  OPTkey = 	 {},
  volume =	 {24},
  OPTnumber = 	 {},
  pages =	 {399-408},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{Sto96,
  author = 	 {Arne Storjohann},
  title = 	 {Near optimal algorithms for computing smith normal forms of integer matrices},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'96},
  OPTpages = 	 {},
  year =	 {1996},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Zurich, Switzerland},
  month =	 {7},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Gie95,
  author = 	 {Mark Giesbrecht},
  title = 	 {Near optimal algorithms for canonical matrix forms},
  journal = 	 {SIAM J. Comput.},
  year = 	 {1995},
  OPTkey = 	 {},
  volume =	 {24},
  number =	 {5},
  pages =	 {948-969},
  month =	 {10},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{EbeGie00,
  author = 	 {W. Ebeley and M. Giesbrecht and G. Villard},
  title = 	 {On computing the determinant and smith form of an integer matrix},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {41st IEEE annual symposium on foundation of computer science, FOCS'2000},
  OPTpages = 	 {},
  year =	 {2000},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  month =	 {11},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@PhdThesis{Gie93,
  author = 	 {Mark Giesbrecht},
  title = 	 {Nearly optimal algorithms for canonical matrix forms},
  school = 	 {University of Toronto},
  year = 	 {993},
  OPTkey = 	 {},
  OPTtype = 	 {},
  address =	 {Toronto, Canada},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{KanBac79,
  author = 	 {R. Kannan and A. Bachem},
  title = 	 {Polynomial algorithms for computing the smith and hermite normal forms of an integer matrix},
  journal = 	 {SIAM J. Comput.},
  year = 	 {1979},
  OPTkey = 	 {},
  volume =	 {8},
  number =	 {4},
  pages =	 {499-507},
  month =	 {11},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{BunHop72,
  author = 	 {J. Bunch and J. Hopcroft},
  title = 	 {Triangularization and inversion by fast matrix multiplication},
  OPTjournal = 	 {},
  year = 	 {1972},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{AbdMal01,
  author = 	 {J. Abdeljaoued and G. Malaschonok},
  title = 	 {Efficient algorithms for computing the characteristic polynomial in a domain},
  journal = 	 {Journal of pure and applied algebra},
  year = 	 {2001},
  OPTkey = 	 {},
  volume =	 {156},
  OPTnumber = 	 {},
  pages =	 {127-145},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{PerSto07,
  author = 	 {Cl\'ement Pernet and Arne Storjohann},
  title = 	 {Faster algorithms for the characteristic polynomial},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'07},
  OPTpages = 	 {},
  year =	 {2007},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Waterloo, Ontario, Canada},
  month =	 {July 29-August 1},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Unpublished{PerWan03,
  author = 	 {Cl\'ement Pernet and Zhendong Wan},
  title = 	 {LU based algorithms for characteristic polynomial over a finite field},
  note = 	 {ISSAC'03 poster},
  OPTkey = 	 {},
  OPTmonth = 	 {},
  OPTyear = 	 {},
  OPTannote = 	 {}
}

@Article{PanMur07,
  author = 	 {V. Pan and B. Murphy and others},
  title = 	 {Numerical computation of determinants with additive preconditioning},
  OPTjournal = 	 {},
  year = 	 {2007},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{GenJoh76,
  author = 	 {W. Gentleman and S. Johnson},
  title = 	 {Analysis of algorithms, a case study: determinants of matrices with polynomial entries},
  journal = 	 {ACM transactions on mathematical software},
  year = 	 {1976},
  OPTkey = 	 {},
  volume =	 {2},
  number =	 {3},
  pages =	 {232-241},
  month =	 {9},
  OPTnote = 	 {},
  OPTannote = 	 {}
}



