\contentsline {chapter}{\numberline {1}Index}{5}{chapter.1}%
\contentsline {chapter}{\numberline {2}Introduction}{7}{chapter.2}%
\contentsline {section}{\numberline {2.1}Notations used in this manual}{7}{section.2.1}%
\contentsline {section}{\numberline {2.2}Interfaces for the \texttt {giac} library}{8}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}The \texttt {Xcas} interface }{8}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}The command-line interface: \texttt {icas} \texttt {giac} }{9}{subsection.2.2.2}%
\contentsline {subsection}{\numberline {2.2.3}The Firefox interface}{10}{subsection.2.2.3}%
\contentsline {subsection}{\numberline {2.2.4}The TeXmacs interface}{10}{subsection.2.2.4}%
\contentsline {subsection}{\numberline {2.2.5}Checking the version of \texttt {giac} that you are using: \texttt {version} \texttt {giac} }{10}{subsection.2.2.5}%
\contentsline {chapter}{\numberline {3}The \texttt {Xcas} interface}{13}{chapter.3}%
\contentsline {section}{\numberline {3.1}The entry levels}{13}{section.3.1}%
\contentsline {section}{\numberline {3.2}The starting window}{14}{section.3.2}%
\contentsline {section}{\numberline {3.3}Getting help}{16}{section.3.3}%
\contentsline {subsubsection}{Tooltips}{16}{section*.2}%
\contentsline {subsubsection}{HTML help}{17}{section*.3}%
\contentsline {subsubsection}{The help index}{17}{section*.4}%
\contentsline {subsubsection}{\texttt {findhelp} }{18}{section*.5}%
\contentsline {section}{\numberline {3.4}The menus}{18}{section.3.4}%
\contentsline {subsection}{\numberline {3.4.1}The \texttt {File} menu}{18}{subsection.3.4.1}%
\contentsline {subsection}{\numberline {3.4.2}The \texttt {Edit} menu}{20}{subsection.3.4.2}%
\contentsline {subsection}{\numberline {3.4.3}The \texttt {Cfg} menu}{21}{subsection.3.4.3}%
\contentsline {subsection}{\numberline {3.4.4}The \texttt {Help} menu}{22}{subsection.3.4.4}%
\contentsline {subsection}{\numberline {3.4.5}The \texttt {Toolbox} menu}{24}{subsection.3.4.5}%
\contentsline {subsection}{\numberline {3.4.6}The \texttt {Expression} menu}{25}{subsection.3.4.6}%
\contentsline {subsection}{\numberline {3.4.7}The \texttt {Cmds} menu}{25}{subsection.3.4.7}%
\contentsline {subsection}{\numberline {3.4.8}The \texttt {Prg} menu}{25}{subsection.3.4.8}%
\contentsline {subsection}{\numberline {3.4.9}The \texttt {Graphic} menu}{25}{subsection.3.4.9}%
\contentsline {subsection}{\numberline {3.4.10}The \texttt {Geo} menu}{25}{subsection.3.4.10}%
\contentsline {subsection}{\numberline {3.4.11}The \texttt {Spreadsheet} menu}{25}{subsection.3.4.11}%
\contentsline {subsection}{\numberline {3.4.12}The \texttt {Phys} menu}{26}{subsection.3.4.12}%
\contentsline {subsection}{\numberline {3.4.13}The \texttt {Highschool} menu}{26}{subsection.3.4.13}%
\contentsline {subsection}{\numberline {3.4.14}The \texttt {Turtle} menu}{26}{subsection.3.4.14}%
\contentsline {section}{\numberline {3.5}Configuring \texttt {Xcas}}{26}{section.3.5}%
\contentsline {subsection}{\numberline {3.5.1}The number of significant digits: \texttt {Digits} \texttt {DIGITS} }{26}{subsection.3.5.1}%
\contentsline {subsection}{\numberline {3.5.2}The language mode: \texttt {xcas\_mode} }{26}{subsection.3.5.2}%
\contentsline {subsection}{\numberline {3.5.3}The units for angles: \texttt {angle\_radian} }{27}{subsection.3.5.3}%
\contentsline {subsection}{\numberline {3.5.4}Exact or approximate values: \texttt {approx\_mode} }{27}{subsection.3.5.4}%
\contentsline {subsection}{\numberline {3.5.5}Complex numbers: \texttt {cfactor} \texttt {complex\_mode} }{27}{subsection.3.5.5}%
\contentsline {subsection}{\numberline {3.5.6}Complex variables: \texttt {complex\_variables} }{28}{subsection.3.5.6}%
\contentsline {subsection}{\numberline {3.5.7}Configuring the computations}{29}{subsection.3.5.7}%
\contentsline {subsection}{\numberline {3.5.8}Configuring the graphics}{31}{subsection.3.5.8}%
\contentsline {subsection}{\numberline {3.5.9}More configuration}{32}{subsection.3.5.9}%
\contentsline {subsection}{\numberline {3.5.10}The configuration file: \texttt {widget\_size} \texttt {cas\_setup} \texttt {xcas\_mode} \texttt {xyztrange} }{33}{subsection.3.5.10}%
\contentsline {section}{\numberline {3.6}Printing and saving}{36}{section.3.6}%
\contentsline {subsection}{\numberline {3.6.1}Saving a session }{36}{subsection.3.6.1}%
\contentsline {subsection}{\numberline {3.6.2}Saving a spreadsheet}{36}{subsection.3.6.2}%
\contentsline {subsection}{\numberline {3.6.3}Saving a program }{36}{subsection.3.6.3}%
\contentsline {subsection}{\numberline {3.6.4}Printing a session}{37}{subsection.3.6.4}%
\contentsline {section}{\numberline {3.7}Translating to other computer languages}{37}{section.3.7}%
\contentsline {subsection}{\numberline {3.7.1}Translating an expression to \LaTeX {}: \texttt {latex} }{37}{subsection.3.7.1}%
\contentsline {subsection}{\numberline {3.7.2}Translating the entire session to \LaTeX {}}{37}{subsection.3.7.2}%
\contentsline {subsection}{\numberline {3.7.3}Translating graphical output to \LaTeX {}: \texttt {graph2tex} \texttt {graph3d2tex} }{37}{subsection.3.7.3}%
\contentsline {subsection}{\numberline {3.7.4}Translating an expression to MathML: \texttt {mathml} }{38}{subsection.3.7.4}%
\contentsline {subsection}{\numberline {3.7.5}Translating a spreadsheet to MathML}{39}{subsection.3.7.5}%
\contentsline {subsection}{\numberline {3.7.6}Indent an XML string: \texttt {xml\_print} }{39}{subsection.3.7.6}%
\contentsline {subsection}{\numberline {3.7.7}Export to presentation or content MathML: \texttt {export\_mathml} }{39}{subsection.3.7.7}%
\contentsline {subsection}{\numberline {3.7.8}Translating a Maple file to Xcas: \texttt {maple2xcas} }{42}{subsection.3.7.8}%
\contentsline {chapter}{\numberline {4}Entry in \texttt {Xcas}}{43}{chapter.4}%
\contentsline {section}{\numberline {4.1}Suppressing output: \texttt {nodisp} \texttt {:;} }{43}{section.4.1}%
\contentsline {section}{\numberline {4.2}Entering comments: \texttt {comment} }{44}{section.4.2}%
\contentsline {section}{\numberline {4.3}Editing expressions}{44}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}Entering expressions in the editor: an example}{44}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}Subexpressions }{45}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}Manipulating subexpressions}{47}{subsection.4.3.3}%
\contentsline {section}{\numberline {4.4}Previous results: \texttt {ans} }{48}{section.4.4}%
\contentsline {section}{\numberline {4.5}Spreadsheet}{48}{section.4.5}%
\contentsline {subsection}{\numberline {4.5.1}Opening a spreadsheet}{48}{subsection.4.5.1}%
\contentsline {subsection}{\numberline {4.5.2}The spreadsheet window}{49}{subsection.4.5.2}%
\contentsline {chapter}{\numberline {5}CAS building blocks}{51}{chapter.5}%
\contentsline {section}{\numberline {5.1}Numbers}{51}{section.5.1}%
\contentsline {section}{\numberline {5.2}Symbolic constants: \texttt {e} \texttt {pi} \texttt {infinity} \texttt {inf} \texttt {i} \texttt {euler\_gamma} }{52}{section.5.2}%
\contentsline {section}{\numberline {5.3}Sequences, sets and lists}{53}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Sequences: \texttt {seq[] ()}}{53}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.3.2}Sets: \texttt {set[]} }{54}{subsection.5.3.2}%
\contentsline {subsection}{\numberline {5.3.3}Lists: \texttt {[ ]}}{54}{subsection.5.3.3}%
\contentsline {subsection}{\numberline {5.3.4}Accessing elements}{55}{subsection.5.3.4}%
\contentsline {section}{\numberline {5.4}Variables}{56}{section.5.4}%
\contentsline {subsection}{\numberline {5.4.1}Variable names}{56}{subsection.5.4.1}%
\contentsline {subsection}{\numberline {5.4.2}Assigning values: \texttt {:=} \texttt {=>} \texttt {=} \texttt {assign} \texttt {sto} \texttt {Store} }{56}{subsection.5.4.2}%
\contentsline {subsection}{\numberline {5.4.3}Assignment by reference: \texttt {=<} }{57}{subsection.5.4.3}%
\contentsline {subsection}{\numberline {5.4.4}Copying values of list: \texttt {copy} }{57}{subsection.5.4.4}%
\contentsline {subsection}{\numberline {5.4.5}Incrementing variables: \texttt {+=} \texttt {-=} \texttt {*=} \texttt {/=} }{58}{subsection.5.4.5}%
\contentsline {subsection}{\numberline {5.4.6}Storing and recalling variables and their values: \texttt {archive} \texttt {unarchive} }{58}{subsection.5.4.6}%
\contentsline {subsection}{\numberline {5.4.7}Copying variables: \texttt {CopyVar} }{59}{subsection.5.4.7}%
\contentsline {subsection}{\numberline {5.4.8}Assumptions on variables: \texttt {about} \texttt {additionally} \texttt {assume} \texttt {purge} \texttt {supposons} \texttt {and} \texttt {or} }{60}{subsection.5.4.8}%
\contentsline {subsection}{\numberline {5.4.9}Unassigning variables: \texttt {VARS} \texttt {purge} \texttt {DelVar} \texttt {del} \texttt {restart} \texttt {rm\_a\_z} \texttt {rm\_all\_vars} }{63}{subsection.5.4.9}%
\contentsline {subsection}{\numberline {5.4.10}The \texttt {CST} variable }{64}{subsection.5.4.10}%
\contentsline {section}{\numberline {5.5}Functions}{65}{section.5.5}%
\contentsline {subsection}{\numberline {5.5.1}Defining functions}{65}{subsection.5.5.1}%
\contentsline {section}{\numberline {5.6}Directories}{66}{section.5.6}%
\contentsline {subsection}{\numberline {5.6.1}Working directories: \texttt {pwd} \texttt {cd} }{66}{subsection.5.6.1}%
\contentsline {subsection}{\numberline {5.6.2}Reading files: \texttt {read} \texttt {load} }{66}{subsection.5.6.2}%
\contentsline {subsection}{\numberline {5.6.3}Internal directories: \texttt {NewFold} \texttt {SetFold} \texttt {GetFold} \texttt {DelFold} \texttt {VARS} }{67}{subsection.5.6.3}%
\contentsline {chapter}{\numberline {6}The CAS functions}{69}{chapter.6}%
\contentsline {section}{\numberline {6.1}Booleans}{69}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}The values of a boolean: \texttt {true} \texttt {false} }{69}{subsection.6.1.1}%
\contentsline {subsection}{\numberline {6.1.2}Tests: \texttt {==} \texttt {!=} \texttt {>} \texttt {>=} \texttt {<} \texttt {=<} }{69}{subsection.6.1.2}%
\contentsline {subsection}{\numberline {6.1.3}Defining functions with boolean tests: \texttt {ifte} \texttt {?:} \texttt {when} }{70}{subsection.6.1.3}%
\contentsline {subsection}{\numberline {6.1.4}Boolean operators: \texttt {or} \texttt {xor} \texttt {and} \texttt {not}}{72}{subsection.6.1.4}%
\contentsline {subsection}{\numberline {6.1.5}Transform a boolean expression to a list: \texttt {exp2list} }{73}{subsection.6.1.5}%
\contentsline {subsection}{\numberline {6.1.6}Transform a list into a boolean expression: \texttt {list2exp} }{74}{subsection.6.1.6}%
\contentsline {subsection}{\numberline {6.1.7}Evaluate booleans: \texttt {evalb} }{75}{subsection.6.1.7}%
\contentsline {section}{\numberline {6.2}Bitwise operators}{75}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}Basic operators: \texttt {bitor} \texttt {bitxor} \texttt {bitand}}{75}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}Bitwise Hamming distance: \texttt {hamdist}}{77}{subsection.6.2.2}%
\contentsline {section}{\numberline {6.3}Strings}{77}{section.6.3}%
\contentsline {subsection}{\numberline {6.3.1}Characters and strings: \texttt {"}}{77}{subsection.6.3.1}%
\contentsline {subsection}{\numberline {6.3.2}The newline character: \texttt {\symbol {92}n}}{78}{subsection.6.3.2}%
\contentsline {subsection}{\numberline {6.3.3}The length of a string: \texttt {size} \texttt {length} }{79}{subsection.6.3.3}%
\contentsline {subsection}{\numberline {6.3.4}The left and right parts of a string: \texttt {left} \texttt {right} }{79}{subsection.6.3.4}%
\contentsline {subsection}{\numberline {6.3.5}First character, middle and end of a string: \texttt {head} \texttt {mid} \texttt {tail} }{80}{subsection.6.3.5}%
\contentsline {subsection}{\numberline {6.3.6}Concatenation of a sequence of words: \texttt {cumSum} }{81}{subsection.6.3.6}%
\contentsline {subsection}{\numberline {6.3.7}ASCII code of a character: \texttt {ord}}{81}{subsection.6.3.7}%
\contentsline {subsection}{\numberline {6.3.8}ASCII code of a string: \texttt {asc} }{82}{subsection.6.3.8}%
\contentsline {subsection}{\numberline {6.3.9}String defined by the ASCII codes of its characters: \texttt {char} }{82}{subsection.6.3.9}%
\contentsline {subsection}{\numberline {6.3.10}Find a character in a string: \texttt {inString} }{83}{subsection.6.3.10}%
\contentsline {subsection}{\numberline {6.3.11}Concat objects into a string: \texttt {cat} }{83}{subsection.6.3.11}%
\contentsline {subsection}{\numberline {6.3.12}Add an object to a string: \texttt {+} }{84}{subsection.6.3.12}%
\contentsline {subsection}{\numberline {6.3.13}Transform a real number into a string: \texttt {cat} \texttt {+} }{85}{subsection.6.3.13}%
\contentsline {subsection}{\numberline {6.3.14}Transform a string into a number: \texttt {expr}}{85}{subsection.6.3.14}%
\contentsline {section}{\numberline {6.4}Write an integer in a different base}{86}{section.6.4}%
\contentsline {subsection}{\numberline {6.4.1}Writing an integer in base 2, 8 or 16}{86}{subsection.6.4.1}%
\contentsline {subsection}{\numberline {6.4.2}Writing an integer in an arbitrary base $b$: \texttt {convert} }{88}{subsection.6.4.2}%
\contentsline {section}{\numberline {6.5}Integers (and Gaussian Integers)}{89}{section.6.5}%
\contentsline {subsection}{\numberline {6.5.1}GCD: \texttt {gcd} \texttt {igcd} \texttt {Gcd} }{89}{subsection.6.5.1}%
