\contentsline {chapter}{Abstract}{9}
\contentsline {chapter}{\numberline {1}Introductory Information}{13}
\contentsline {chapter}{\numberline {2}Structure of Programs}{17}
\contentsline {section}{\numberline {2.1}The {REDUCE} Standard Character Set}{17}
\contentsline {section}{\numberline {2.2}Numbers}{17}
\contentsline {section}{\numberline {2.3}Identifiers}{19}
\contentsline {section}{\numberline {2.4}Variables}{20}
\contentsline {section}{\numberline {2.5}Strings}{21}
\contentsline {section}{\numberline {2.6}Comments}{21}
\contentsline {section}{\numberline {2.7}Operators}{22}
\contentsline {chapter}{\numberline {3}Expressions}{25}
\contentsline {section}{\numberline {3.1}Scalar Expressions}{25}
\contentsline {section}{\numberline {3.2}Integer Expressions}{26}
\contentsline {section}{\numberline {3.3}Boolean Expressions}{27}
\contentsline {section}{\numberline {3.4}Equations}{28}
\contentsline {section}{\numberline {3.5}Proper Statements as Expressions}{29}
\contentsline {chapter}{\numberline {4}Lists}{31}
\contentsline {section}{\numberline {4.1}Operations on Lists}{31}
\contentsline {subsection}{\numberline {4.1.1}LIST}{32}
\contentsline {subsection}{\numberline {4.1.2}FIRST}{32}
\contentsline {subsection}{\numberline {4.1.3}SECOND}{32}
\contentsline {subsection}{\numberline {4.1.4}THIRD}{32}
\contentsline {subsection}{\numberline {4.1.5}REST}{32}
\contentsline {subsection}{\numberline {4.1.6}$.$ (Cons) Operator}{32}
\contentsline {subsection}{\numberline {4.1.7}APPEND}{32}
\contentsline {subsection}{\numberline {4.1.8}REVERSE}{33}
\contentsline {subsection}{\numberline {4.1.9}List Arguments of Other Operators}{33}
\contentsline {subsection}{\numberline {4.1.10}Caveats and Examples}{33}
\contentsline {chapter}{\numberline {5}Statements}{35}
\contentsline {section}{\numberline {5.1}Assignment Statements}{36}
\contentsline {subsection}{\numberline {5.1.1}Set Statement}{36}
\contentsline {section}{\numberline {5.2}Group Statements}{37}
\contentsline {section}{\numberline {5.3}Conditional Statements}{37}
\contentsline {section}{\numberline {5.4}FOR Statements}{38}
\contentsline {section}{\numberline {5.5}WHILE \dots DO}{40}
\contentsline {section}{\numberline {5.6}REPEAT \dots UNTIL}{41}
\contentsline {section}{\numberline {5.7}Compound Statements}{41}
\contentsline {subsection}{\numberline {5.7.1}Compound Statements with GO TO}{43}
\contentsline {subsection}{\numberline {5.7.2}Labels and GO TO Statements}{43}
\contentsline {subsection}{\numberline {5.7.3}RETURN Statements}{44}
\contentsline {chapter}{\numberline {6}Commands and Declarations}{47}
\contentsline {section}{\numberline {6.1}Array Declarations}{47}
\contentsline {section}{\numberline {6.2}Mode Handling Declarations}{48}
\contentsline {section}{\numberline {6.3}END}{49}
\contentsline {section}{\numberline {6.4}BYE Command}{49}
\contentsline {section}{\numberline {6.5}SHOWTIME Command}{49}
\contentsline {section}{\numberline {6.6}DEFINE Command}{49}
\contentsline {chapter}{\numberline {7}Built-in Prefix Operators}{51}
\contentsline {section}{\numberline {7.1}Numerical Operators}{51}
\contentsline {subsection}{\numberline {7.1.1}ABS}{51}
\contentsline {subsection}{\numberline {7.1.2}CEILING}{52}
\contentsline {subsection}{\numberline {7.1.3}CONJ}{52}
\contentsline {subsection}{\numberline {7.1.4}FACTORIAL}{52}
\contentsline {subsection}{\numberline {7.1.5}FIX}{52}
\contentsline {subsection}{\numberline {7.1.6}FLOOR}{53}
\contentsline {subsection}{\numberline {7.1.7}IMPART}{53}
\contentsline {subsection}{\numberline {7.1.8}MAX/MIN}{53}
\contentsline {subsection}{\numberline {7.1.9}NEXTPRIME}{53}
\contentsline {subsection}{\numberline {7.1.10}RANDOM}{54}
\contentsline {subsection}{\numberline {7.1.11}RANDOM\_NEW\_SEED}{54}