@InProceedings{Vil03,
  author = 	 {Gilles Villard},
  title = 	 {Computation of the inverse and determinant of a matrix},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Algorithms seminar, 2001-2002},
  pages =	 {29-32},
  year =	 {2003},
  editor =	 {Chyzak, F.},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  month =	 {11},
  organization = {Institut national de recherche en informatique et en automatique},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{KalVil04,
  author = 	 {Erich Kaltofen and Gilles Villard},
  title = 	 {Computing the sign or the value of the determinant of an integer matrix, a complexity survey},
  journal = 	 {Journal of computational and applied mathematics},
  year = 	 {2004},
  OPTkey = 	 {},
  volume =	 {162},
  OPTnumber = 	 {},
  pages =	 {133-146},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{JeaVil05,
  author = 	 {C. Jeannerod and G. Villard},
  title = 	 {Essentially optimal computation of the inverse of generic polynomial matrices},
  journal = 	 {Journal of complexity},
  year = 	 {2005},
  OPTkey = 	 {},
  volume =	 {21},
  OPTnumber = 	 {},
  pages =	 {72-86},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


@Article{EmiPan05,
  author = 	 {I. Emiris and V. Pan},
  title = 	 {Improved algorithms for computing determinants and resultants},
  journal = 	 {Journal of complexity},
  year = 	 {2005},
  OPTkey = 	 {},
  volume =	 {21},
  OPTnumber = 	 {},
  pages =	 {43-71},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{AbbBro99,
  author = 	 {J. Abbott and M. Bronstein and T. Mulders},
  title = 	 {Fast deterministic computation of determinants of dense matrices},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {ISSAC'99},
  OPTpages = 	 {},
  year =	 {1999},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Vancouver, British Columbia, Canada},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{VilDum02,
  author = 	 {Gilles Villard and Jean-Guillaume Dumas},
  title = 	 {Computing the rank of large sparse matrices over finite fields},
  journal = 	 {Computer Algebra in Scientific Computing},
  year = 	 {2002},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTpages = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Unpublished{DumPer08,
  author = 	 {Jean-Guillaume Dumas and Cl\'ement Pernet and Zhendong Wan},
  title = 	 {Efficient computation of the characteristic polynomial},
  note = 	 {arXiv:cs/0501074v2},
  OPTkey = 	 {},
  OPTmonth = 	 {},
  OPTyear = 	 {},
  OPTannote = 	 {}
}

@Article{Lanczos-gamma,
  author = 	 {Lanczos, Cornelius},
  title = 	 {A Precision Approximation of the Gamma Function},
  journal = 	 {SIAM Journal on Numerical Analysis series B},
  year = 	 {1964},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  number =	 {1},
  pages =	 {86-96},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Spouge-gamma,
  author = 	 {Spouge, John L.},
  title = 	 {Computation of the gamma, digamma, and trigamma functions},
  journal = 	 {SIAM Journal on Numerical Analysis},
  year = 	 {1994},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  number =	 {3},
  pages =	 {931-944},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{Gutierrez-fastinverse,
  author = {C. Gutierrez and M. Monsalve}, 
  title = {Fast inverse for big numbers: Picarte's iteration}, 
  OPThowpublished = {},
  month =	 {October},
  year =	 {2005},
  note =	 {Technical Report},
  OPTannote =	 {}
}

@Misc{Burnikel-fastrecursive,
  author = {C. Burnikel and J. Ziegler and I. Stadtwald and D. Saarbrucken},
  title = {Fast Recursive Division},
  OPThowpublished = {},
  month =	 {October},
  year =	 {1998},
  note =	 {Research Report},
  OPTannote =	 {}
}

@Article{Karatsuba-multiply,
  author = 	 {A. Karatsuba and P. Ofman},
  title = 	 {Multiplication of Many-Digital Numbers by Automatic Computers},
  journal = 	 {Proceedings of the USSR Academy of Sciences},
  year = 	 {1962},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  number =	 {145},
  pages =	 {293-294},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Schoenhage-multiply,
  author = 	 {A. Schoenhage and V. Strassen},
  title = 	 {Schnelle Multiplikation groeer Zahlen},
  journal = 	 {Computing},
  year = 	 {1971},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  number =	 {7},
  pages =	 {281-292},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}