\contentsline {subsection}{\numberline {6.5.2}GCD of a list of integers: \texttt {lgcd} }{92}{subsection.6.5.2}%
\contentsline {subsection}{\numberline {6.5.3}The least common multiple: \texttt {lcm}}{92}{subsection.6.5.3}%
\contentsline {subsection}{\numberline {6.5.4}Decomposition into prime factors: \texttt {ifactor} }{93}{subsection.6.5.4}%
\contentsline {subsection}{\numberline {6.5.5}List of prime factors: \texttt {ifactors} }{94}{subsection.6.5.5}%
\contentsline {subsection}{\numberline {6.5.6}Matrix of factors: \texttt {maple\_ifactors} }{95}{subsection.6.5.6}%
\contentsline {subsection}{\numberline {6.5.7}The divisors of a number: \texttt {idivis} \texttt {divisors} }{95}{subsection.6.5.7}%
\contentsline {subsection}{\numberline {6.5.8}The integer Euclidean quotient: \texttt {iquo} \texttt {intDiv} \texttt {div} }{96}{subsection.6.5.8}%
\contentsline {subsection}{\numberline {6.5.9}The integer Euclidean remainder: \texttt {irem} \texttt {remain} \texttt {smod} \texttt {mods} \texttt {mod} \texttt {\%} }{97}{subsection.6.5.9}%
\contentsline {subsection}{\numberline {6.5.10}Euclidean quotient and Euclidean remainder of two integers: \texttt {iquorem} }{98}{subsection.6.5.10}%
\contentsline {subsection}{\numberline {6.5.11}Test of evenness: \texttt {even} }{99}{subsection.6.5.11}%
\contentsline {subsection}{\numberline {6.5.12}Test of oddness: \texttt {odd} }{100}{subsection.6.5.12}%
\contentsline {subsection}{\numberline {6.5.13}Test of pseudo-primality: \texttt {is\_pseudoprime} }{100}{subsection.6.5.13}%
\contentsline {subsection}{\numberline {6.5.14}Test of primality: \texttt {is\_prime} \texttt {isprime} \texttt {isPrime} }{101}{subsection.6.5.14}%
\contentsline {subsection}{\numberline {6.5.15}The smallest pseudo-prime greater than $n$: \texttt {nextprime} }{103}{subsection.6.5.15}%
\contentsline {subsection}{\numberline {6.5.16}The greatest pseudo-prime less than \texttt {n}: \texttt {prevprime} }{103}{subsection.6.5.16}%
\contentsline {subsection}{\numberline {6.5.17}The $n$th pseudo-prime number: \texttt {ithprime}}{103}{subsection.6.5.17}%
\contentsline {subsection}{\numberline {6.5.18}The number of pseudo-primes less than or equal to $n$: \texttt {nprimes} }{104}{subsection.6.5.18}%
\contentsline {subsection}{\numberline {6.5.19}B\'{e}zout's Identity: \texttt {iegcd} \texttt {igcdex} }{104}{subsection.6.5.19}%
\contentsline {subsection}{\numberline {6.5.20}Solving $au+bv=c$ in $\mathbb {Z}$: \texttt {iabcuv} }{105}{subsection.6.5.20}%
\contentsline {subsection}{\numberline {6.5.21}Chinese remainders: \texttt {ichinrem} \texttt {ichrem} \texttt {chrem} }{105}{subsection.6.5.21}%
\contentsline {subsection}{\numberline {6.5.22}Solving $a^2+b^2=p$ in $\mathbb {Z}$: \texttt {pa2b2} }{108}{subsection.6.5.22}%
\contentsline {subsection}{\numberline {6.5.23}The Euler indicatrix: \texttt {euler} \texttt {phi} }{108}{subsection.6.5.23}%
\contentsline {subsection}{\numberline {6.5.24}Legendre symbol: \texttt {legendre\_symbol} }{109}{subsection.6.5.24}%
\contentsline {subsection}{\numberline {6.5.25}Jacobi symbol: \texttt {jacobi\_symbol} }{110}{subsection.6.5.25}%
\contentsline {subsection}{\numberline {6.5.26}Listing all compositions of an integer into $k$ parts: \texttt {icomp} }{110}{subsection.6.5.26}%
\contentsline {section}{\numberline {6.6}Combinatorial analysis}{111}{section.6.6}%
\contentsline {subsection}{\numberline {6.6.1}Factorial: \texttt {factorial} \texttt {!} }{111}{subsection.6.6.1}%
\contentsline {subsection}{\numberline {6.6.2}Binomial coefficients: \texttt {binomial} \texttt {comb} \texttt {nCr} }{112}{subsection.6.6.2}%
\contentsline {subsection}{\numberline {6.6.3}Permutations: \texttt {perm} \texttt {nPr} }{113}{subsection.6.6.3}%
\contentsline {subsection}{\numberline {6.6.4}Wilf-Zeilberger pairs: \texttt {wz\_certificate} }{113}{subsection.6.6.4}%
\contentsline {section}{\numberline {6.7}Rational numbers}{115}{section.6.7}%
\contentsline {subsection}{\numberline {6.7.1}Transform a floating point number into a rational: \texttt {exact} \texttt {float2rational} }{115}{subsection.6.7.1}%
\contentsline {subsection}{\numberline {6.7.2}Integer and fractional part: \texttt {propfrac} \texttt {propFrac} }{116}{subsection.6.7.2}%
\contentsline {subsection}{\numberline {6.7.3}Numerator of a fraction after simplification: \texttt {numer} \texttt {getNum} }{117}{subsection.6.7.3}%
\contentsline {subsection}{\numberline {6.7.4}Denominator of a fraction after simplification: \texttt {denom} \texttt {getDenom} }{117}{subsection.6.7.4}%
\contentsline {subsection}{\numberline {6.7.5}Numerator and denominator of a fraction: \texttt {f2nd} \texttt {fxnd} }{118}{subsection.6.7.5}%
\contentsline {subsection}{\numberline {6.7.6}Simplification of a pair of integers: \texttt {simp2} }{118}{subsection.6.7.6}%
\contentsline {subsection}{\numberline {6.7.7}Continued fraction representation of a real: \texttt {dfc} }{119}{subsection.6.7.7}%
\contentsline {subsection}{\numberline {6.7.8}Transform a continued fraction representation into a real: \texttt {dfc2f} }{122}{subsection.6.7.8}%
\contentsline {subsection}{\numberline {6.7.9}The $n$-th Bernoulli number: \texttt {bernoulli} }{123}{subsection.6.7.9}%
\contentsline {subsection}{\numberline {6.7.10}Access to PARI/GP commands: \texttt {pari} }{124}{subsection.6.7.10}%
\contentsline {section}{\numberline {6.8}Real numbers}{124}{section.6.8}%
\contentsline {subsection}{\numberline {6.8.1}Evaluate a real at a given precision: \texttt {evalf} \texttt {Digits} \texttt {DIGITS} }{124}{subsection.6.8.1}%
\contentsline {subsection}{\numberline {6.8.2}The standard infixed operators on real numbers: \texttt {+} \texttt {-} \texttt {*} \texttt {/} \texttt {\^{}} }{126}{subsection.6.8.2}%
\contentsline {subsection}{\numberline {6.8.3}Prefixed division on reals: \texttt {rdiv} }{127}{subsection.6.8.3}%
\contentsline {subsection}{\numberline {6.8.4}$n$-th root: \texttt {root} }{128}{subsection.6.8.4}%
\contentsline {subsection}{\numberline {6.8.5}The exponential integral function: \texttt {Ei} }{128}{subsection.6.8.5}%
\contentsline {subsection}{\numberline {6.8.6}The logarithmic integral function: \texttt {Li} }{130}{subsection.6.8.6}%
\contentsline {subsection}{\numberline {6.8.7}The cosine integral function: \texttt {Ci} }{131}{subsection.6.8.7}%
\contentsline {subsection}{\numberline {6.8.8}The sine integral function: \texttt {Si} }{131}{subsection.6.8.8}%
\contentsline {subsection}{\numberline {6.8.9}The Heaviside function: \texttt {Heaviside} }{132}{subsection.6.8.9}%
\contentsline {subsection}{\numberline {6.8.10}The Dirac distribution: \texttt {Dirac} }{133}{subsection.6.8.10}%
\contentsline {subsection}{\numberline {6.8.11}Error function: \texttt {erf} }{134}{subsection.6.8.11}%
\contentsline {subsection}{\numberline {6.8.12}Complementary error function: \texttt {erfc} }{135}{subsection.6.8.12}%
\contentsline {subsection}{\numberline {6.8.13}The $\Gamma $ function: \texttt {Gamma} }{136}{subsection.6.8.13}%
\contentsline {subsection}{\numberline {6.8.14}The upper incomplete $\gamma $ function: \texttt {ugamma} }{137}{subsection.6.8.14}%
\contentsline {subsection}{\numberline {6.8.15}The lower incomplete $\gamma $ function: \texttt {igamma} }{137}{subsection.6.8.15}%
\contentsline {subsection}{\numberline {6.8.16}The $\beta $ function: \texttt {Beta} }{138}{subsection.6.8.16}%
\contentsline {subsection}{\numberline {6.8.17}Derivatives of the DiGamma function: \texttt {Psi} }{139}{subsection.6.8.17}%
\contentsline {subsection}{\numberline {6.8.18}The $\zeta $ function: \texttt {Zeta} }{140}{subsection.6.8.18}%
\contentsline {subsection}{\numberline {6.8.19}Airy functions: \texttt {Airy\_Ai} and \texttt {Airy\_Bi} }{140}{subsection.6.8.19}%
\contentsline {section}{\numberline {6.9}Permutations}{142}{section.6.9}%
\contentsline {subsection}{\numberline {6.9.1}Random permutation: \texttt {randperm} \texttt {shuffle} }{142}{subsection.6.9.1}%
\contentsline {subsection}{\numberline {6.9.2}Previous and next permutation: \texttt {prevperm} \texttt {nextperm} }{142}{subsection.6.9.2}%
\contentsline {subsection}{\numberline {6.9.3}Decomposition of a permuation as a product of disjoint cycles: \texttt {permu2cycles} }{143}{subsection.6.9.3}%
\contentsline {subsection}{\numberline {6.9.4}Product of cycles to permutation: \texttt {cycles2permu} }{144}{subsection.6.9.4}%
\contentsline {subsection}{\numberline {6.9.5}Transform a cycle into a permutation: \texttt {cycle2perm} }{145}{subsection.6.9.5}%
\contentsline {subsection}{\numberline {6.9.6}Transform a permutation into a matrix: \texttt {permu2mat} }{145}{subsection.6.9.6}%
\contentsline {subsection}{\numberline {6.9.7}Checking for a permutation: \texttt {is\_permu} }{145}{subsection.6.9.7}%
\contentsline {subsection}{\numberline {6.9.8}Checking for a cycle: \texttt {is\_cycle} }{146}{subsection.6.9.8}%
\contentsline {subsection}{\numberline {6.9.9}Product of two permutations: \texttt {p1op2} \texttt {c1op2} \texttt {p1oc2} \texttt {c1oc2} }{147}{subsection.6.9.9}%
\contentsline {subsection}{\numberline {6.9.10}Signature of a permutation: \texttt {signature} }{148}{subsection.6.9.10}%
\contentsline {subsection}{\numberline {6.9.11}Inverse of a permutation: \texttt {perminv} }{148}{subsection.6.9.11}%
\contentsline {subsection}{\numberline {6.9.12}Inverse of a cycle: \texttt {cycleinv} }{149}{subsection.6.9.12}%
\contentsline {subsection}{\numberline {6.9.13}Order of a permutation: \texttt {permuorder} }{149}{subsection.6.9.13}%
\contentsline {subsection}{\numberline {6.9.14}The group generated by two permutations: \texttt {groupermu} }{150}{subsection.6.9.14}%
\contentsline {section}{\numberline {6.10}Complex numbers}{150}{section.6.10}%
\contentsline {subsection}{\numberline {6.10.1}The usual complex operators: \texttt {+} \texttt {-} \texttt {*} \texttt {/} \texttt {\^{}} }{150}{subsection.6.10.1}%
\contentsline {subsection}{\numberline {6.10.2}The real and imaginary parts of a complex number: \texttt {re} \texttt {real} \texttt {im} \texttt {imag} }{150}{subsection.6.10.2}%
\contentsline {subsection}{\numberline {6.10.3}Write a complex number $z$ in rectangular form: \texttt {evalc} }{151}{subsection.6.10.3}%
\contentsline {subsection}{\numberline {6.10.4}The modulus and argument of a complex number: \texttt {abs} \texttt {arg} }{151}{subsection.6.10.4}%
\contentsline {subsection}{\numberline {6.10.5}The normalized complex number: \texttt {normalize} \texttt {unitV} }{152}{subsection.6.10.5}%
\contentsline {subsection}{\numberline {6.10.6}Conjugate of a complex number: \texttt {conj}}{153}{subsection.6.10.6}%
\contentsline {subsection}{\numberline {6.10.7}Multiplication by the complex conjugate: \texttt {mult\_c\_conjugate} }{153}{subsection.6.10.7}%
\contentsline {subsection}{\numberline {6.10.8}Barycenter of complex numbers: \texttt {barycenter} }{154}{subsection.6.10.8}%
\contentsline {section}{\numberline {6.11}Algebraic numbers}{154}{section.6.11}%
\contentsline {subsection}{\numberline {6.11.1}Definition}{154}{subsection.6.11.1}%
\contentsline {subsection}{\numberline {6.11.2}Minimum polynomial of an algebraic number: \texttt {pmin} }{154}{subsection.6.11.2}%
\contentsline {section}{\numberline {6.12}Algebraic expressions}{156}{section.6.12}%
\contentsline {subsection}{\numberline {6.12.1}Evaluate an expression: \texttt {eval} }{156}{subsection.6.12.1}%
\contentsline {subsection}{\numberline {6.12.2}Change the evaluation level: \texttt {eval\_level} }{156}{subsection.6.12.2}%
\contentsline {subsection}{\numberline {6.12.3}Evaluate algebraic expressions: \texttt {evala} }{157}{subsection.6.12.3}%
\contentsline {subsection}{\numberline {6.12.4}Prevent evaluation: \texttt {quote} \texttt {hold} \texttt {'} }{157}{subsection.6.12.4}%
\contentsline {subsection}{\numberline {6.12.5}Force evaluation: \texttt {unquote} }{158}{subsection.6.12.5}%
\contentsline {subsection}{\numberline {6.12.6}Distribution: \texttt {expand} \texttt {fdistrib} }{158}{subsection.6.12.6}%
\contentsline {subsection}{\numberline {6.12.7}Canonical form: \texttt {canonical\_form} }{159}{subsection.6.12.7}%
\contentsline {subsection}{\numberline {6.12.8}Multiplication by the conjugate quantity: \texttt {mult\_conjugate} }{159}{subsection.6.12.8}%
\contentsline {subsection}{\numberline {6.12.9}Separation of variables: \texttt {split} }{160}{subsection.6.12.9}%
\contentsline {subsection}{\numberline {6.12.10}Factorization: \texttt {factor} \texttt {cfactor} }{161}{subsection.6.12.10}%
\contentsline {subsection}{\numberline {6.12.11}Zeros of an expression: \texttt {zeros} }{162}{subsection.6.12.11}%
\contentsline {subsection}{\numberline {6.12.12}Regrouping expressions: \texttt {regroup} }{164}{subsection.6.12.12}%
\contentsline {subsection}{\numberline {6.12.13}Normal form: \texttt {normal} }{164}{subsection.6.12.13}%
\contentsline {subsection}{\numberline {6.12.14}Simplify: \texttt {simplify}}{165}{subsection.6.12.14}%
\contentsline {subsection}{\numberline {6.12.15}Automatic simplification: \texttt {autosimplify} }{165}{subsection.6.12.15}%
\contentsline {subsection}{\numberline {6.12.16}Normal form for rational functions: \texttt {ratnormal} }{167}{subsection.6.12.16}%
\contentsline {subsection}{\numberline {6.12.17}Substitute a variable by a value: \texttt {|} }{167}{subsection.6.12.17}%
