\contentsline {chapter}{\numberline {1}Integration}{1}{chapter.1}
\contentsline {section}{\numberline {1.1}Rational Functions}{2}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}The full partial-fraction algorithm}{2}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}The Hermite reduction}{3}{subsection.1.1.2}
\contentsline {subsection}{\numberline {1.1.3}The Rothstein-Trager and Lazard-Rioboo-Trager algorithms}{5}{subsection.1.1.3}
\contentsline {section}{\numberline {1.2}Algebraic Functions}{5}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}The Hermite reduction}{6}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}Simple radical extensions}{10}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}Liouville's Theorem}{12}{subsection.1.2.3}
\contentsline {subsection}{\numberline {1.2.4}The integral part}{12}{subsection.1.2.4}
\contentsline {subsection}{\numberline {1.2.5}The logarithmic part}{14}{subsection.1.2.5}
\contentsline {section}{\numberline {1.3}Elementary Functions}{16}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Differential algebra}{17}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}The Hermite reduction}{18}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}The polynomial reduction}{19}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}The residue criterion}{20}{subsection.1.3.4}
\contentsline {subsection}{\numberline {1.3.5}The transcendental logarithmic case}{22}{subsection.1.3.5}
\contentsline {subsection}{\numberline {1.3.6}The transcendental exponential case}{23}{subsection.1.3.6}
\contentsline {subsection}{\numberline {1.3.7}The transcendental tangent case}{24}{subsection.1.3.7}
\contentsline {subsection}{\numberline {1.3.8}The algebraic logarithmic case}{24}{subsection.1.3.8}
\contentsline {subsection}{\numberline {1.3.9}The algebraic exponential case}{27}{subsection.1.3.9}
\contentsline {chapter}{\numberline {2}Singular Value Decomposition}{31}{chapter.2}
\contentsline {section}{\numberline {2.1}Singular Value Decomposition Tutorial}{31}{section.2.1}
\contentsline {chapter}{\numberline {3}Quaternions}{37}{chapter.3}
\contentsline {section}{Preface}{37}{section*.2}
\contentsline {section}{\numberline {3.1}Quaternions}{38}{section.3.1}
\contentsline {section}{\numberline {3.2}Vectors, and their Composition}{38}{section.3.2}
\contentsline {section}{\numberline {3.3}Examples To Chapter 1.}{65}{section.3.3}
\contentsline {section}{\numberline {3.4}Products And Quotients of Vectors}{67}{section.3.4}
\contentsline {section}{\numberline {3.5}Examples To Chapter 2.}{93}{section.3.5}
\contentsline {section}{\numberline {3.6}Interpretations And Transformations}{94}{section.3.6}
\contentsline {section}{\numberline {3.7}Examples to Chapter 3}{124}{section.3.7}
\contentsline {section}{\numberline {3.8}Axiom Examples}{130}{section.3.8}
\contentsline {chapter}{\numberline {4}Clifford Algebra\cite {Fl09}}{133}{chapter.4}
\contentsline {section}{\numberline {4.1}Introduction}{133}{section.4.1}
\contentsline {section}{\numberline {4.2}Clifford Basis Matrix Theory}{134}{section.4.2}
\contentsline {section}{\numberline {4.3}Calculation of the inverse of a Clifford number}{136}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Example 1: Clifford (2)}{137}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}Example 2: Clifford (3)}{137}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}Example 3: Clifford (2,2)}{139}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}Conclusion}{142}{subsection.4.3.4}
\contentsline {chapter}{\numberline {5}Package for Algebraic Function Fields}{143}{chapter.5}
\contentsline {chapter}{\numberline {6}Groebner Basis}{145}{chapter.6}
\contentsline {chapter}{\numberline {7}Greatest Common Divisor}{147}{chapter.7}
\contentsline {chapter}{\numberline {8}Polynomial Factorization}{149}{chapter.8}
\contentsline {chapter}{\numberline {9}Cylindrical Algebraic Decomposition}{151}{chapter.9}
\contentsline {chapter}{\numberline {10}Pade approximant}{153}{chapter.10}
\contentsline {chapter}{\numberline {11}Schwartz-Zippel lemma and testing polynomial identities}{155}{chapter.11}
\contentsline {chapter}{\numberline {12}Chinese Remainder Theorem}{157}{chapter.12}
\contentsline {chapter}{\numberline {13}Gaussian Elimination}{159}{chapter.13}
\contentsline {chapter}{\numberline {14}Diophantine Equations}{161}{chapter.14}
\contentsline {chapter}{\numberline {15}Index}{167}{chapter.15}