%rw added

@InProceedings{cmf72,
  author = 	 {Charles M. Fiduccia},
  title = 	 {Polynomial evaluation via the division algorithm: the fast Fourier transform revisited},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {In Proceedings of the Fourth Annual ACM Symposium on the Theory of Computing, Denver CO},
  pages =	 {88-93},
  year =	 {1972},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{eh72,
  author = 	 {Ellis Horowitz},
  title = 	 {A fast method for interpoloation using preconditioning},
  journal = 	 {Infomation Processing Letters 1},
  year = 	 {1972},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  pages =	 {157-163},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{wsb71,
  author = 	 {W. S. Brown},
  title = 	 {On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors},
  journal = 	 {Journal of the ACM},
  year = 	 {1971},
  OPTkey = 	 {},
  volume =	 {18},
  number =	 {4},
  pages =	 {478-504},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{dhl38,
  author = 	 {D. H. Lehmer},
  title = 	 {Euclid's algorithm for large numbers},
  journal = 	 {The American Mathematical Monthly},
  year = 	 {1938},
  OPTkey = 	 {},
  volume =	 {45},
  OPTnumber = 	 {},
  pages =	 {227-233},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{dek70,
  author = 	 {Donald E. Knuth},
  title = 	 {The analysis of algorithms},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {In proceedings of the International Congress of Mathematicians},
  pages =	 {269-274},
  year =	 {1970},
  OPTeditor = 	 {},
  volume =	 {3},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  OPTpublisher = {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{as71,
  author = 	 {A. Sch\"onhage},
  title = 	 {Schnelle Berechnung von Kettenbruchentwicklungen},
  journal = 	 {Acta Informatica},
  year = 	 {1971},
  OPTkey = 	 {},
  volume =	 {1},
  OPTnumber = 	 {},
  pages =	 {139-144},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{rtm73,
  author = 	 {R. T. Moenck},
  title = 	 {Fast computation of gcd's},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {In Proceedings of the Fifth Annual ACM Symposium on the Theory of Computing, Austin TX},
  pages =	 {142-151},
  year =	 {1973},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jts80,
  author = 	 {J. T. Schwartz},
  title = 	 {Fast Probabilistic Algorithms for Verification of Polynomial Indentities},
  journal = 	 {Journal of the ACM},
  year = 	 {1980},
  OPTkey = 	 {},
  volume =	 {27},
  number =	 {4},
  pages =	 {701-717},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{vs83,
  author = 	 {V. Strassen},
  title = 	 {The computational complexity of continued fractions},
  journal = 	 {SIAM journal on Computing},
  year = 	 {1983},
  OPTkey = 	 {},
  volume =	 {12},
  number =	 {1},
  pages =	 {1-27},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{hz69,
  author = 	 {Hans Zassenhaus},
  title = 	 {On Hensel Factorization},
  journal = 	 {Journal of Number Theory},
  year = 	 {1969},
  OPTkey = 	 {},
  volume =	 {1},
  OPTnumber = 	 {},
  pages =	 {291-311},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{dgchz81,
  author = 	 {David G. Cantor and Hans Zassenhaus},
  title = 	 {A New Algorithm for Factoring Polynomials Over Finite Fields},
  journal = 	 {Mathematics of Computation},
  year = 	 {1981},
  OPTkey = 	 {},
  volume =	 {36},
  number =	 {154},
  pages =	 {587-592},
  month =	 {April},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jvzgvs92,
  author = 	 {J. von zur Gathen and V. Shoup},
  title = 	 {Computing Frobenius maps and factoring polynomials},
  journal = 	 {Computational Complexity},
  year = 	 {1992},
  OPTkey = 	 {},
  volume =	 {2},
  OPTnumber = 	 {},
  pages =	 {187-224},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{dyyy76,
  author = 	 {David Y. Y. Yun},
  title = 	 {On Square-free Decomposition Algorithms},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {In Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation ISSAC'76},
  pages =	 {26-35},
  OPTyear = 	 {},
  editor =	 {R. D. Jenks},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Yorktown Heights NY},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM Press},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{berlekamp67,
  author = 	 {E. R. Berlekamp},
  title = 	 {Factoring polynomials over finite fields},
  journal = 	 {Bell System Technical Journal},
  year = 	 {1967},
  OPTkey = 	 {},
  volume =	 {46},
  OPTnumber = 	 {},
  pages =	 {1853-1859},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{berlekamp70,
  author = 	 {E. R. Berlekamp},
  title = 	 {Factoring polynomials over large finite fields},
  journal = 	 {Mathematics of Computation},
  year = 	 {1970},
  OPTkey = 	 {},
  volume =	 {24},
  number =	 {11},
  pages =	 {713-735},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{hensel18,
  author = 	 {Kurt Hensel},
  title = 	 {Eine neue Theorie der algebraischen Zablen},
  journal = 	 {Mathematicsche Zeitschift},
  year = 	 {1918},
  OPTkey = 	 {},
  volume =	 {2},
  OPTnumber = 	 {},
  pages =	 {433-452},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jvzg84,
  author = 	 {J. von zur Gathen},
  title = 	 {Hensel and Newton methods in valuation rings},
  journal = 	 {Mathematics of Computation},
  year = 	 {1984},
  OPTkey = 	 {},
  volume =	 {42},
  number =	 {166},
  pages =	 {637-661},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{jt70real,
  author = 	 {M. A. Jenkins and J. F. Traub},
  title = 	 {A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration},
  journal = 	 {SIAM Journal on Numerical Analysis},
  year = 	 {1970},
  OPTkey = 	 {},
  volume =	 {7},
  number =	 {4},
  pages =	 {545-566},
  pages =	 {545-566},
  month =	 {Dec.},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{pan66,
  author = 	 {V. Ya. Pan},
  title = 	 {Methods of computing values of polynomials},
  journal = 	 {Russian Mathematical Surveys},
  year = 	 {1966},
  OPTkey = 	 {},
  volume =	 {21},
  OPTnumber = 	 {},
  pages =	 {105-136},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