\contentsline {subsection}{\numberline {6.12.18}Substitute a variable by a value: \texttt {subst}}{168}{subsection.6.12.18}%
\contentsline {subsection}{\numberline {6.12.19}Substitute a variable by a value: \texttt {()} }{170}{subsection.6.12.19}%
\contentsline {subsection}{\numberline {6.12.20}Substitute a variable by a value (Maple and Mupad compatibility): \texttt {subs} }{170}{subsection.6.12.20}%
\contentsline {subsection}{\numberline {6.12.21}Substitute a subexpression by another expression: \texttt {algsubs} }{172}{subsection.6.12.21}%
\contentsline {subsection}{\numberline {6.12.22}Eliminate one or more variables from a list of equations: \texttt {eliminate} }{172}{subsection.6.12.22}%
\contentsline {subsection}{\numberline {6.12.23}Evaluate a primitive at boundaries: \texttt {preval} }{174}{subsection.6.12.23}%
\contentsline {subsection}{\numberline {6.12.24}Sub-expression of an expression: \texttt {part} }{174}{subsection.6.12.24}%
\contentsline {section}{\numberline {6.13}Values of a sequence $u_n$}{175}{section.6.13}%
\contentsline {subsection}{\numberline {6.13.1}Array of values of a sequence : \texttt {tablefunc} }{175}{subsection.6.13.1}%
\contentsline {subsection}{\numberline {6.13.2}Values of a recurrence relation or a system: \texttt {seqsolve} }{176}{subsection.6.13.2}%
\contentsline {subsection}{\numberline {6.13.3}Values of a recurrence relation or a system: \texttt {rsolve} }{177}{subsection.6.13.3}%
\contentsline {subsection}{\numberline {6.13.4}Table of values and graph of a recurrent sequence: \texttt {tableseq}}{179}{subsection.6.13.4}%
\contentsline {section}{\numberline {6.14}Operators or infixed functions}{180}{section.6.14}%
\contentsline {subsection}{\numberline {6.14.1}\texttt {Xcas} operators: \texttt {\spacefactor \@m {}} \texttt {\spacefactor \@m {}\spacefactor \@m {}} \texttt {\$} \texttt {\%} }{180}{subsection.6.14.1}%
\contentsline {subsection}{\numberline {6.14.2}Define an operator: \texttt {user\_operator} }{183}{subsection.6.14.2}%
\contentsline {section}{\numberline {6.15}Functions and expressions with symbolic variables}{186}{section.6.15}%
\contentsline {subsection}{\numberline {6.15.1}The difference between a function and an expression }{186}{subsection.6.15.1}%
\contentsline {subsection}{\numberline {6.15.2}Transform an expression into a function: \texttt {unapply} }{186}{subsection.6.15.2}%
\contentsline {subsection}{\numberline {6.15.3}Top and leaves of an expression: \texttt {sommet} \texttt {feuille} \texttt {op} \texttt {left} \texttt {right} }{188}{subsection.6.15.3}%
\contentsline {section}{\numberline {6.16}Functions}{191}{section.6.16}%
\contentsline {subsection}{\numberline {6.16.1}Context-dependent functions.}{191}{subsection.6.16.1}%
\contentsline {subsubsection}{The \texttt {+} operator }{191}{section*.6}%
\contentsline {subsubsection}{The \texttt {-},\texttt {*} and \texttt {/} operators }{193}{section*.7}%
\contentsline {subsection}{\numberline {6.16.2}Standard functions}{197}{subsection.6.16.2}%
\contentsline {subsection}{\numberline {6.16.3}Defining algebraic functions}{209}{subsection.6.16.3}%
\contentsline {subsubsection}{Defining a function from $\mathbb {R}^{p}$ to $\mathbb {R}^{q}$}{209}{section*.8}%
\contentsline {subsubsection}{Defining families of function from $\mathbb {R}^{p-1}$ to $\mathbb {R}^q$ using a function from $\mathbb {R}^p$ to $\mathbb {R}^q$}{211}{section*.9}%
\contentsline {subsection}{\numberline {6.16.4}Composition of two functions: \texttt {@}}{212}{subsection.6.16.4}%
\contentsline {subsection}{\numberline {6.16.5}Repeated function composition: \texttt {@@} }{212}{subsection.6.16.5}%
\contentsline {subsection}{\numberline {6.16.6}Define a function with history: \texttt {as\_function\_of} }{213}{subsection.6.16.6}%
\contentsline {section}{\numberline {6.17}Getting information about functions from $\mathbb {R}$ to $\mathbb {R}$}{214}{section.6.17}%
\contentsline {subsection}{\numberline {6.17.1}The domain of a function: \texttt {domain} }{214}{subsection.6.17.1}%
\contentsline {subsection}{\numberline {6.17.2}Table of variations of a function: \texttt {tabvar} }{215}{subsection.6.17.2}%
\contentsline {section}{\numberline {6.18}Limits: \texttt {limit} }{216}{section.6.18}%
\contentsline {section}{\numberline {6.19}Derivation and applications}{218}{section.6.19}%
\contentsline {subsection}{\numberline {6.19.1}Functional derivative: \texttt {function\_diff} }{218}{subsection.6.19.1}%
\contentsline {subsection}{\numberline {6.19.2}Length of an arc: \texttt {arcLen} }{219}{subsection.6.19.2}%
\contentsline {subsection}{\numberline {6.19.3}Maximum and minimum of an expression: \texttt {fMax} \texttt {fMin} }{221}{subsection.6.19.3}%
\contentsline {subsection}{\numberline {6.19.4}Derivatives and partial derivatives }{223}{subsection.6.19.4}%
\contentsline {subsection}{\numberline {6.19.5}Implicit differentiation: \texttt {implicitdiff} }{225}{subsection.6.19.5}%
\contentsline {section}{\numberline {6.20}Integration}{227}{section.6.20}%
\contentsline {subsection}{\numberline {6.20.1}Antiderivative and definite integral: \texttt {integrate} \texttt {int} \texttt {Int} }{227}{subsection.6.20.1}%
\contentsline {subsection}{\numberline {6.20.2}Primitive and definite integral: \texttt {risch} }{230}{subsection.6.20.2}%
\contentsline {subsection}{\numberline {6.20.3}Discrete summation: \texttt {sum} }{231}{subsection.6.20.3}%
\contentsline {subsection}{\numberline {6.20.4}Riemann sum: \texttt {sum\_riemann} }{233}{subsection.6.20.4}%
\contentsline {subsection}{\numberline {6.20.5}Integration by parts}{235}{subsection.6.20.5}%
\contentsline {subsubsection}{\texttt {ibpdv} }{235}{section*.10}%
\contentsline {subsubsection}{\texttt {ibpu} }{237}{section*.11}%
\contentsline {subsection}{\numberline {6.20.6}Change of variables: \texttt {subst}}{238}{subsection.6.20.6}%
\contentsline {subsection}{\numberline {6.20.7}Integrals and limits}{238}{subsection.6.20.7}%
\contentsline {section}{\numberline {6.21}Multivariate calculus}{239}{section.6.21}%
\contentsline {subsection}{\numberline {6.21.1}Gradient: \texttt {derive} \texttt {deriver} \texttt {diff} \texttt {grad} }{239}{subsection.6.21.1}%
\contentsline {subsection}{\numberline {6.21.2}Laplacian: \texttt {laplacian} }{240}{subsection.6.21.2}%
\contentsline {subsection}{\numberline {6.21.3}Hessian matrix: \texttt {hessian} }{242}{subsection.6.21.3}%
\contentsline {subsection}{\numberline {6.21.4}Divergence: \texttt {divergence} }{243}{subsection.6.21.4}%
\contentsline {subsection}{\numberline {6.21.5}Rotational: \texttt {curl} }{243}{subsection.6.21.5}%
\contentsline {subsection}{\numberline {6.21.6}Potential: \texttt {potential} }{244}{subsection.6.21.6}%
\contentsline {subsection}{\numberline {6.21.7}Conservative flux field: \texttt {vpotential} }{244}{subsection.6.21.7}%
\contentsline {subsection}{\numberline {6.21.8}Determining where a function is convex: \texttt {convex} }{245}{subsection.6.21.8}%
\contentsline {section}{\numberline {6.22}Calculus of variations}{247}{section.6.22}%
\contentsline {subsection}{\numberline {6.22.1}The Brachistochrone Problem}{247}{subsection.6.22.1}%
\contentsline {subsection}{\numberline {6.22.2}Euler-Lagrange equation(s): \texttt {euler\_lagrange} }{248}{subsection.6.22.2}%
\contentsline {subsection}{\numberline {6.22.3}Solution of the Brachistochrone Problem}{252}{subsection.6.22.3}%
\contentsline {subsection}{\numberline {6.22.4}Jacobi equation: \texttt {jacobi\_equation} }{253}{subsection.6.22.4}%
\contentsline {subsection}{\numberline {6.22.5}Finding conjugate points: \texttt {conjugate\_equation} }{254}{subsection.6.22.5}%
\contentsline {subsection}{\numberline {6.22.6}An example: Finding the surface of revolution with minimal area}{255}{subsection.6.22.6}%
\contentsline {section}{\numberline {6.23}Trigonometry}{258}{section.6.23}%
\contentsline {subsection}{\numberline {6.23.1}Expand a trigonometric expression: \texttt {trigexpand} }{259}{subsection.6.23.1}%
\contentsline {subsection}{\numberline {6.23.2}Linearize a trigonometric expression: \texttt {tlin} }{259}{subsection.6.23.2}%
\contentsline {subsection}{\numberline {6.23.3}Increase the phase by $\pi /2$ in a trigonometric expression: \texttt {shift\_phase} }{260}{subsection.6.23.3}%
\contentsline {subsection}{\numberline {6.23.4}Put together sine and cosine of the same angle: \texttt {tcollect} \texttt {tCollect} }{261}{subsection.6.23.4}%
\contentsline {subsection}{\numberline {6.23.5}Simplify: \texttt {simplify} }{261}{subsection.6.23.5}%
\contentsline {subsection}{\numberline {6.23.6}Simplify trigonometric expressions: \texttt {trigsimplify} }{262}{subsection.6.23.6}%
\contentsline {subsection}{\numberline {6.23.7}Transform arccos into arcsin: \texttt {acos2asin} }{262}{subsection.6.23.7}%
\contentsline {subsection}{\numberline {6.23.8}Transform arccos into arctan: \texttt {acos2atan} }{263}{subsection.6.23.8}%
\contentsline {subsection}{\numberline {6.23.9}Transform arcsin into arccos: \texttt {asin2acos} }{263}{subsection.6.23.9}%
\contentsline {subsection}{\numberline {6.23.10}Transform arcsin into arctan: \texttt {asin2atan} }{263}{subsection.6.23.10}%
\contentsline {subsection}{\numberline {6.23.11}Transform arctan into arcsin: \texttt {atan2asin} }{264}{subsection.6.23.11}%
\contentsline {subsection}{\numberline {6.23.12}Transform arctan into arccos: \texttt {atan2acos} }{264}{subsection.6.23.12}%
\contentsline {subsection}{\numberline {6.23.13}Transform complex exponentials into sin and cos: \texttt {sincos} \texttt {exp2trig} }{265}{subsection.6.23.13}%
\contentsline {subsection}{\numberline {6.23.14}Transform tan(x) into sin(x)/cos(x): \texttt {tan2sincos} }{266}{subsection.6.23.14}%
\contentsline {subsection}{\numberline {6.23.15}Transform sin(x) into cos(x)*tan(x): \texttt {sin2costan} }{266}{subsection.6.23.15}%
\contentsline {subsection}{\numberline {6.23.16}Transform cos(x) into sin(x)/tan(x): \texttt {cos2sintan} }{266}{subsection.6.23.16}%
\contentsline {subsection}{\numberline {6.23.17}Rewrite tan(x) with sin(2x) and cos(2x): \texttt {tan2sincos2} }{267}{subsection.6.23.17}%
\contentsline {subsection}{\numberline {6.23.18}Rewrite tan(x) with cos(2x) and sin(2x): \texttt {tan2cossin2} }{267}{subsection.6.23.18}%
\contentsline {subsection}{\numberline {6.23.19}Rewrite sin, cos, tan in terms of tan(x/2): \texttt {halftan} }{267}{subsection.6.23.19}%
\contentsline {subsection}{\numberline {6.23.20}Rewrite trigonometric functions as function of tan(x/2) and hyperbolic functions as function of exp(x): \texttt {halftan\_hyp2exp} }{268}{subsection.6.23.20}%
\contentsline {subsection}{\numberline {6.23.21}Transform trigonometric functions into complex exponentials : \texttt {trig2exp} }{269}{subsection.6.23.21}%
\contentsline {subsection}{\numberline {6.23.22}Transform inverse trigonometric functions into logarithms: \texttt {atrig2ln} }{270}{subsection.6.23.22}%
\contentsline {subsection}{\numberline {6.23.23}Simplify and express preferentially with sines: \texttt {trigsin} }{270}{subsection.6.23.23}%
\contentsline {subsection}{\numberline {6.23.24}Simplify and express preferentially with cosines: \texttt {trigcos} }{270}{subsection.6.23.24}%
\contentsline {subsection}{\numberline {6.23.25}Simplify and express preferentially with tangents: \texttt {trigtan} }{271}{subsection.6.23.25}%
\contentsline {subsection}{\numberline {6.23.26}Rewrite an expression with different options: \texttt {convert} \texttt {convertir} \texttt {=>} }{271}{subsection.6.23.26}%
\contentsline {section}{\numberline {6.24}Exponentials and Logarithms}{273}{section.6.24}%
\contentsline {subsection}{\numberline {6.24.1}Rewrite hyperbolic functions as exponentials: \texttt {hyp2exp} }{273}{subsection.6.24.1}%
\contentsline {subsection}{\numberline {6.24.2}Expand exponentials: \texttt {expexpand} }{274}{subsection.6.24.2}%
\contentsline {subsection}{\numberline {6.24.3}Expand logarithms: \texttt {lnexpand} }{274}{subsection.6.24.3}%
\contentsline {subsection}{\numberline {6.24.4}Linearize exponentials: \texttt {lin} }{274}{subsection.6.24.4}%
\contentsline {subsection}{\numberline {6.24.5}Collect logarithms: \texttt {lncollect} }{275}{subsection.6.24.5}%
\contentsline {subsection}{\numberline {6.24.6}Expand powers: \texttt {powexpand} }{276}{subsection.6.24.6}%
\contentsline {subsection}{\numberline {6.24.7}Rewrite a power as an exponential: \texttt {pow2exp} }{276}{subsection.6.24.7}%
\contentsline {subsection}{\numberline {6.24.8}Rewrite exp(n*ln(x)) as a power: \texttt {exp2pow} }{276}{subsection.6.24.8}%
\contentsline {subsection}{\numberline {6.24.9}Simplify complex exponentials: \texttt {tsimplify} }{277}{subsection.6.24.9}%
\contentsline {section}{\numberline {6.25}Rewriting transcendental and trigonometric expressions}{277}{section.6.25}%
\contentsline {subsection}{\numberline {6.25.1}Expand a transcendental and trigonometric expression: \texttt {texpand} \texttt {tExpand} }{277}{subsection.6.25.1}%
\contentsline {subsection}{\numberline {6.25.2}Combine terms of the same type: \texttt {combine} }{279}{subsection.6.25.2}%
\contentsline {section}{\numberline {6.26}Fourier transformation}{281}{section.6.26}%