\contentsline {subsection}{\numberline {7.1.12}REPART}{54}
\contentsline {subsection}{\numberline {7.1.13}ROUND}{55}
\contentsline {subsection}{\numberline {7.1.14}SIGN}{55}
\contentsline {section}{\numberline {7.2}Mathematical Functions}{55}
\contentsline {section}{\numberline {7.3}DF Operator}{59}
\contentsline {subsection}{\numberline {7.3.1}Adding Differentiation Rules}{59}
\contentsline {section}{\numberline {7.4}INT Operator}{60}
\contentsline {subsection}{\numberline {7.4.1}Options}{61}
\contentsline {subsection}{\numberline {7.4.2}Advanced Use}{61}
\contentsline {subsection}{\numberline {7.4.3}References}{61}
\contentsline {section}{\numberline {7.5}LENGTH Operator}{62}
\contentsline {section}{\numberline {7.6}MAP Operator}{62}
\contentsline {section}{\numberline {7.7}MKID Operator}{63}
\contentsline {section}{\numberline {7.8}PF Operator}{64}
\contentsline {section}{\numberline {7.9}SELECT Operator}{65}
\contentsline {section}{\numberline {7.10}SOLVE Operator}{65}
\contentsline {subsection}{\numberline {7.10.1}Handling of Undetermined Solutions}{67}
\contentsline {subsection}{\numberline {7.10.2}Solutions of Equations Involving Cubics and Quartics}{68}
\contentsline {subsection}{\numberline {7.10.3}Other Options}{70}
\contentsline {subsection}{\numberline {7.10.4}Parameters and Variable Dependency}{71}
\contentsline {section}{\numberline {7.11}Even and Odd Operators}{74}
\contentsline {section}{\numberline {7.12}Linear Operators}{75}
\contentsline {section}{\numberline {7.13}Non-Commuting Operators}{76}
\contentsline {section}{\numberline {7.14}Symmetric and Antisymmetric Operators}{76}
\contentsline {section}{\numberline {7.15}Declaring New Prefix Operators}{77}
\contentsline {section}{\numberline {7.16}Declaring New Infix Operators}{77}
\contentsline {section}{\numberline {7.17}Creating/Removing Variable Dependency}{78}
\contentsline {chapter}{\numberline {8}Display and Structuring of Expressions}{81}
\contentsline {section}{\numberline {8.1}Kernels}{81}
\contentsline {section}{\numberline {8.2}The Expression Workspace}{82}
\contentsline {section}{\numberline {8.3}Output of Expressions}{83}
\contentsline {subsection}{\numberline {8.3.1}LINELENGTH Operator}{84}
\contentsline {subsection}{\numberline {8.3.2}Output Declarations}{84}
\contentsline {subsubsection}{ORDER Declaration}{84}
\contentsline {subsubsection}{FACTOR Declaration}{85}
\contentsline {subsection}{\numberline {8.3.3}Output Control Switches}{85}
\contentsline {subsubsection}{ALLFAC Switch}{85}
\contentsline {subsubsection}{DIV Switch}{86}
\contentsline {subsubsection}{LIST Switch}{86}
\contentsline {subsubsection}{NOSPLIT Switch}{86}
\contentsline {subsubsection}{RAT Switch}{87}
\contentsline {subsubsection}{RATPRI Switch}{88}
\contentsline {subsubsection}{REVPRI Switch}{88}
\contentsline {subsection}{\numberline {8.3.4}WRITE Command}{89}
\contentsline {subsection}{\numberline {8.3.5}Suppression of Zeros}{91}
\contentsline {subsection}{\numberline {8.3.6}{FORTRAN} Style Output Of Expressions}{91}
\contentsline {subsubsection}{{FORTRAN} Output Options}{93}
\contentsline {subsection}{\numberline {8.3.7}Saving Expressions for Later Use as Input}{94}
\contentsline {subsection}{\numberline {8.3.8}Displaying Expression Structure}{94}
\contentsline {section}{\numberline {8.4}Changing the Internal Order of Variables}{97}
\contentsline {section}{\numberline {8.5}Obtaining Parts of Algebraic Expressions}{97}
\contentsline {subsection}{\numberline {8.5.1}COEFF Operator}{97}
\contentsline {subsection}{\numberline {8.5.2}COEFFN Operator}{98}
\contentsline {subsection}{\numberline {8.5.3}PART Operator}{98}
\contentsline {subsection}{\numberline {8.5.4}Substituting for Parts of Expressions}{100}