% new of lc
@MastersThesis{kah53,
  author = 	 {H. G. Kahrimanian},
  title = 	 {Analytical Differentiation by a Digital Computer},
  school = 	 {Temple University},
  year = 	 {1953},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTaddress = 	 {},
  month =	 {5},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@MastersThesis{nol53,
  author = 	 {J. Nolan},
  title = 	 {Analytical Differentiation on a Digital Computer},
  school = 	 {Massachusetts Institute of Technology},
  year = 	 {1953},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTaddress = 	 {},
  month =	 {5},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{sla63,
  author = 	 {James R. Slagle},
  title = 	 {A Heuristic Program that Solves Symbolic Integration Problems in Freshman Calculus},
  journal = 	 {Journal of the ACM},
  year = 	 {1963},
  OPTkey = 	 {},
  volume =	 {10},
  number =	 {4},
  pages =	 {507 - 520},
  month =	 {11},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@PhdThesis{mos67,
  author = 	 {Joel Moses},
  title = 	 {Symbolic Integration},
  school = 	 {Massachusetts Institute of Technology},
  year = 	 {1967},
  OPTkey = 	 {},
  OPTtype = 	 {},
  OPTaddress = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{mos71,
  author = 	 {Joel Moses},
  title = 	 {Symbolic integration: the stormy decade},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proceedings of the second ACM symposium on Symbolic and algebraic manipulation},
  pages =	 {427 - 440},
  year =	 {1971},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {Symposium on Symbolic and Algebraic Manipulation},
  address =	 {New York},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{che85,
  author = 	 {G. W. Cherry},
  title = 	 {Integration in finite terms with special functions: the error function},
  journal = 	 {Journal of Symbolic Computation},
  year = 	 {1985},
  OPTkey = 	 {},
  volume =	 {1},
  number =	 {3},
  pages =	 {283 - 302},
  month =	 {9},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{che86,
  author = 	 {G. W. Cherry},
  title = 	 {Integration in finite terms with special functions: the logarithmic integral},
  journal = 	 {SIAM Journal on Computing},
  year = 	 {1986},
  OPTkey = 	 {},
  volume =	 {15},
  number =	 {1},
  pages =	 {1 - 21},
  month =	 {2},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{ber94,
  author = 	 {Laurent Bertrand},
  title = 	 {On the implementation of a new algorithm for the computation of hyperelliptic integrals},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proceedings of the international symposium on Symbolic and algebraic computation},
  pages =	 {211 - 215},
  year =	 {1994},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {International Conference on Symbolic and Algebraic Computation},
  address =	 {New York},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{ber95,
  author = 	 {Laurent Bertrand},
  title = 	 {Computing a hyperelliptic integral using arithmetic in the Jacobian of the curve},
  journal = 	 {Applicable Algebra in Engineering, Communication and Computing},
  year = 	 {1995},
  OPTkey = 	 {},
  volume =	 {6},
  number =	 {4-5},
  pages =	 {275-298},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{kno86,
  author = 	 {P. H. Knowles},
  title = 	 {Integration of Liouvillian functions with special functions},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proceedings of the fifth ACM symposium on Symbolic and algebraic computation},
  pages =	 {179 - 184},