\contentsline {subsection}{\numberline {6.26.1}Fourier coefficients: \texttt {fourier\_an} and \texttt {fourier\_bn} or \texttt {fourier\_cn} }{281}{subsection.6.26.1}%
\contentsline {subsubsection}{\texttt {fourier\_an} }{281}{section*.12}%
\contentsline {subsubsection}{\texttt {fourier\_bn} }{282}{section*.13}%
\contentsline {subsubsection}{\texttt {fourier\_cn} }{282}{section*.14}%
\contentsline {subsection}{\numberline {6.26.2}Continuous Fourier Transform: \texttt {fourier} \texttt {ifourier} \texttt {addtable} }{284}{subsection.6.26.2}%
\contentsline {subsection}{\numberline {6.26.3}Discrete Fourier Transform and the Fast Fourier Transform}{291}{subsection.6.26.3}%
\contentsline {subsubsection}{The properties of the Discrete Fourier Transform}{293}{section*.15}%
\contentsline {subsubsection}{Applications}{293}{section*.16}%
\contentsline {subsection}{\numberline {6.26.4}An \textbf {exercise} with \texttt {fft}}{297}{subsection.6.26.4}%
\contentsline {section}{\numberline {6.27}Polynomials}{299}{section.6.27}%
\contentsline {subsection}{\numberline {6.27.1}Polynomials of a single variable: \texttt {poly1} }{299}{subsection.6.27.1}%
\contentsline {subsection}{\numberline {6.27.2}Polynomials of several variables: \texttt {\%\%\%\{ \%\%\%\}}}{299}{subsection.6.27.2}%
\contentsline {subsection}{\numberline {6.27.3}Apply a function to the internal sparse format of a polynomial: \texttt {map} }{300}{subsection.6.27.3}%
\contentsline {subsection}{\numberline {6.27.4}Convert to a symbolic polynomial: \texttt {r2e} \texttt {poly2symb} }{300}{subsection.6.27.4}%
\contentsline {subsection}{\numberline {6.27.5}Convert from a symbolic polynomial: \texttt {e2r} \texttt {symb2poly} }{301}{subsection.6.27.5}%
\contentsline {subsection}{\numberline {6.27.6}Transform a polynomial in internal format into a list, and conversely: \texttt {convert} }{302}{subsection.6.27.6}%
\contentsline {subsection}{\numberline {6.27.7}Coefficients of a polynomial: \texttt {coeff} \texttt {coeffs} }{304}{subsection.6.27.7}%
\contentsline {subsection}{\numberline {6.27.8}Polynomial degree: \texttt {degree} }{304}{subsection.6.27.8}%
\contentsline {subsection}{\numberline {6.27.9}Polynomial valuation: \texttt {valuation} \texttt {ldegree} }{305}{subsection.6.27.9}%
\contentsline {subsection}{\numberline {6.27.10}Leading coefficient of a polynomial: \texttt {lcoeff} }{305}{subsection.6.27.10}%
\contentsline {subsection}{\numberline {6.27.11}Trailing coefficient degree of a polynomial: \texttt {tcoeff} }{306}{subsection.6.27.11}%
\contentsline {subsection}{\numberline {6.27.12}Evaluation of a polynomial: \texttt {peval} \texttt {polyEval} }{307}{subsection.6.27.12}%
\contentsline {subsection}{\numberline {6.27.13}Factorize $x^n$ in a polynomial: \texttt {factor\_xn} }{307}{subsection.6.27.13}%
\contentsline {subsection}{\numberline {6.27.14}GCD of the coefficients of a polynomial: \texttt {content} }{308}{subsection.6.27.14}%
\contentsline {subsection}{\numberline {6.27.15}Primitive part of a polynomial: \texttt {primpart} }{308}{subsection.6.27.15}%
\contentsline {subsection}{\numberline {6.27.16}Factoring: \texttt {collect} }{309}{subsection.6.27.16}%
\contentsline {subsection}{\numberline {6.27.17}Square-free factorization: \texttt {sqrfree} }{310}{subsection.6.27.17}%
\contentsline {subsection}{\numberline {6.27.18}List of factors: \texttt {factors} }{311}{subsection.6.27.18}%
\contentsline {subsection}{\numberline {6.27.19}Evaluate a polynomial: \texttt {horner} }{311}{subsection.6.27.19}%
\contentsline {subsection}{\numberline {6.27.20}Rewrite in terms of the powers of (x-a): \texttt {ptayl} }{312}{subsection.6.27.20}%
\contentsline {subsection}{\numberline {6.27.21}Compute with the exact root of a polynomial: \texttt {rootof} }{312}{subsection.6.27.21}%
\contentsline {subsection}{\numberline {6.27.22}Exact roots of a polynomial: \texttt {roots} }{313}{subsection.6.27.22}%
\contentsline {subsection}{\numberline {6.27.23}Coefficients of a polynomial defined by its roots: \texttt {pcoeff} \texttt {pcoef} }{314}{subsection.6.27.23}%
\contentsline {subsection}{\numberline {6.27.24}Truncate of order $n$: \texttt {truncate} }{315}{subsection.6.27.24}%
\contentsline {subsection}{\numberline {6.27.25}Convert a series expansion into a polynomial: \texttt {convert} \texttt {convertir} }{315}{subsection.6.27.25}%
\contentsline {subsection}{\numberline {6.27.26}Random polynomial: \texttt {randpoly} \texttt {randPoly} }{316}{subsection.6.27.26}%
\contentsline {subsection}{\numberline {6.27.27}Change the order of variables: \texttt {reorder} }{317}{subsection.6.27.27}%
\contentsline {subsection}{\numberline {6.27.28}Random list: \texttt {ranm} }{317}{subsection.6.27.28}%
\contentsline {subsection}{\numberline {6.27.29}Lagrange polynomial: \texttt {lagrange interp} }{317}{subsection.6.27.29}%
\contentsline {subsection}{\numberline {6.27.30}Trigonometric interpolation: \texttt {triginterp}}{319}{subsection.6.27.30}%
\contentsline {subsection}{\numberline {6.27.31}Natural splines: \texttt {spline} }{320}{subsection.6.27.31}%
\contentsline {subsubsection}{Definition}{320}{section*.17}%
\contentsline {subsubsection}{Theorem}{320}{section*.18}%
\contentsline {subsubsection}{Types of spline functions}{321}{section*.19}%
\contentsline {subsection}{\numberline {6.27.32}Natural interpolation: \texttt {spline}}{322}{subsection.6.27.32}%
\contentsline {subsection}{\numberline {6.27.33}Rational interpolation: \texttt {thiele} }{323}{subsection.6.27.33}%
\contentsline {section}{\numberline {6.28}Arithmetic and polynomials}{324}{section.6.28}%
\contentsline {subsection}{\numberline {6.28.1}The divisors of a polynomial: \texttt {divis} }{325}{subsection.6.28.1}%
\contentsline {subsection}{\numberline {6.28.2}Euclidean quotient: \texttt {quo} \texttt {Quo} }{325}{subsection.6.28.2}%
\contentsline {subsection}{\numberline {6.28.3}Euclidean remainder: \texttt {rem} \texttt {Rem} }{327}{subsection.6.28.3}%
\contentsline {subsection}{\numberline {6.28.4}Quotient and remainder: \texttt {quorem} \texttt {divide} }{328}{subsection.6.28.4}%
\contentsline {subsection}{\numberline {6.28.5}GCD of two polynomials with the Euclidean algorithm: \texttt {gcd} \texttt {Gcd} }{328}{subsection.6.28.5}%
\contentsline {subsection}{\numberline {6.28.6}GCD of two polynomials with the Euclidean algorithm: \texttt {Gcd} }{329}{subsection.6.28.6}%
\contentsline {subsection}{\numberline {6.28.7}Choosing the GCD algorithm of two polynomials: \texttt {ezgcd} \texttt {heugcd} \texttt {modgcd} \texttt {psrgcd} }{330}{subsection.6.28.7}%
\contentsline {subsection}{\numberline {6.28.8}LCM of two polynomials: \texttt {lcm} }{331}{subsection.6.28.8}%
\contentsline {subsection}{\numberline {6.28.9}B\'ezout's Identity: \texttt {egcd} \texttt {gcdex} }{332}{subsection.6.28.9}%
\contentsline {subsection}{\numberline {6.28.10}Solving au+bv=c over polynomials: \texttt {abcuv} }{333}{subsection.6.28.10}%
\contentsline {subsection}{\numberline {6.28.11}Chinese remainders: \texttt {chinrem} }{334}{subsection.6.28.11}%
\contentsline {subsection}{\numberline {6.28.12}Cyclotomic polynomial: \texttt {cyclotomic} }{335}{subsection.6.28.12}%
\contentsline {subsection}{\numberline {6.28.13}Sturm sequences and number of sign changes of $P$ on $(a,\ b]$: \texttt {sturm} \texttt {sturmseq} \texttt {sturmab} }{336}{subsection.6.28.13}%
\contentsline {subsection}{\numberline {6.28.14}Sylvester matrix of two polynomials and resultant: \texttt {sylvester} \texttt {resultant} }{340}{subsection.6.28.14}%
\contentsline {section}{\numberline {6.29}Exact bounds for roots of a polynomial}{343}{section.6.29}%
\contentsline {subsection}{\numberline {6.29.1}Exact bounds for real roots of a polynomial: \texttt {realroot} }{343}{subsection.6.29.1}%
\contentsline {subsection}{\numberline {6.29.2}Exact bounds for complex roots of a polynomial: \texttt {complexroot} }{345}{subsection.6.29.2}%
\contentsline {subsection}{\numberline {6.29.3}Exact bounds for real roots of a polynomial: \texttt {VAS} }{346}{subsection.6.29.3}%
\contentsline {subsection}{\numberline {6.29.4}Exact bounds for positive real roots of a polynomial: \texttt {VAS\_positive} }{347}{subsection.6.29.4}%
\contentsline {subsection}{\numberline {6.29.5}An upper bound for the positive real roots of a polynomial: \texttt {posubLMQ} }{347}{subsection.6.29.5}%
\contentsline {subsection}{\numberline {6.29.6}A lower bound for the positive real roots of a polynomial: \texttt {poslbdLMQ} }{348}{subsection.6.29.6}%
\contentsline {subsection}{\numberline {6.29.7}Exact values of rational roots of a polynomial: \texttt {rationalroot} }{349}{subsection.6.29.7}%
\contentsline {subsection}{\numberline {6.29.8}Exact values of the complex rational roots of a polynomial: \texttt {crationalroot} }{350}{subsection.6.29.8}%
\contentsline {section}{\numberline {6.30}Orthogonal polynomials}{350}{section.6.30}%
\contentsline {subsection}{\numberline {6.30.1}Legendre polynomials: \texttt {legendre} }{350}{subsection.6.30.1}%
\contentsline {subsection}{\numberline {6.30.2}Hermite polynomial: \texttt {hermite} }{351}{subsection.6.30.2}%
\contentsline {subsection}{\numberline {6.30.3}Laguerre polynomials: \texttt {laguerre} }{352}{subsection.6.30.3}%
\contentsline {subsection}{\numberline {6.30.4}Tchebychev polynomials of the first kind: \texttt {tchebyshev1} }{353}{subsection.6.30.4}%
\contentsline {subsection}{\numberline {6.30.5}Tchebychev polynomial of the second kind: \texttt {tchebyshev2} }{354}{subsection.6.30.5}%
\contentsline {section}{\numberline {6.31}Gr\"obner basis and Gr\"obner reduction}{355}{section.6.31}%
\contentsline {subsection}{\numberline {6.31.1}Gr\"obner basis: \texttt {gbasis} }{355}{subsection.6.31.1}%
\contentsline {subsection}{\numberline {6.31.2}Gr\"obner reduction: \texttt {greduce} }{356}{subsection.6.31.2}%
\contentsline {subsection}{\numberline {6.31.3}Test if a polynomial or list of polynomials belongs to an ideal given by a Gr\"obner basis: \texttt {in\_ideal} }{357}{subsection.6.31.3}%
\contentsline {subsection}{\numberline {6.31.4}Build a polynomial from its evaluation: \texttt {genpoly} }{358}{subsection.6.31.4}%
\contentsline {section}{\numberline {6.32}Rational functions}{359}{section.6.32}%
\contentsline {subsection}{\numberline {6.32.1}Numerator: \texttt {getNum} }{359}{subsection.6.32.1}%
\contentsline {subsection}{\numberline {6.32.2}Numerator after simplification: \texttt {numer} }{360}{subsection.6.32.2}%
\contentsline {subsection}{\numberline {6.32.3}Denominator: \texttt {getDenom}}{360}{subsection.6.32.3}%
\contentsline {subsection}{\numberline {6.32.4}Denominator after simplification: \texttt {denom} }{361}{subsection.6.32.4}%
\contentsline {subsection}{\numberline {6.32.5}Numerator and denominator: \texttt {f2nd} \texttt {fxnd} }{361}{subsection.6.32.5}%
\contentsline {subsection}{\numberline {6.32.6}Simplify: \texttt {simp2} }{362}{subsection.6.32.6}%
\contentsline {subsection}{\numberline {6.32.7}Common denominator: \texttt {comDenom} }{362}{subsection.6.32.7}%
\contentsline {subsection}{\numberline {6.32.8}Polynomial and fractional part: \texttt {propfrac} }{362}{subsection.6.32.8}%
\contentsline {subsection}{\numberline {6.32.9}Partial fraction expansion: \texttt {partfrac} \texttt {cpartfrac} }{363}{subsection.6.32.9}%
\contentsline {section}{\numberline {6.33}Exact roots and poles}{364}{section.6.33}%
\contentsline {subsection}{\numberline {6.33.1}Roots and poles of a rational function: \texttt {froot} }{364}{subsection.6.33.1}%
\contentsline {subsection}{\numberline {6.33.2}Rational function given by roots and poles: \texttt {fcoeff} }{365}{subsection.6.33.2}%
\contentsline {section}{\numberline {6.34}Computing in $\mathbb {Z}/p\mathbb {Z}$ or in $\mathbb {Z}/p\mathbb {Z}[x]$ }{365}{section.6.34}%
\contentsline {subsection}{\numberline {6.34.1}Expand and reduce: \texttt {normal} }{366}{subsection.6.34.1}%
\contentsline {subsection}{\numberline {6.34.2}Addition in $\mathbb {Z}/p\mathbb {Z}$ or in $ \mathbb {Z}/p\mathbb {Z}[x]$: \texttt {+} }{366}{subsection.6.34.2}%
\contentsline {subsection}{\numberline {6.34.3}Subtraction in $\mathbb {Z}/p\mathbb {Z}$ or in $ \mathbb {Z}/p\mathbb {Z}[x]$: \texttt {-} }{367}{subsection.6.34.3}%
\contentsline {subsection}{\numberline {6.34.4}Multiplication in $\mathbb {Z}/p\mathbb {Z}$ or in $ \mathbb {Z}/p\mathbb {Z}[x]$: \texttt {*} }{367}{subsection.6.34.4}%
\contentsline {subsection}{\numberline {6.34.5}Euclidean quotient : \texttt {quo} }{368}{subsection.6.34.5}%
\contentsline {subsection}{\numberline {6.34.6}Euclidean remainder: \texttt {rem} }{368}{subsection.6.34.6}%
\contentsline {subsection}{\numberline {6.34.7}Euclidean quotient and euclidean remainder: \texttt {quorem} }{369}{subsection.6.34.7}%
\contentsline {subsection}{\numberline {6.34.8}Division in $\mathbb {Z}/p\mathbb {Z}$ or in $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {/} }{369}{subsection.6.34.8}%
\contentsline {subsection}{\numberline {6.34.9}Power in $\mathbb {Z}/p\mathbb {Z}$ and in $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {\^{}}}{370}{subsection.6.34.9}%
\contentsline {subsection}{\numberline {6.34.10}Compute $a^n\ \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip \ p$: \texttt {powmod powermod} }{370}{subsection.6.34.10}%
\contentsline {subsection}{\numberline {6.34.11}Inverse in $\mathbb {Z}/p\mathbb {Z}$: \texttt {inv} \texttt {inverse} \texttt {/} }{371}{subsection.6.34.11}%