\contentsline {chapter}{\numberline {9}Polynomials and Rationals}{101}
\contentsline {section}{\numberline {9.1}Controlling the Expansion of Expressions}{102}
\contentsline {section}{\numberline {9.2}Factorization of Polynomials}{102}
\contentsline {section}{\numberline {9.3}Cancellation of Common Factors}{104}
\contentsline {subsection}{\numberline {9.3.1}Determining the GCD of Two Polynomials}{105}
\contentsline {section}{\numberline {9.4}Working with Least Common Multiples}{106}
\contentsline {section}{\numberline {9.5}Controlling Use of Common Denominators}{106}
\contentsline {section}{\numberline {9.6}REMAINDER Operator}{106}
\contentsline {section}{\numberline {9.7}RESULTANT Operator}{107}
\contentsline {section}{\numberline {9.8}DECOMPOSE Operator}{108}
\contentsline {section}{\numberline {9.9}INTERPOL operator}{109}
\contentsline {section}{\numberline {9.10}Obtaining Parts of Polynomials and Rationals}{109}
\contentsline {subsection}{\numberline {9.10.1}DEG Operator}{109}
\contentsline {subsection}{\numberline {9.10.2}DEN Operator}{110}
\contentsline {subsection}{\numberline {9.10.3}LCOF Operator}{110}
\contentsline {subsection}{\numberline {9.10.4}LPOWER Operator}{111}
\contentsline {subsection}{\numberline {9.10.5}LTERM Operator}{111}
\contentsline {subsection}{\numberline {9.10.6}MAINVAR Operator}{111}
\contentsline {subsection}{\numberline {9.10.7}NUM Operator}{112}
\contentsline {subsection}{\numberline {9.10.8}REDUCT Operator}{112}
\contentsline {section}{\numberline {9.11}Polynomial Coefficient Arithmetic}{113}
\contentsline {subsection}{\numberline {9.11.1}Rational Coefficients in Polynomials}{113}
\contentsline {subsection}{\numberline {9.11.2}Real Coefficients in Polynomials}{113}
\contentsline {subsection}{\numberline {9.11.3}Modular Number Coefficients in Polynomials}{115}
\contentsline {subsection}{\numberline {9.11.4}Complex Number Coefficients in Polynomials}{115}
\contentsline {chapter}{\numberline {10}Substitution Commands}{117}
\contentsline {section}{\numberline {10.1}SUB Operator}{117}
\contentsline {section}{\numberline {10.2}LET Rules}{118}
\contentsline {subsection}{\numberline {10.2.1}FOR ALL \dots LET}{120}
\contentsline {subsection}{\numberline {10.2.2}FOR ALL \dots SUCH THAT \dots LET}{121}
\contentsline {subsection}{\numberline {10.2.3}Removing Assignments and Substitution Rules}{122}
\contentsline {subsection}{\numberline {10.2.4}Overlapping LET Rules}{122}
\contentsline {subsection}{\numberline {10.2.5}Substitutions for General Expressions}{123}
\contentsline {section}{\numberline {10.3}Rule Lists}{125}
\contentsline {section}{\numberline {10.4}Asymptotic Commands}{131}
\contentsline {chapter}{\numberline {11}File Handling Commands}{133}
\contentsline {section}{\numberline {11.1}IN Command}{133}
\contentsline {section}{\numberline {11.2}OUT Command}{134}
\contentsline {section}{\numberline {11.3}SHUT Command}{134}
\contentsline {chapter}{\numberline {12}Commands for Interactive Use}{135}
\contentsline {section}{\numberline {12.1}Referencing Previous Results}{135}
\contentsline {section}{\numberline {12.2}Interactive Editing}{136}
\contentsline {section}{\numberline {12.3}Interactive File Control}{137}
\contentsline {chapter}{\numberline {13}Matrix Calculations}{139}
\contentsline {section}{\numberline {13.1}MAT Operator}{139}
\contentsline {section}{\numberline {13.2}Matrix Variables}{140}
\contentsline {section}{\numberline {13.3}Matrix Expressions}{140}
\contentsline {section}{\numberline {13.4}Operators with Matrix Arguments}{141}
\contentsline {subsection}{\numberline {13.4.1}DET Operator}{141}
\contentsline {subsection}{\numberline {13.4.2}MATEIGEN Operator}{142}
\contentsline {subsection}{\numberline {13.4.3}TP Operator}{143}
\contentsline {subsection}{\numberline {13.4.4}Trace Operator}{143}