  year =	 {1986},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {Symposium on Symbolic and Algebraic Manipulation},
  address =	 {New York},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {ACM},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{yan08,
  author =	 {颜松远},
  ALTeditor = 	 {},
  title = 	 {计算数论},
  publisher = 	 {清华大学出版社},
  year = 	 {2008},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  edition =	 {2},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{shuxueshouce,
  author =	 {数学手册编写组},
  ALTeditor = 	 {},
  title = 	 {数学手册（第1版）},
  publisher = 	 {高等教育出版社},
  year = 	 {1979},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{pei02,
  author =	 {裴定一，祝跃飞},
  ALTeditor = 	 {},
  title = 	 {算法数论},
  publisher = 	 {科学出版社},
  year = 	 {2002},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  series =	 {中国科学院研究生教学丛书},
  address =	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{pol74,
  author = 	 {J. M. Pollard},
  title = 	 {Theorems on factorization and primality testing},
  journal = 	 {Mathematical Proceedings of the Cambridge Philosophical Society},
  year = 	 {1974},
  OPTkey = 	 {},
  volume =	 {76},
  number =	 {3},
  pages =	 {521-528},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{pol75,
  author = 	 {J. M. Pollard},
  title = 	 {A Monte Carlo method for factorization},
  journal = 	 {BIT Numerical Mathematics},
  year = 	 {1975},
  OPTkey = 	 {},
  volume =	 {15},
  number =	 {3},
  pages =	 {331-334},
  month =	 {9},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{len87,
  author = 	 {Lenstra, Jr., H. W.},
  title = 	 {Factoring Integers with Elliptic Curves},
  journal = 	 {The Annals of Mathematics},
  year = 	 {1987},
  OPTkey = 	 {},
  volume =	 {126},
  number =	 {3},
  pages =	 {649-673},
  month =	 {11},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{mon87,
  author = 	 {Peter L. Montgomery},
  title = 	 {Speeding the Pollard and Elliptic Curve Methods of Factorization},
  journal = 	 {Mathematics of Computation},
  year = 	 {1987},
  OPTkey = 	 {},
  volume =	 {48},
  number =	 {177},
  pages =	 {243-264},
  month =	 {1},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InCollection{pom82,
  author = 	 {Carl Pomerance},
  title = 	 {Analysis and comparison of some integer factoring algorithms},
  booktitle = 	 {Computational Methods in Number Theory},
  OPTcrossref =  {},
  OPTkey = 	 {},
  pages =	 {89-139},
  publisher =	 {Mathematisch Centrum},
  year =	 {1982},
  OPTeditor = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTtype = 	 {},
  OPTchapter = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{llmp,
  author = 	 {A. K. Lenstra and Lenstra, Jr., H. W. and M. S. Manasse and J. M. Pollard },
  title = 	 {The Factorization of the Ninth Fermat Number},
  journal = 	 {Mathematics of Computation},
  year = 	 {1993},
  OPTkey = 	 {},
  volume =	 {61},
  number =	 {203},
  pages =	 {319-349},
  month =	 {7},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{har01,
  author =	 {R. Hartshorne著，冯克勤，刘木兰，胥鸣伟译},
  ALTeditor = 	 {},
  title = 	 {代数几何},
  publisher = 	 {科学出版社},
  year = 	 {2001},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {北京},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Misc{seiler,
  OPTkey = 	 {},
  author =	 {Werner M. Seiler},
  title =	 {Computer Algebra and Differential Equations - An Overview},
  OPThowpublished = {},
  OPTmonth = 	 {},
  OPTyear = 	 {},
  note =	 {\url{http://iaks-www.ira.uka.de/iaks-calmet/werner/Papers/CADE.ps.gz}},
  OPTannote = 	 {}
}