\contentsline {subsection}{\numberline {6.34.12}Rebuild a fraction from its value modulo $p$: \texttt {fracmod} \texttt { iratrecon} }{371}{subsection.6.34.12}%
\contentsline {subsection}{\numberline {6.34.13}GCD in $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {gcd} }{372}{subsection.6.34.13}%
\contentsline {subsection}{\numberline {6.34.14}Factorization over $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {factor} \texttt {factoriser} }{373}{subsection.6.34.14}%
\contentsline {subsection}{\numberline {6.34.15}Determinant of a matrix in $\mathbb {Z}/p\mathbb {Z}$: \texttt {det} }{373}{subsection.6.34.15}%
\contentsline {subsection}{\numberline {6.34.16}Inverse of a matrix with coefficients in $\mathbb {Z}/p\mathbb {Z}$: \texttt {inv} \texttt {inverse} }{374}{subsection.6.34.16}%
\contentsline {subsection}{\numberline {6.34.17}Row reduction to echelon form in $\mathbb {Z}/p\mathbb {Z}$: \texttt {rref} }{374}{subsection.6.34.17}%
\contentsline {subsection}{\numberline {6.34.18}Construction of a Galois field: \texttt {GF} }{375}{subsection.6.34.18}%
\contentsline {subsection}{\numberline {6.34.19}Factorize a polynomial with coefficients in a Galois field: \texttt {factor} }{378}{subsection.6.34.19}%
\contentsline {section}{\numberline {6.35}Compute in $\mathbb {Z}/p\mathbb {Z}[x]$ using Maple syntax }{379}{section.6.35}%
\contentsline {subsection}{\numberline {6.35.1}Euclidean quotient: \texttt {Quo} }{379}{subsection.6.35.1}%
\contentsline {subsection}{\numberline {6.35.2}Euclidean remainder: \texttt {Rem} }{380}{subsection.6.35.2}%
\contentsline {subsection}{\numberline {6.35.3}GCD in $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {Gcd} }{381}{subsection.6.35.3}%
\contentsline {subsection}{\numberline {6.35.4}Factorization in $\mathbb {Z}/p\mathbb {Z}[x]$: \texttt {Factor} }{382}{subsection.6.35.4}%
\contentsline {subsection}{\numberline {6.35.5}Determinant of a matrix with coefficients in $\mathbb {Z}/p\mathbb {Z}$: \texttt {Det} }{382}{subsection.6.35.5}%
\contentsline {subsection}{\numberline {6.35.6}Inverse of a matrix in $\mathbb {Z}/p\mathbb {Z}$: \texttt {Inverse} }{383}{subsection.6.35.6}%
\contentsline {subsection}{\numberline {6.35.7}Row reduction to echelon form in $\mathbb {Z}/p\mathbb {Z}$: \texttt {Rref} }{384}{subsection.6.35.7}%
\contentsline {section}{\numberline {6.36}Taylor and asymptotic expansions}{385}{section.6.36}%
\contentsline {subsection}{\numberline {6.36.1}Division by increasing power order: \texttt {divpc} }{385}{subsection.6.36.1}%
\contentsline {subsection}{\numberline {6.36.2}Series expansion: \texttt {taylor} \texttt {series} }{385}{subsection.6.36.2}%
\contentsline {subsection}{\numberline {6.36.3}The inverse of a series: \texttt {revert} }{388}{subsection.6.36.3}%
\contentsline {subsection}{\numberline {6.36.4}The residue of an expression at a point: \texttt {residue} }{389}{subsection.6.36.4}%
\contentsline {subsection}{\numberline {6.36.5}Pad\'e expansion: \texttt {pade} }{389}{subsection.6.36.5}%
\contentsline {section}{\numberline {6.37}Ranges of values}{391}{section.6.37}%
\contentsline {subsection}{\numberline {6.37.1}Definition of a range of values: \texttt {..} }{391}{subsection.6.37.1}%
\contentsline {subsection}{\numberline {6.37.2}Center of a range of values: \texttt {interval2center} }{392}{subsection.6.37.2}%
\contentsline {subsection}{\numberline {6.37.3}Ranges of values defined by their center: \texttt {center2interval} }{392}{subsection.6.37.3}%
\contentsline {section}{\numberline {6.38}Intervals}{393}{section.6.38}%
\contentsline {subsection}{\numberline {6.38.1}Defining intervals: \texttt {i[]} }{393}{subsection.6.38.1}%
\contentsline {subsection}{\numberline {6.38.2}The endpoints of an interval: \texttt {left} \texttt {right} }{394}{subsection.6.38.2}%
\contentsline {subsection}{\numberline {6.38.3}Interval arithmetic: \texttt {+ - * /} }{394}{subsection.6.38.3}%
\contentsline {subsection}{\numberline {6.38.4}The midpoint of an interval: \texttt {midpoint} }{396}{subsection.6.38.4}%
\contentsline {subsection}{\numberline {6.38.5}The union of intervals: \texttt {union} }{396}{subsection.6.38.5}%
\contentsline {subsection}{\numberline {6.38.6}The intersection of intervals: \texttt {intersect} }{397}{subsection.6.38.6}%
\contentsline {subsection}{\numberline {6.38.7}Test if an object is in an interval: \texttt {contains} }{397}{subsection.6.38.7}%
\contentsline {subsection}{\numberline {6.38.8}Convert a number to an interval: \texttt {convert} }{398}{subsection.6.38.8}%
\contentsline {section}{\numberline {6.39}Sequences and lists}{398}{section.6.39}%
\contentsline {subsection}{\numberline {6.39.1}Defining a sequence or a list: \texttt {seq[ ]} \texttt {()} }{398}{subsection.6.39.1}%
\contentsline {subsection}{\numberline {6.39.2}Make a sequence or a list: \texttt {seq} \texttt {\$} }{399}{subsection.6.39.2}%
\contentsline {subsection}{\numberline {6.39.3}Length of a sequence or list: \texttt {size} \texttt {nops} \texttt {length} }{403}{subsection.6.39.3}%
\contentsline {subsection}{\numberline {6.39.4}Get the first element of a sequence or list: \texttt {head} }{404}{subsection.6.39.4}%
\contentsline {subsection}{\numberline {6.39.5}Get a sequence or list without the first element: \texttt {tail} }{404}{subsection.6.39.5}%
\contentsline {subsection}{\numberline {6.39.6}Get an element of a sequence or a list: \texttt {[]} \texttt {[[]]} \texttt {at} }{405}{subsection.6.39.6}%
\contentsline {subsection}{\numberline {6.39.7}Find a subsequence or a sublist }{406}{subsection.6.39.7}%
\contentsline {subsection}{\numberline {6.39.8}Concatenate two sequences: \texttt {,} }{407}{subsection.6.39.8}%
\contentsline {subsection}{\numberline {6.39.9}The \texttt {+} operator applied on sequences and lists }{407}{subsection.6.39.9}%
\contentsline {subsection}{\numberline {6.39.10}Transforming sequences into lists and lists into sequences: \texttt {[]} \texttt {nop} \texttt {op} \texttt {makesuite} }{408}{subsection.6.39.10}%
\contentsline {section}{\numberline {6.40}Operations on lists }{409}{section.6.40}%
\contentsline {subsection}{\numberline {6.40.1}Sizes of a list of lists: \texttt {sizes} }{409}{subsection.6.40.1}%
\contentsline {subsection}{\numberline {6.40.2}Make a list with a function: \texttt {makelist} }{409}{subsection.6.40.2}%
\contentsline {subsection}{\numberline {6.40.3}Make a list with zeros: \texttt {newList} }{410}{subsection.6.40.3}%
\contentsline {subsection}{\numberline {6.40.4}Make a list of integers: \texttt {range} }{411}{subsection.6.40.4}%
\contentsline {subsection}{\numberline {6.40.5}Select the elements of a list: \texttt {select} }{412}{subsection.6.40.5}%
\contentsline {subsection}{\numberline {6.40.6}The left and right portions of a list: \texttt {left} \texttt {right} }{413}{subsection.6.40.6}%
\contentsline {subsection}{\numberline {6.40.7}Modifying the elements of a list: \texttt {subsop} }{413}{subsection.6.40.7}%
\contentsline {subsection}{\numberline {6.40.8}Remove an element in a list: \texttt {suppress} }{415}{subsection.6.40.8}%
\contentsline {subsection}{\numberline {6.40.9}Remove elements of a list: \texttt {remove} }{415}{subsection.6.40.9}%
\contentsline {subsection}{\numberline {6.40.10}Insert an element into a list or a string: \texttt {insert} }{416}{subsection.6.40.10}%
\contentsline {subsection}{\numberline {6.40.11}Append an element at the end of a list: \texttt {append} }{417}{subsection.6.40.11}%
\contentsline {subsection}{\numberline {6.40.12}Prepend an element at the beginning of a list: \texttt {prepend} }{417}{subsection.6.40.12}%
\contentsline {subsection}{\numberline {6.40.13}Concatenate two lists or a list and an element: \texttt {concat} \texttt {augment} }{418}{subsection.6.40.13}%
\contentsline {subsection}{\numberline {6.40.14}Flatten a list: \texttt {flatten} }{419}{subsection.6.40.14}%
\contentsline {subsection}{\numberline {6.40.15}Reverse order in a list: \texttt {revlist} }{419}{subsection.6.40.15}%
\contentsline {subsection}{\numberline {6.40.16}Rotate a list: \texttt {rotate} }{420}{subsection.6.40.16}%
\contentsline {subsection}{\numberline {6.40.17}Shifting the elements of a list: \texttt {shift} }{420}{subsection.6.40.17}%
\contentsline {subsection}{\numberline {6.40.18}Sort: \texttt {sort} }{421}{subsection.6.40.18}%
\contentsline {subsection}{\numberline {6.40.19}Sort a list by increasing order: \texttt {SortA} }{422}{subsection.6.40.19}%
\contentsline {subsection}{\numberline {6.40.20}Sort a list by decreasing order: \texttt {SortD} }{423}{subsection.6.40.20}%
\contentsline {subsection}{\numberline {6.40.21}Number of elements equal to a given value: \texttt {count\_eq} }{423}{subsection.6.40.21}%
\contentsline {subsection}{\numberline {6.40.22}Number of elements smaller than a given value: \texttt {count\_inf} }{424}{subsection.6.40.22}%
\contentsline {subsection}{\numberline {6.40.23}Number of elements greater than a given value: \texttt {count\_sup} }{424}{subsection.6.40.23}%
\contentsline {subsection}{\numberline {6.40.24}Sum of elements of a list: \texttt {sum add} }{424}{subsection.6.40.24}%
\contentsline {subsection}{\numberline {6.40.25}Sum of list (or matrix) elements transformed by a function: \texttt {count} }{425}{subsection.6.40.25}%
\contentsline {subsection}{\numberline {6.40.26}Cumulated sum of the elements of a list: \texttt {cumSum} }{426}{subsection.6.40.26}%
\contentsline {subsection}{\numberline {6.40.27}Product: \texttt {product} \texttt {mul} }{427}{subsection.6.40.27}%
\contentsline {subsubsection}{Product of values of an expression: \texttt {product}}{427}{section*.20}%
\contentsline {subsubsection}{Product of elements of a list: \texttt {product} }{428}{section*.21}%
\contentsline {subsection}{\numberline {6.40.28}Apply a function of one variable to the elements of a list: \texttt {map} \texttt {apply} }{429}{subsection.6.40.28}%
\contentsline {subsection}{\numberline {6.40.29}Apply a bivariate function to the elements of two lists: \texttt {zip} }{431}{subsection.6.40.29}%
\contentsline {subsection}{\numberline {6.40.30}Fold operators: \texttt {foldl} \texttt {foldr} }{432}{subsection.6.40.30}%
\contentsline {subsection}{\numberline {6.40.31}List of differences of consecutive terms: \texttt {deltalist} }{433}{subsection.6.40.31}%
\contentsline {subsection}{\numberline {6.40.32}Make a matrix with a list: \texttt {list2mat} }{433}{subsection.6.40.32}%
\contentsline {subsection}{\numberline {6.40.33}Make a list with a matrix: \texttt {mat2list} }{434}{subsection.6.40.33}%
\contentsline {section}{\numberline {6.41}Operations on sets and lists}{434}{section.6.41}%
\contentsline {subsection}{\numberline {6.41.1}Defining a set or list: \texttt {set[ ]} \texttt {\%\{ \%\}} }{434}{subsection.6.41.1}%
\contentsline {subsection}{\numberline {6.41.2}Test if a value is in a list or a set: \texttt {member} \texttt {contains} }{435}{subsection.6.41.2}%
\contentsline {subsection}{\numberline {6.41.3}Union of two sets or of two lists: \texttt {union} }{436}{subsection.6.41.3}%
\contentsline {subsection}{\numberline {6.41.4}Intersection of two sets or of two lists: \texttt {intersect} }{436}{subsection.6.41.4}%
\contentsline {subsection}{\numberline {6.41.5}Difference of two sets or of two lists: \texttt {minus} }{437}{subsection.6.41.5}%
\contentsline {subsubsection}{Cartesian products}{437}{section*.22}%
\contentsline {subsubsection}{Defining an $n$-tuple: \texttt {tuple} }{437}{section*.23}%
\contentsline {subsubsection}{The Cartesian product of two sets: \texttt {*}}{437}{section*.24}%
\contentsline {section}{\numberline {6.42}Functions for vectors}{438}{section.6.42}%
\contentsline {subsection}{\numberline {6.42.1}Norms of a vector: \texttt {maxnorm} \texttt {l1norm} \texttt {l2norm} \texttt {norm} }{438}{subsection.6.42.1}%
\contentsline {subsection}{\numberline {6.42.2}Normalize a vector: \texttt {normalize unitV} }{438}{subsection.6.42.2}%
\contentsline {subsection}{\numberline {6.42.3}Term by term sum of two lists: \texttt {+} \texttt {.+} }{439}{subsection.6.42.3}%
\contentsline {subsection}{\numberline {6.42.4}Term by term difference of two lists: \texttt {-} \texttt {.-} }{440}{subsection.6.42.4}%
\contentsline {subsection}{\numberline {6.42.5}Term by term product of two lists: \texttt {.*} }{440}{subsection.6.42.5}%
\contentsline {subsection}{\numberline {6.42.6}Term by term quotient of two lists: \texttt {./} }{440}{subsection.6.42.6}%
\contentsline {subsection}{\numberline {6.42.7}Scalar product : \texttt {scalar\_product} \texttt {*} \texttt {dotprod} \texttt {dot} \texttt {dotP} \texttt {scalar\_Product} }{441}{subsection.6.42.7}%
\contentsline {subsection}{\numberline {6.42.8}Cross product: \texttt {cross} \texttt {crossP} \texttt {crossproduct} }{441}{subsection.6.42.8}%
\contentsline {subsection}{\numberline {6.42.9}Statistics on lists: \texttt {mean} \texttt {variance} \texttt {stddev} \texttt {stddevp} \texttt {median} \texttt {quantile} \texttt {quartiles} \texttt {boxwhisker} }{442}{subsection.6.42.9}%
\contentsline {section}{\numberline {6.43}Table with strings as indices: \texttt {table} }{446}{section.6.43}%
\contentsline {section}{\numberline {6.44}Matrices}{447}{section.6.44}%