\contentsline {subsection}{\numberline {13.4.5}Matrix Cofactors}{143}
\contentsline {subsection}{\numberline {13.4.6}NULLSPACE Operator}{143}
\contentsline {subsection}{\numberline {13.4.7}RANK Operator}{144}
\contentsline {section}{\numberline {13.5}Matrix Assignments}{145}
\contentsline {section}{\numberline {13.6}Evaluating Matrix Elements}{145}
\contentsline {chapter}{\numberline {14}Procedures}{147}
\contentsline {section}{\numberline {14.1}Procedure Heading}{148}
\contentsline {section}{\numberline {14.2}Procedure Body}{149}
\contentsline {section}{\numberline {14.3}Using LET Inside Procedures}{151}
\contentsline {section}{\numberline {14.4}LET Rules as Procedures}{152}
\contentsline {section}{\numberline {14.5}REMEMBER Statement}{153}
\contentsline {chapter}{\numberline {15}Symbolic Mode}{155}
\contentsline {section}{\numberline {15.1}Symbolic Infix Operators}{157}
\contentsline {section}{\numberline {15.2}Symbolic Expressions}{157}
\contentsline {section}{\numberline {15.3}Quoted Expressions}{157}
\contentsline {section}{\numberline {15.4}Lambda Expressions}{157}
\contentsline {section}{\numberline {15.5}Symbolic Assignment Statements}{158}
\contentsline {section}{\numberline {15.6}FOR EACH Statement}{159}
\contentsline {section}{\numberline {15.7}Symbolic Procedures}{159}
\contentsline {section}{\numberline {15.8}Standard Lisp Equivalent of Reduce Input}{160}
\contentsline {section}{\numberline {15.9}Communicating with Algebraic Mode}{160}
\contentsline {subsection}{\numberline {15.9.1}Passing Algebraic Mode Values to Symbolic Mode}{161}
\contentsline {subsection}{\numberline {15.9.2}Passing Symbolic Mode Values to Algebraic Mode}{163}
\contentsline {subsection}{\numberline {15.9.3}Complete Example}{164}
\contentsline {subsection}{\numberline {15.9.4}Defining Procedures for Intermode Communication}{164}
\contentsline {section}{\numberline {15.10}Rlisp '88}{165}
\contentsline {section}{\numberline {15.11}References}{166}
\contentsline {chapter}{\numberline {16}Calculations in High Energy Physics}{167}
\contentsline {section}{\numberline {16.1}High Energy Physics Operators}{167}
\contentsline {subsection}{\numberline {16.1.1}. (Cons) Operator}{167}
\contentsline {subsection}{\numberline {16.1.2}G Operator for Gamma Matrices}{168}
\contentsline {subsection}{\numberline {16.1.3}EPS Operator}{169}
\contentsline {section}{\numberline {16.2}Vector Variables}{169}
\contentsline {section}{\numberline {16.3}Additional Expression Types}{170}
\contentsline {subsection}{\numberline {16.3.1}Vector Expressions}{170}
\contentsline {subsection}{\numberline {16.3.2}Dirac Expressions}{170}
\contentsline {section}{\numberline {16.4}Trace Calculations}{171}
\contentsline {section}{\numberline {16.5}Mass Declarations}{171}
\contentsline {section}{\numberline {16.6}Example}{172}
\contentsline {section}{\numberline {16.7}Extensions to More Than Four Dimensions}{173}
\contentsline {chapter}{\numberline {17}{REDUCE} and Rlisp Utilities}{175}
\contentsline {section}{\numberline {17.1}The Standard Lisp Compiler}{175}
\contentsline {section}{\numberline {17.2}Fast Loading Code Generation Program}{176}
\contentsline {section}{\numberline {17.3}The Standard Lisp Cross Reference Program}{177}
\contentsline {subsection}{\numberline {17.3.1}Restrictions}{178}
\contentsline {subsection}{\numberline {17.3.2}Usage}{178}
\contentsline {subsection}{\numberline {17.3.3}Options}{178}
\contentsline {section}{\numberline {17.4}Prettyprinting Reduce Expressions}{179}
\contentsline {section}{\numberline {17.5}Prettyprinting Standard Lisp S-Expressions}{179}
\contentsline {chapter}{\numberline {18}Maintaining {REDUCE}}{181}
\contentsline {chapter}{\numberline {A}Reserved Identifiers}{183}