@Book{kol73,
  author =	 {E. Kolchin},
  ALTeditor = 	 {},
  title = 	 {Differential Algebra and Algebraic Groups},
  publisher = 	 {Academic Press },
  year = 	 {1973},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {New York},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{New72,
  author =	 {Morris Newman},
  ALTeditor = 	 {},
  title = 	 {Integral matrices},
  publisher = 	 {Academic Press},
  year = 	 {1972},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {New York and London},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Gra03,
  ALTauthor =	 {},
  editor =	 {Johannes Grabmeier and Erich Kaltofen and Volker Weispfenning},
  title = 	 {Computer algebra handbook},
  publisher = 	 {Springer Verlag},
  year = 	 {2003},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Bur97,
  author =	 {Peter B\"urgisser and Michael Clausen and M. Amin Shokrollahi},
  ALTeditor = 	 {},
  title = 	 {Algebraic complexity theory},
  publisher = 	 {Springer Verlag},
  year = 	 {1997},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Article{Dem07,
  author = 	 {James Demmel and others},
  title = 	 {Fast matrix multiplication is stable},
  journal = 	 {Numerische Mathematik},
  year = 	 {2007},
  OPTkey = 	 {},
  volume =	 {106},
  OPTnumber = 	 {},
  pages =	 {199-224},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@InProceedings{MoeCar79,
  author = 	 {Robert T. Moeck and John H. Carter},
  title = 	 {Approximate algorithms to derive exact solutions to systems of linear equations},
  OPTcrossref =  {},
  OPTkey = 	 {},
  booktitle =	 {Proc. EUROSAM '79},
  pages =	 {65-73},
  year =	 {1979},
  OPTeditor = 	 {},
  volume =	 {72},
  OPTnumber = 	 {},
  series =	 {Lecture notes in computer science},
  address =	 {},
  OPTmonth = 	 {},
  OPTorganization = {},
  publisher =	 {Springer-Verlag},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{BinPan94,
  author =	 {Dario Bini and Victor Pan},
  ALTeditor = 	 {},
  title = 	 {Polynomial and matrix computations},
  publisher = 	 {Birkh\"auser},
  year = 	 {1994},
  OPTkey = 	 {},
  volume =	 {1},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  address =	 {Boston},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{fftca,
  author = 	 {H．J．Nussbaumer著，胡光锐译},
  ALTeditor = 	 {},
  title = 	 {快速付里叶变换和卷积算法},
  publisher = 	 {上海科学技术文献出版社},
  year = 	 {1984},
  address =      {上海},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{Nussbaumer82,
  author = 	 {H．J．Nussbaumer},
  ALTeditor = 	 {},
  title = 	 {Fast Fourier Transform and Convolution Algorithms},
  publisher = 	 {Springer-Verlag},
  year = 	 {1982},
  address =      {Berlin},
  OPTkey = 	 {},
  OPTvolume = 	 {},
  OPTnumber = 	 {},
  OPTseries = 	 {},
  OPTaddress = 	 {},
  OPTedition = 	 {},
  OPTmonth = 	 {},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@PhdThesis{gruntz96,
  author = 	 {D. Gruntz},
  title = 	 {On Computing Limits in a Symbolic Manipulation System},
  school = 	 {ETH Z\"urich},
  year = 	 {1996},
  OPTkey = 	 {},
  OPTtype = 	 {},
  address = 	 {Institute of Scientific Computing},
  month = 	 {Jun},
  OPTnote = 	 {},
  OPTannote = 	 {}
}