\contentsline {subsection}{\numberline {6.44.1}Matrices}{447}{subsection.6.44.1}%
\contentsline {subsection}{\numberline {6.44.2}Special matrices}{447}{subsection.6.44.2}%
\contentsline {subsubsection}{Identity matrix: \texttt {idn} \texttt {identity} }{447}{section*.25}%
\contentsline {subsubsection}{Zero matrix: \texttt {newMat} }{448}{section*.26}%
\contentsline {subsubsection}{Diagonals of matrices and diagonal matrices: \texttt {BlockDiagonal} \texttt {diag} }{448}{section*.27}%
\contentsline {subsubsection}{Jordan block: \texttt {JordanBlock} }{449}{section*.28}%
\contentsline {subsubsection}{Hilbert matrix: \texttt {hilbert} }{449}{section*.29}%
\contentsline {subsubsection}{Vandermonde matrix: \texttt {vandermonde} }{450}{section*.30}%
\contentsline {subsection}{\numberline {6.44.3}Combining matrices}{450}{subsection.6.44.3}%
\contentsline {subsubsection}{Make a matrix with a list of matrices: \texttt {blockmatrix} }{450}{section*.31}%
\contentsline {subsubsection}{Make a matrix from two matrices: \texttt {semi\_augment} }{452}{section*.32}%
\contentsline {subsubsection}{Make a matrix from two matrices: \texttt {augment} \texttt {concat} }{453}{section*.33}%
\contentsline {subsubsection}{Append a column to a matrix: \texttt {border} }{454}{section*.34}%
\contentsline {subsection}{\numberline {6.44.4}Creating a matrix with a formula or function: \texttt {makemat} \texttt {matrix} }{455}{subsection.6.44.4}%
\contentsline {subsection}{\numberline {6.44.5}Getting the parts of a matrix}{456}{subsection.6.44.5}%
\contentsline {subsubsection}{Accessing parts of a matrix: \texttt {[]} \texttt {at} }{456}{section*.35}%
\contentsline {subsubsection}{Extract rows or columns of a matrix (Maple compatibility): \texttt {row} \texttt {col} }{458}{section*.36}%
\contentsline {subsubsection}{Extract a sub-matrix of a matrix (TI compatibility): \texttt {subMat} }{459}{section*.37}%
\contentsline {subsection}{\numberline {6.44.6}Modifying matrices}{461}{subsection.6.44.6}%
\contentsline {subsubsection}{Modifying matrix elements by assignment: \texttt {:=} }{461}{section*.38}%
\contentsline {subsubsection}{Modifying matrix elements by reference: \texttt {::=} \texttt {=<} }{462}{section*.39}%
\contentsline {subsubsection}{Modify an element or a row of a matrix: \texttt {subsop} }{464}{section*.40}%
\contentsline {paragraph}{In \texttt {Xcas}, \texttt {Mupad} and \texttt {TI} modes:}{464}{section*.41}%
\contentsline {paragraph}{In \texttt {Maple} mode:}{466}{section*.42}%
\contentsline {paragraph}{In all modes:}{466}{section*.43}%
\contentsline {subsubsection}{Remove rows or columns of a matrix: \texttt {delrows} \texttt {delcols} }{467}{section*.44}%
\contentsline {subsubsection}{Resize a matrix or vector: \texttt {REDIM} \texttt {redim} }{468}{section*.45}%
\contentsline {subsubsection}{Replacing part of a matrix or vector: \texttt {REPLACE} \texttt {replace} }{469}{section*.46}%
\contentsline {subsubsection}{Applying a function to the elements of a matrix: \texttt {apply} }{470}{section*.47}%
\contentsline {section}{\numberline {6.45}Arithmetic and matrices}{470}{section.6.45}%
\contentsline {subsection}{\numberline {6.45.1}Evaluate a matrix: \texttt {evalm} }{470}{subsection.6.45.1}%
\contentsline {subsection}{\numberline {6.45.2}Addition and subtraction of two matrices: \texttt {+} \texttt {-} \texttt {.+} \texttt {.-} }{471}{subsection.6.45.2}%
\contentsline {subsection}{\numberline {6.45.3}Multiplication of two matrices: \texttt {*} \texttt {\&*} }{471}{subsection.6.45.3}%
\contentsline {subsection}{\numberline {6.45.4}Addition of elements of a column of a matrix: \texttt {sum} }{472}{subsection.6.45.4}%
\contentsline {subsection}{\numberline {6.45.5}Cumulated sum of elements of each column of a matrix: \texttt {cumSum} }{472}{subsection.6.45.5}%
\contentsline {subsection}{\numberline {6.45.6}Multiplication of elements of each column of a matrix: \texttt {product} }{472}{subsection.6.45.6}%
\contentsline {subsection}{\numberline {6.45.7}Power of a matrix: \texttt {\^{}} \texttt {\&\^{}} }{473}{subsection.6.45.7}%
\contentsline {subsection}{\numberline {6.45.8}Hadamard product: \texttt {hadamard} \texttt {product} \texttt {.*} }{473}{subsection.6.45.8}%
\contentsline {subsection}{\numberline {6.45.9}Hadamard division: \texttt {./} }{474}{subsection.6.45.9}%
\contentsline {subsection}{\numberline {6.45.10}Hadamard power: \texttt {.\^{}}}{474}{subsection.6.45.10}%
\contentsline {subsection}{\numberline {6.45.11}The elementary row operations}{474}{subsection.6.45.11}%
\contentsline {subsubsection}{Add a row to another row: \texttt {rowAdd} }{474}{section*.48}%
\contentsline {subsubsection}{Multiply a row by an expression: \texttt {mRow} \texttt {scale} \texttt {SCALE} }{475}{section*.49}%
\contentsline {subsubsection}{Add $k$ times a row to an another row: \texttt {mRowAdd} \texttt {scaleadd} \texttt {SCALEADD} }{476}{section*.50}%
\contentsline {subsubsection}{Exchange two rows: \texttt {rowSwap} \texttt {rowswap} \texttt {swaprow} }{477}{section*.51}%
\contentsline {subsubsection}{Exchange two columns: \texttt {colSwap} \texttt {colswap} \texttt {swapcol} }{477}{section*.52}%
\contentsline {subsection}{\numberline {6.45.12}Count the elements of a matrix satisfying a property: \texttt {count} }{477}{subsection.6.45.12}%
\contentsline {subsection}{\numberline {6.45.13}Count the elements equal to a given value: \texttt {count\_eq} }{478}{subsection.6.45.13}%
\contentsline {subsection}{\numberline {6.45.14}Count the elements smaller than a given value: \texttt {count\_inf} }{478}{subsection.6.45.14}%
\contentsline {subsection}{\numberline {6.45.15}Count the elements greater than a given value: \texttt {count\_sup} }{479}{subsection.6.45.15}%
\contentsline {subsection}{\numberline {6.45.16}Statistics functions acting on column matrices: \texttt {mean} \texttt {stddev} \texttt {variance} \texttt {median} \texttt {quantile} \texttt {quartiles} \texttt {boxwhisker} }{479}{subsection.6.45.16}%
\contentsline {subsection}{\numberline {6.45.17}Dimension of a matrix: \texttt {dim} }{482}{subsection.6.45.17}%
\contentsline {subsection}{\numberline {6.45.18}Number of rows: \texttt {rowdim} \texttt {rowDim} \texttt {nrows} }{482}{subsection.6.45.18}%
\contentsline {subsection}{\numberline {6.45.19}Number of columns: \texttt {coldim} \texttt {colDim} \texttt {ncols} }{483}{subsection.6.45.19}%
\contentsline {section}{\numberline {6.46}Sparse matrices}{483}{section.6.46}%
\contentsline {subsection}{\numberline {6.46.1}Defining sparse matrices }{483}{subsection.6.46.1}%
\contentsline {subsection}{\numberline {6.46.2}Operations on sparse matrices}{484}{subsection.6.46.2}%
\contentsline {section}{\numberline {6.47}Linear algebra}{485}{section.6.47}%
\contentsline {subsection}{\numberline {6.47.1}Transpose of a matrix: \texttt {tran} \texttt {transpose} }{485}{subsection.6.47.1}%
\contentsline {subsection}{\numberline {6.47.2}Inverse of a matrix: \texttt {inv} \texttt {/} }{485}{subsection.6.47.2}%
\contentsline {subsection}{\numberline {6.47.3}Trace of a matrix: \texttt {trace} }{486}{subsection.6.47.3}%
\contentsline {subsection}{\numberline {6.47.4}Determinant of a matrix: \texttt {det} }{486}{subsection.6.47.4}%
\contentsline {subsection}{\numberline {6.47.5}Determinant of a sparse matrix: \texttt {det\_minor} }{487}{subsection.6.47.5}%
\contentsline {subsection}{\numberline {6.47.6}Rank of a matrix: \texttt {rank} }{488}{subsection.6.47.6}%
\contentsline {subsection}{\numberline {6.47.7}Transconjugate of a matrix: \texttt {trn} }{488}{subsection.6.47.7}%
\contentsline {subsection}{\numberline {6.47.8}Equivalent matrix: \texttt {changebase} }{488}{subsection.6.47.8}%
\contentsline {subsection}{\numberline {6.47.9}Basis of a linear subspace : \texttt {basis} }{489}{subsection.6.47.9}%
\contentsline {subsection}{\numberline {6.47.10}Basis of the intersection of two subspaces: \texttt {ibasis} }{489}{subsection.6.47.10}%
\contentsline {subsection}{\numberline {6.47.11}Image of a linear function: \texttt {image} }{490}{subsection.6.47.11}%
\contentsline {subsection}{\numberline {6.47.12}Kernel of a linear function: \texttt {kernel} \texttt {nullspace} \texttt {ker} }{490}{subsection.6.47.12}%
\contentsline {subsection}{\numberline {6.47.13}Kernel of a linear function: \texttt {Nullspace} }{490}{subsection.6.47.13}%
\contentsline {subsection}{\numberline {6.47.14}Subspace generated by the columns of a matrix: \texttt {colspace} }{491}{subsection.6.47.14}%
\contentsline {subsection}{\numberline {6.47.15}Subspace generated by the rows of a matrix: \texttt {rowspace} }{492}{subsection.6.47.15}%
\contentsline {section}{\numberline {6.48}Matrix reduction}{493}{section.6.48}%
\contentsline {subsection}{\numberline {6.48.1}Eigenvalues: \texttt {eigenvals} }{493}{subsection.6.48.1}%
\contentsline {subsection}{\numberline {6.48.2}Eigenvalues: \texttt {egvl} \texttt {eigenvalues} \texttt {eigVl} }{493}{subsection.6.48.2}%
\contentsline {subsection}{\numberline {6.48.3}Eigenvectors: \texttt {egv} \texttt {eigenvectors} \texttt {eigenvects} \texttt {eigVc} }{494}{subsection.6.48.3}%
\contentsline {subsection}{\numberline {6.48.4}Rational Jordan matrix: \texttt {rat\_jordan} }{495}{subsection.6.48.4}%
\contentsline {subsection}{\numberline {6.48.5}Jordan normal form: \texttt {jordan} }{497}{subsection.6.48.5}%
\contentsline {subsection}{\numberline {6.48.6}Powers of a square matrix: \texttt {matpow} }{498}{subsection.6.48.6}%
\contentsline {subsection}{\numberline {6.48.7}Characteristic polynomial: \texttt {charpoly} }{498}{subsection.6.48.7}%
\contentsline {subsection}{\numberline {6.48.8}Characteristic polynomial using Hessenberg algorithm: \texttt {pcar\_hessenberg} }{499}{subsection.6.48.8}%
\contentsline {subsection}{\numberline {6.48.9}Minimal polynomial: \texttt {pmin} }{500}{subsection.6.48.9}%
\contentsline {subsection}{\numberline {6.48.10}Adjoint matrix: \texttt {adjoint\_matrix} }{501}{subsection.6.48.10}%
\contentsline {subsection}{\numberline {6.48.11}Companion matrix of a polynomial: \texttt {companion} }{502}{subsection.6.48.11}%
\contentsline {subsection}{\numberline {6.48.12}Hessenberg matrix reduction: \texttt {hessenberg} \texttt {SCHUR} }{503}{subsection.6.48.12}%
\contentsline {subsection}{\numberline {6.48.13}Hermite normal form: \texttt {ihermite} }{504}{subsection.6.48.13}%
\contentsline {paragraph}{Application: Compute a $\mathbb {Z}$-basis of the kernel of a matrix having integer coefficients}{505}{section*.53}%
\contentsline {subsection}{\numberline {6.48.14}Smith normal form in $\mathbb {Z}$: \texttt {ismith} }{505}{subsection.6.48.14}%
\contentsline {subsection}{\numberline {6.48.15}Smith normal form: \texttt {smith} }{506}{subsection.6.48.15}%
\contentsline {section}{\numberline {6.49}Matrix factorizations}{507}{section.6.49}%
\contentsline {subsection}{\numberline {6.49.1}Cholesky decomposition: \texttt {cholesky} }{508}{subsection.6.49.1}%
\contentsline {subsection}{\numberline {6.49.2}QR decomposition: \texttt {qr} }{509}{subsection.6.49.2}%
\contentsline {subsection}{\numberline {6.49.3}QR decomposition (for TI compatibility): \texttt {QR} }{509}{subsection.6.49.3}%
\contentsline {subsection}{\numberline {6.49.4}LQ decomposition (HP compatible): \texttt {LQ} }{510}{subsection.6.49.4}%
\contentsline {subsection}{\numberline {6.49.5}LU decomposition: \texttt {lu} }{510}{subsection.6.49.5}%
\contentsline {subsection}{\numberline {6.49.6}LU decomposition (for TI compatibility): \texttt {LU} }{512}{subsection.6.49.6}%
\contentsline {subsection}{\numberline {6.49.7}Singular values (HP compatible): \texttt {SVL} \texttt {svl} }{513}{subsection.6.49.7}%
\contentsline {subsection}{\numberline {6.49.8}Singular value decomposition: \texttt {svd} }{514}{subsection.6.49.8}%
\contentsline {subsection}{\numberline {6.49.9}Short basis of a lattice: \texttt {lll} }{514}{subsection.6.49.9}%
\contentsline {section}{\numberline {6.50}Different matrix norms}{516}{section.6.50}%
\contentsline {subsection}{\numberline {6.50.1}The Frobenius norm: \texttt {frobenius\_norm} }{516}{subsection.6.50.1}%
\contentsline {subsection}{\numberline {6.50.2}$\ell ^2$ matrix norm: \texttt {norm} \texttt {l2norm} }{516}{subsection.6.50.2}%
\contentsline {subsection}{\numberline {6.50.3}$\ell ^\infty $ matrix norm: \texttt {maxnorm} }{517}{subsection.6.50.3}%
\contentsline {subsection}{\numberline {6.50.4}Matrix row norm: \texttt {rownorm} \texttt {rowNorm} }{517}{subsection.6.50.4}%
\contentsline {subsection}{\numberline {6.50.5}Matrix column norm: \texttt {colnorm} \texttt {colNorm} \texttt {l1norm} }{518}{subsection.6.50.5}%
\contentsline {subsection}{\numberline {6.50.6}The operator norm of a matrix: \texttt {matrix\_norm} \texttt {l1norm} \texttt {l2norm} \texttt {norm} \texttt {specnorm} \texttt {linfnorm} }{518}{subsection.6.50.6}%
\contentsline {subsubsection}{Operator norms}{518}{section*.54}%
\contentsline {subsubsection}{Computing operator norms}{519}{section*.55}%
\contentsline {section}{\numberline {6.51}Isometries}{520}{section.6.51}%
\contentsline {subsection}{\numberline {6.51.1}Recognize an isometry: \texttt {isom} }{521}{subsection.6.51.1}%
\contentsline {subsection}{\numberline {6.51.2}Find the matrix of an isometry: \texttt {mkisom} }{522}{subsection.6.51.2}%
\contentsline {section}{\numberline {6.52}Linear Programming}{523}{section.6.52}%
\contentsline {subsection}{\numberline {6.52.1}Simplex algorithm: \texttt {simplex\_reduce} }{524}{subsection.6.52.1}%
\contentsline {subsubsection}{The simple case}{524}{section*.56}%
\contentsline {subsubsection}{The simplex method}{524}{section*.57}%
\contentsline {subsubsection}{\texttt {simplex\_reduce}}{525}{section*.58}%
\contentsline {subsubsection}{Reducing a more complicated case to the simple case}{526}{section*.59}%
\contentsline {subsubsection}{The general case}{526}{section*.60}%
\contentsline {paragraph}{Finding a starting solution}{527}{section*.61}%
\contentsline {paragraph}{Finding a solution of the LP problem}{527}{section*.62}%
\contentsline {subsection}{\numberline {6.52.2}Solving general linear programming problems: \texttt {lpsolve} }{528}{subsection.6.52.2}%
\contentsline {subsubsection}{Solving an LP problem in symbolic form}{528}{section*.63}%
\contentsline {subsubsection}{Solving an LP problem in matrix form}{530}{section*.64}%
\contentsline {subsubsection}{Solving MIP (Mixed Integer Programming) problems}{531}{section*.65}%
\contentsline {subsubsection}{About \texttt {lpsolve}}{532}{section*.66}%
\contentsline {paragraph}{Implementation details.}{532}{section*.67}%
\contentsline {paragraph}{Stopping criteria.}{533}{section*.68}%
\contentsline {paragraph}{Branching strategies.}{533}{section*.69}%
\contentsline {paragraph}{Cutting planes.}{534}{section*.70}%
\contentsline {paragraph}{Displaying detailed output.}{534}{section*.71}%
\contentsline {subsubsection}{Solving problems in floating-point arithmetic}{535}{section*.72}%
\contentsline {subsubsection}{Loading a problem from a file}{537}{section*.73}%
\contentsline {subsection}{\numberline {6.52.3}Solving the transportation problems: \texttt {tpsolve} }{538}{subsection.6.52.3}%
\contentsline {section}{\numberline {6.53}Nonlinear optimization}{539}{section.6.53}%
\contentsline {subsection}{\numberline {6.53.1}Global extrema: \texttt {minimize} \texttt {maximize} }{539}{subsection.6.53.1}%
\contentsline {subsection}{\numberline {6.53.2}Local extrema: \texttt {extrema} }{542}{subsection.6.53.2}%
\contentsline {subsection}{\numberline {6.53.3}Global extrema without using derivatives: \texttt {nlpsolve} }{544}{subsection.6.53.3}%
\contentsline {subsection}{\numberline {6.53.4}Minimax polynomial approximation: \texttt {minimax} }{546}{subsection.6.53.4}%
\contentsline {section}{\numberline {6.54}Quadratic forms}{547}{section.6.54}%
\contentsline {subsection}{\numberline {6.54.1}Matrix of a quadratic form: \texttt {q2a} }{547}{subsection.6.54.1}%
\contentsline {subsection}{\numberline {6.54.2}Transform a matrix into a quadratic form: \texttt {a2q} }{547}{subsection.6.54.2}%
\contentsline {subsection}{\numberline {6.54.3}Reduction of a quadratic form: \texttt {gauss} }{548}{subsection.6.54.3}%
\contentsline {subsection}{\numberline {6.54.4}The conjugate gradient algorithm: \texttt {conjugate\_gradient} }{548}{subsection.6.54.4}%
\contentsline {subsection}{\numberline {6.54.5}Gram-Schmidt orthonormalization: \texttt {gramschmidt} }{549}{subsection.6.54.5}%
\contentsline {subsection}{\numberline {6.54.6}Graph of a conic: \texttt {conic} }{550}{subsection.6.54.6}%
\contentsline {subsection}{\numberline {6.54.7}Conic reduction: \texttt {reduced\_conic} }{550}{subsection.6.54.7}%
\contentsline {subsection}{\numberline {6.54.8}Graph of a quadric: \texttt {quadric} }{552}{subsection.6.54.8}%
\contentsline {subsection}{\numberline {6.54.9}Quadric reduction: \texttt {reduced\_quadric} }{552}{subsection.6.54.9}%
\contentsline {section}{\numberline {6.55}Equations}{554}{section.6.55}%
\contentsline {subsection}{\numberline {6.55.1}Define an equation: \texttt {equal} }{554}{subsection.6.55.1}%
\contentsline {subsection}{\numberline {6.55.2}Transform an equation into a difference: \texttt {equal2diff} }{555}{subsection.6.55.2}%
\contentsline {subsection}{\numberline {6.55.3}Transform an equation into a list: \texttt {equal2list} }{555}{subsection.6.55.3}%
\contentsline {subsection}{\numberline {6.55.4}The left member of an equation: \texttt {left} \texttt {gauche} \texttt {lhs} }{555}{subsection.6.55.4}%
\contentsline {subsection}{\numberline {6.55.5}The right member of an equation: \texttt {right} \texttt {droit} \texttt {rhs} }{556}{subsection.6.55.5}%
\contentsline {subsection}{\numberline {6.55.6}Solving equation(s): \texttt {solve} \texttt {cSolve} }{556}{subsection.6.55.6}%
\contentsline {section}{\numberline {6.56}Linear systems}{559}{section.6.56}%
\contentsline {subsection}{\numberline {6.56.1}Matrix of a system: \texttt {syst2mat} }{559}{subsection.6.56.1}%
\contentsline {subsection}{\numberline {6.56.2}Gauss reduction of a matrix: \texttt {ref} }{560}{subsection.6.56.2}%
\contentsline {subsection}{\numberline {6.56.3}Gauss-Jordan reduction: \texttt {rref} \texttt {gaussjord} }{560}{subsection.6.56.3}%
\contentsline {subsection}{\numberline {6.56.4}Solving $AX=b$: \texttt {simult} }{562}{subsection.6.56.4}%
\contentsline {subsection}{\numberline {6.56.5}Step by step Gauss-Jordan reduction of a matrix: \texttt {pivot} }{563}{subsection.6.56.5}%
\contentsline {subsection}{\numberline {6.56.6}Linear system solving: \texttt {linsolve} }{563}{subsection.6.56.6}%
\contentsline {subsection}{\numberline {6.56.7}Solving a linear system using the Jacobi iteration method: \texttt {jacobi\_linsolve} }{567}{subsection.6.56.7}%
\contentsline {subsection}{\numberline {6.56.8}Solving a linear system using the Gauss-Seidel iteration method: \texttt {gauss\_seidel\_linsolve} }{568}{subsection.6.56.8}%
\contentsline {subsection}{\numberline {6.56.9}The least squares solution of a linear system: \texttt {LSQ} \texttt {lsq} }{569}{subsection.6.56.9}%
\contentsline {subsection}{\numberline {6.56.10}Finding linear recurrences: \texttt {reverse\_rsolve} }{570}{subsection.6.56.10}%
\contentsline {section}{\numberline {6.57}Differential equations}{571}{section.6.57}%
\contentsline {subsection}{\numberline {6.57.1}Solving differential equations: \texttt {desolve} \texttt {deSolve} \texttt {dsolve} }{571}{subsection.6.57.1}%
\contentsline {subsection}{\numberline {6.57.2}Laplace transform and inverse Laplace transform: \texttt {laplace} \texttt {ilaplace} \texttt {invlaplace} }{579}{subsection.6.57.2}%
\contentsline {subsection}{\numberline {6.57.3}Solving linear homogeneous second-order ODE with rational coefficients: \texttt {kovacicsols} }{581}{subsection.6.57.3}%
\contentsline {section}{\numberline {6.58}The Z-transform}{586}{section.6.58}%
\contentsline {subsection}{\numberline {6.58.1}The Z-transform of a sequence: \texttt {ztrans} }{586}{subsection.6.58.1}%
\contentsline {subsection}{\numberline {6.58.2}The inverse Z-transform of a rational function: \texttt {invztrans} }{587}{subsection.6.58.2}%
\contentsline {section}{\numberline {6.59}Other functions}{588}{section.6.59}%
\contentsline {subsection}{\numberline {6.59.1}Replace small values by 0: \texttt {epsilon2zero} }{588}{subsection.6.59.1}%
\contentsline {subsection}{\numberline {6.59.2}List of variables: \texttt {lname} \texttt {indets} }{589}{subsection.6.59.2}%
\contentsline {subsection}{\numberline {6.59.3}List of variables and of expressions: \texttt {lvar} }{589}{subsection.6.59.3}%
\contentsline {subsection}{\numberline {6.59.4}List of variables of an algebraic expressions: \texttt {algvar} }{590}{subsection.6.59.4}%
\contentsline {subsection}{\numberline {6.59.5}Test if a variable is in an expression: \texttt {has} }{591}{subsection.6.59.5}%
\contentsline {subsection}{\numberline {6.59.6}Numeric evaluation: \texttt {evalf} }{592}{subsection.6.59.6}%
\contentsline {subsection}{\numberline {6.59.7}Rational approximation: \texttt {float2rational} \texttt {exact} }{592}{subsection.6.59.7}%
\contentsline {section}{\numberline {6.60}The day of the week: \texttt {dayofweek} }{593}{section.6.60}%
\contentsline {chapter}{\numberline {7}Metric properties of curves}{595}{chapter.7}%
\contentsline {section}{\numberline {7.1}The center of curvature}{595}{section.7.1}%
\contentsline {section}{\numberline {7.2}Computing the curvature and related values: \texttt {curvature} \texttt {osculating\_circle} \texttt {evolute} }{595}{section.7.2}%
\contentsline {chapter}{\numberline {8}Graphs}{599}{chapter.8}%
\contentsline {section}{\numberline {8.1}Generalities}{599}{section.8.1}%
\contentsline {section}{\numberline {8.2}The graphic screen }{600}{section.8.2}%
\contentsline {section}{\numberline {8.3}Graph and geometric objects attributes}{601}{section.8.3}%
\contentsline {subsection}{\numberline {8.3.1}Individual attributes }{601}{subsection.8.3.1}%
\contentsline {subsection}{\numberline {8.3.2}Global attributes }{603}{subsection.8.3.2}%
\contentsline {section}{\numberline {8.4}Graph of a function: \texttt {plotfunc} \texttt {funcplot} \texttt {DrawFunc} \texttt {Graph} }{605}{section.8.4}%
\contentsline {subsection}{\numberline {8.4.1}2-d graph}{605}{subsection.8.4.1}%
\contentsline {subsection}{\numberline {8.4.2}3-d graph}{606}{subsection.8.4.2}%
\contentsline {subsubsection}{Two variable functions}{606}{section*.74}%
\contentsline {subsubsection}{3-d graph with rainbow colors}{609}{section*.75}%
\contentsline {subsubsection}{4-d graph}{609}{section*.76}%
\contentsline {section}{\numberline {8.5}2d graph for Maple compatibility: \texttt {plot} }{611}{section.8.5}%
\contentsline {section}{\numberline {8.6}3d surfaces for Maple compatibility \texttt {plot3d} }{612}{section.8.6}%
\contentsline {section}{\numberline {8.7}Graph of a line and tangent to a graph}{614}{section.8.7}%
\contentsline {subsection}{\numberline {8.7.1}Draw a line: \texttt {line} }{614}{subsection.8.7.1}%
\contentsline {subsection}{\numberline {8.7.2}Draw a 2D horizontal line: \texttt {LineHorz} }{616}{subsection.8.7.2}%
\contentsline {subsection}{\numberline {8.7.3}Draw a 2D vertical line: \texttt {LineVert} }{617}{subsection.8.7.3}%
\contentsline {subsection}{\numberline {8.7.4}Tangent to a 2D graph: \texttt {LineTan} }{617}{subsection.8.7.4}%
\contentsline {subsection}{\numberline {8.7.5}Tangent to a 2D graph: \texttt {tangent} }{618}{subsection.8.7.5}%
\contentsline {subsection}{\numberline {8.7.6}Plot a line with a point and the slope: \texttt {DrawSlp}}{619}{subsection.8.7.6}%
\contentsline {subsection}{\numberline {8.7.7}Intersection of a 2D graph with the axis }{620}{subsection.8.7.7}%
\contentsline {section}{\numberline {8.8}Graph of inequalities with 2 variables: \texttt {plotinequation} \texttt {inequationplot}}{620}{section.8.8}%
\contentsline {section}{\numberline {8.9}The area under a curve: \texttt {area} }{621}{section.8.9}%
\contentsline {section}{\numberline {8.10}Graph of the area below a curve: \texttt {plotarea} \texttt {areaplot}}{622}{section.8.10}%
\contentsline {section}{\numberline {8.11}Contour lines: \texttt {plotcontour} \texttt {contourplot} \texttt {DrwCtour} }{624}{section.8.11}%
\contentsline {section}{\numberline {8.12}2-d graph of a 2-d function with colors: \texttt {plotdensity} \texttt {densityplot} }{625}{section.8.12}%
\contentsline {section}{\numberline {8.13}Implicit graph: \texttt {plotimplicit} \texttt {implicitplot}}{626}{section.8.13}%
\contentsline {subsection}{\numberline {8.13.1}2D implicit curve}{627}{subsection.8.13.1}%
\contentsline {subsection}{\numberline {8.13.2}3D implicit surface}{628}{subsection.8.13.2}%
\contentsline {section}{\numberline {8.14}Parametric curves and surfaces: \texttt {plotparam} \texttt {paramplot} \texttt {DrawParm} }{630}{section.8.14}%
\contentsline {subsection}{\numberline {8.14.1}2D parametric curve}{630}{subsection.8.14.1}%
\contentsline {subsection}{\numberline {8.14.2}3D parametric surface: \texttt {plotparam} \texttt {paramplot} \texttt {DrawParm} }{631}{subsection.8.14.2}%
\contentsline {section}{\numberline {8.15}Bezier curves: \texttt {bezier} }{632}{section.8.15}%
\contentsline {section}{\numberline {8.16}Curve defined in polar coordinates: \texttt {plotpolar} \texttt {polarplot} \texttt {DrawPol} \texttt {courbe\_polaire} }{634}{section.8.16}%
\contentsline {section}{\numberline {8.17}Graph of a recurrent sequence: \texttt {plotseq} \texttt {seqplot} \texttt {graphe\_suite} }{635}{section.8.17}%
\contentsline {section}{\numberline {8.18}Tangent field: \texttt {plotfield} \texttt {fieldplot} }{636}{section.8.18}%
\contentsline {section}{\numberline {8.19}Plotting a solution of a differential equation: \texttt {plotode} \texttt {odeplot} }{637}{section.8.19}%
\contentsline {section}{\numberline {8.20}Interactive plotting of solutions of a differential equation: \texttt {interactive\_plotode} \texttt {interactive\_odeplot} }{639}{section.8.20}%
\contentsline {section}{\numberline {8.21}Animated graphs (2D, 3D or "4D")}{639}{section.8.21}%
\contentsline {subsection}{\numberline {8.21.1}Animation of a 2D graph: \texttt {animate} }{640}{subsection.8.21.1}%
\contentsline {subsection}{\numberline {8.21.2}Animation of a 3D graph: \texttt {animate3d} }{640}{subsection.8.21.2}%
\contentsline {subsection}{\numberline {8.21.3}Animation of a sequence of graphic objects: \texttt {animation} }{641}{subsection.8.21.3}%
\contentsline {chapter}{\numberline {9}Statistics}{649}{chapter.9}%
\contentsline {section}{\numberline {9.1}One variable statistics}{649}{section.9.1}%