@Book{aeqb,      
  author =       {Marko Petkovsek and Herbert Wilf and Doron Zeilberger},      
  ALTeditor =    {},      
  title =        {A=B},      
  publisher =    {A K Peters, Ltd.},      
  year =         {1996},      
  OPTkey =       {},      
  OPTvolume =    {},      
  OPTnumber =    {},      
  OPTseries =    {},      
  OPTaddress =   {},      
  OPTedition =   {},      
  OPTmonth =     {},      
  OPTnote =      {},      
  OPTannote =    {}      
}

@Article{karr81,
 author = {M. Karr},
 title = {Summation in Finite Terms},
 journal = {Journal of the ACM},
 volume = {28},
 number = {2},
 year = {1981},
 issn = {0004-5411},
 pages = {305--350},
 doi = {http://doi.acm.org/10.1145/322248.322255},
 publisher = {ACM},
 address = {New York, NY, USA},
}

@Article{paule95,
 author = {P. Paule},
 title = {Greatest factorial factorization and symbolic summation},
 journal = {Journal of Symbolic Computation},
 volume = {20},
 number = {3},
 year = {1995},
 issn = {0747-7171},
 pages = {235--268},
 doi = {http://dx.doi.org/10.1006/jsco.1995.1049},
 publisher = {Academic Press, Inc.},
 address = {Duluth, MN, USA},
}

@InProceedings{schneider04,
 author = {C. Schneider},
 title = {Symbolic summation with single-nested sum extensions},
 booktitle = {ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation},
 year = {2004},
 isbn = {1-58113-827-X},
 pages = {282--289},
 location = {Santander, Spain},
 doi = {http://doi.acm.org/10.1145/1005285.1005326},
 publisher = {ACM},
 address = {New York, NY, USA},
}

@PhdThesis{pirastu96,
author = {R. Pirastu},
title = {{On Combinatorial Identities: Symbolic Summation and Umbral Calculus}},
language = {english},
year = {1996},
translation = {0},
school = {RISC, J. Kepler University Linz},
length = {128}
}

@InProceedings{kauers07,
 author = {M. Kauers and C. Schneider},
 title = {Symbolic summation with radical expressions},
 booktitle = {ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation},
 year = {2007},
 isbn = {978-1-59593-743-8},
 pages = {219--226},
 location = {Waterloo, Ontario, Canada},
 doi = {http://doi.acm.org/10.1145/1277548.1277579},
 publisher = {ACM},
 address = {New York, NY, USA},
}

@Book{wangzhuxi,      
  author =       {王竹溪，郭敦仁},      
  ALTeditor =    {},      
  title =        {特殊函数概论},      
  publisher =    {北京大学出版社},      
  year =         {1973},      
  OPTkey =       {},      
  OPTvolume =    {},      
  OPTnumber =    {},      
  OPTseries =    {},      
  address =      {北京},      
  OPTedition =   {},      
  OPTmonth =     {},      
  OPTnote =      {},      
  OPTannote =    {}      
}