\contentsline {subsection}{\numberline {9.1.1}The mean: \texttt {mean} }{649}{subsection.9.1.1}%
\contentsline {subsection}{\numberline {9.1.2}Variance: \texttt {variance} }{650}{subsection.9.1.2}%
\contentsline {subsection}{\numberline {9.1.3}Standard deviation: \texttt {stdev} }{651}{subsection.9.1.3}%
\contentsline {subsection}{\numberline {9.1.4}The population standard deviation: \texttt {stddevp} \texttt {stdDev} }{651}{subsection.9.1.4}%
\contentsline {subsection}{\numberline {9.1.5}The median: \texttt {median} }{653}{subsection.9.1.5}%
\contentsline {subsection}{\numberline {9.1.6}Quartiles: \texttt {quartiles} \texttt {quartile1} \texttt {quartile3} }{653}{subsection.9.1.6}%
\contentsline {subsection}{\numberline {9.1.7}Quantiles: \texttt {quantile} }{654}{subsection.9.1.7}%
\contentsline {subsection}{\numberline {9.1.8}The boxwhisker: \texttt {boxwhisker} \texttt {mustache} }{655}{subsection.9.1.8}%
\contentsline {subsection}{\numberline {9.1.9}Classes: \texttt {classes} }{655}{subsection.9.1.9}%
\contentsline {subsection}{\numberline {9.1.10}Histograms: \texttt {histogram} \texttt {histogramme} }{657}{subsection.9.1.10}%
\contentsline {subsection}{\numberline {9.1.11}Accumulating terms: \texttt {accumulate\_head\_tail} }{658}{subsection.9.1.11}%
\contentsline {subsection}{\numberline {9.1.12}Frequencies: \texttt {frequencies} \texttt {frequences} }{658}{subsection.9.1.12}%
\contentsline {subsection}{\numberline {9.1.13}Cumulative frequencies: \texttt {cumulated\_frequencies} \texttt {frequences\_cumulees} }{659}{subsection.9.1.13}%
\contentsline {subsection}{\numberline {9.1.14}Bar graphs: \texttt {bar\_plot} }{661}{subsection.9.1.14}%
\contentsline {subsection}{\numberline {9.1.15}Pie charts: \texttt {camembert} }{662}{subsection.9.1.15}%
\contentsline {section}{\numberline {9.2}Two variable statistics}{663}{section.9.2}%
\contentsline {subsection}{\numberline {9.2.1}Covariance and correlation: \texttt {covariance} \texttt {correlation} \texttt {covariance\_correlation} }{663}{subsection.9.2.1}%
\contentsline {subsection}{\numberline {9.2.2}Scatterplots: \texttt {scatterplot} \texttt {nuaged\_points} \texttt {batons}}{665}{subsection.9.2.2}%
\contentsline {subsection}{\numberline {9.2.3}Polygonal paths: \texttt {polygonplot} \texttt {ligne\_polygonale} \texttt {linear\_interpolate} \texttt {listplot} \texttt {plotlist} }{666}{subsection.9.2.3}%
\contentsline {subsection}{\numberline {9.2.4}Linear regression: \texttt {linear\_regression} \texttt {linear\_regression\_plot} }{669}{subsection.9.2.4}%
\contentsline {subsection}{\numberline {9.2.5}Exponential regression: \texttt {exponential\_regression} \texttt {exponential\_regression\_plot} }{670}{subsection.9.2.5}%
\contentsline {subsection}{\numberline {9.2.6}Logarithmic regression: \texttt {logarithmic\_regression} \texttt {logarithmic\_regression\_plot} }{672}{subsection.9.2.6}%
\contentsline {subsection}{\numberline {9.2.7}Power regression: \texttt {power\_regression} \texttt {power\_regression\_plot} }{673}{subsection.9.2.7}%
\contentsline {subsection}{\numberline {9.2.8}Polynomial regression: \texttt {polynomial\_regression} \texttt {polynomial\_regression\_plot} }{674}{subsection.9.2.8}%
\contentsline {subsection}{\numberline {9.2.9}Logistic regression: \texttt {logistic\_regression} \texttt {logistic\_regression\_plot} }{675}{subsection.9.2.9}%
\contentsline {section}{\numberline {9.3}Random numbers}{677}{section.9.3}%
\contentsline {subsection}{\numberline {9.3.1}Producing uniformly distributed random numbers: \texttt {rand} \texttt {random} \texttt {alea} \texttt {hasard} \texttt {sample} }{677}{subsection.9.3.1}%
\contentsline {subsection}{\numberline {9.3.2}Initializing the random number generator: \texttt {srand} \texttt {randseed} \texttt {RandSeed} }{680}{subsection.9.3.2}%
\contentsline {subsection}{\numberline {9.3.3}Producing random numbers with the binomial distribution: \texttt {randbinomial} }{680}{subsection.9.3.3}%
\contentsline {subsection}{\numberline {9.3.4}Producing random numbers with a multinomial distribution: \texttt {randmultinomial} }{680}{subsection.9.3.4}%
\contentsline {subsection}{\numberline {9.3.5}Producing random numbers with a Poisson distribution: \texttt {randpoisson} }{681}{subsection.9.3.5}%
\contentsline {subsection}{\numberline {9.3.6}Producing random numbers with a normal distribution: \texttt {randnorm} \texttt {randNorm} }{682}{subsection.9.3.6}%
\contentsline {subsection}{\numberline {9.3.7}Producing random numbers with Student's distribution: \texttt {randstudent} }{682}{subsection.9.3.7}%
\contentsline {subsection}{\numberline {9.3.8}Producing random numbers with the $\chi ^{2}$ distribution: \texttt {randchisquare} }{682}{subsection.9.3.8}%
\contentsline {subsection}{\numberline {9.3.9}Producing random numbers with the Fisher-Sn\'{e}d\'{e}cor distribution: \texttt {randfisher} }{683}{subsection.9.3.9}%
\contentsline {subsection}{\numberline {9.3.10}Producing random numbers with the gamma distribution: \texttt {randgammad} }{683}{subsection.9.3.10}%
\contentsline {subsection}{\numberline {9.3.11}Producing random numbers with the beta distribution: \texttt {randbetad} }{684}{subsection.9.3.11}%
\contentsline {subsection}{\numberline {9.3.12}Producing random numbers with the geometric distribution: \texttt {randgeometric} }{684}{subsection.9.3.12}%
\contentsline {subsection}{\numberline {9.3.13}Producing random numbers with the exponential distribution: \texttt {randexp} }{684}{subsection.9.3.13}%
\contentsline {subsection}{\numberline {9.3.14}Random variables: \texttt {random\_variable} \texttt {randvar} }{685}{subsection.9.3.14}%
\contentsline {subsection}{\numberline {9.3.15}Make a random vector or list: \texttt {randvector} }{693}{subsection.9.3.15}%
\contentsline {subsection}{\numberline {9.3.16}Producing random matrices: \texttt {randmatrix} \texttt {ranm} \texttt {randMat} }{694}{subsection.9.3.16}%
\contentsline {section}{\numberline {9.4}Density and distribution functions}{696}{section.9.4}%
\contentsline {subsection}{\numberline {9.4.1}Distributions and inverse distributions}{696}{subsection.9.4.1}%
\contentsline {subsection}{\numberline {9.4.2}The uniform distribution }{696}{subsection.9.4.2}%
\contentsline {subsubsection}{The probability density function for the uniform distribution: \texttt {uniform} \texttt {uniformd} }{696}{section*.77}%
\contentsline {subsubsection}{The cumulative distribution function for the uniform distribution: \texttt {uniform\_cdf} \texttt {uniformd\_cdf} }{697}{section*.78}%
\contentsline {subsubsection}{The inverse distribution function for the uniform distribution: \texttt {uniform\_icdf} \texttt {uniformd\_icdf} }{698}{section*.79}%
\contentsline {subsection}{\numberline {9.4.3}The binomial distribution }{698}{subsection.9.4.3}%
\contentsline {subsubsection}{The probability density function for the binomial distribution: \texttt {binomial} }{698}{section*.80}%
\contentsline {subsubsection}{The cumulative distribution function for the binomial distribution: \texttt {binomial\_cdf} }{699}{section*.81}%
\contentsline {subsubsection}{The inverse distribution function for the binomial distribution: \texttt {binomial\_icdf} }{700}{section*.82}%
\contentsline {subsection}{\numberline {9.4.4}The negative binomial distribution }{700}{subsection.9.4.4}%
\contentsline {subsubsection}{The probability density function for the negative binomial distribution: \texttt {negbinomial} }{700}{section*.83}%
\contentsline {subsubsection}{The cumulative distribution function for the negative binomial distribution: \texttt {negbinomial\_cdf} }{701}{section*.84}%
\contentsline {subsubsection}{The inverse distribution function for the negative binomial distribution: \texttt {negbinomial\_icdf} }{702}{section*.85}%
\contentsline {subsection}{\numberline {9.4.5}The multinomial probability function: \texttt {multinomial} }{702}{subsection.9.4.5}%
\contentsline {subsection}{\numberline {9.4.6}The Poisson distribution }{703}{subsection.9.4.6}%
\contentsline {subsubsection}{The probability density function for the Poisson distribution: \texttt {poisson} }{703}{section*.86}%
\contentsline {subsubsection}{The cumulative distribution function for the Poisson distribution: \texttt {poisson\_cdf} }{703}{section*.87}%
\contentsline {subsubsection}{The inverse distribution function for the Poisson distribution: \texttt {poisson\_icdf} }{704}{section*.88}%
\contentsline {subsection}{\numberline {9.4.7}Normal distributions }{704}{subsection.9.4.7}%
\contentsline {subsubsection}{The probability density function for a normal distribution: \texttt {normald} \texttt {loi\_normal} }{704}{section*.89}%
\contentsline {subsubsection}{The cumulative distribution function for normal distributions: \texttt {normal\_cdf} \texttt {normald\_cdf} }{705}{section*.90}%
\contentsline {subsubsection}{The inverse distribution function for normal distributions: \texttt {normal\_icdf} \texttt {normald\_icdf} }{706}{section*.91}%
\contentsline {subsubsection}{The upper tail cumulative function for normal distributions: \texttt {UTPN} }{707}{section*.92}%
\contentsline {subsection}{\numberline {9.4.8}Student's distribution }{707}{subsection.9.4.8}%
\contentsline {subsubsection}{The probability density function for Student's distribution: \texttt {student} \texttt {studentd} }{707}{section*.93}%
\contentsline {subsubsection}{The cumulative distribution function for Student's distribution: \texttt {student\_cdf} }{708}{section*.94}%
\contentsline {subsubsection}{The inverse distribution function for Student's distribution: \texttt {student\_icdf} }{709}{section*.95}%
\contentsline {subsubsection}{The upper tail cumulative function for Student's distribution: \texttt {UTPT} }{709}{section*.96}%
\contentsline {subsection}{\numberline {9.4.9}The $\chi ^2$ distribution }{709}{subsection.9.4.9}%
\contentsline {subsubsection}{The probability density function for the $\chi ^2$ distribution: \texttt {chisquare} }{709}{section*.97}%
\contentsline {subsubsection}{The cumulative distribution function for the $\chi ^2$ distribution: \texttt {chisquare\_cdf} }{710}{section*.98}%
\contentsline {subsubsection}{The inverse distribution function for the $\chi ^2$ distribution: \texttt {chisquare\_icdf} }{711}{section*.99}%
\contentsline {subsubsection}{The upper tail cumulative function for the $\chi ^2$ distribution: \texttt {UTPC} }{711}{section*.100}%
\contentsline {subsection}{\numberline {9.4.10}The Fisher-Sn\'{e}d\'{e}cor distribution }{711}{subsection.9.4.10}%
\contentsline {subsubsection}{The probability density function for the Fisher-Sn\'{e}d\'{e}cor distribution: \texttt {fisher} \texttt {fisherd} \texttt {snedecor} \texttt {snedecord} }{711}{section*.101}%
\contentsline {subsubsection}{The cumulative distribution function for the Fisher-Sn\'{e}d\'{e}cor distribution: \texttt {fisher\_cdf} \texttt {snedecor\_cdf} }{712}{section*.102}%
\contentsline {subsubsection}{The inverse distribution function for the Fisher-Sn\'{e}d\'{e}cor distribution: \texttt {fisher\_icdf} \texttt {snedecor\_icdf} }{713}{section*.103}%
\contentsline {subsubsection}{The upper tail cumulative function for the Fisher-Sn\'{e}d\'{e}cor distribution: \texttt {UTPF} }{713}{section*.104}%
\contentsline {subsection}{\numberline {9.4.11}The gamma distribution }{713}{subsection.9.4.11}%
\contentsline {subsubsection}{The probability density function for the gamma distribution: \texttt {gammad} }{713}{section*.105}%
\contentsline {subsubsection}{The cumulative distribution function for the gamma distribution: \texttt {gammad\_cdf} }{714}{section*.106}%
\contentsline {subsubsection}{The inverse distribution function for the gamma distribution: \texttt {gammad\_icdf} }{715}{section*.107}%
\contentsline {subsection}{\numberline {9.4.12}The beta distribution }{715}{subsection.9.4.12}%
\contentsline {subsubsection}{The probability density function for the beta distribution: \texttt {betad} }{715}{section*.108}%
\contentsline {subsubsection}{The cumulative distribution function for the beta distribution: \texttt {betad\_cdf} }{716}{section*.109}%
\contentsline {subsubsection}{The inverse distribution function for the beta distribution: \texttt {betad\_icdf} }{716}{section*.110}%
\contentsline {subsection}{\numberline {9.4.13}The geometric distribution }{717}{subsection.9.4.13}%
\contentsline {subsubsection}{The probability density function for the geometric distribution: \texttt {geometric} }{717}{section*.111}%
\contentsline {subsubsection}{The cumulative distribution function of the geometric distribution: \texttt {geometric\_cdf} }{717}{section*.112}%
\contentsline {subsubsection}{The inverse distribution function for the geometric distribution: \texttt {geometric\_icdf} }{718}{section*.113}%
\contentsline {subsection}{\numberline {9.4.14}The Cauchy distribution }{718}{subsection.9.4.14}%
\contentsline {subsubsection}{The probability density function for the Cauchy distribution: \texttt {cauchy} \texttt {cauchyd} }{718}{section*.114}%
\contentsline {subsubsection}{The cumulative distribution function for the Cauchy distribution: \texttt {cauchy\_cdf} \texttt {cauchyd\_cdf} }{719}{section*.115}%
\contentsline {subsubsection}{The inverse distribution function for the Cauchy distribution: \texttt {cauchy\_icdf} \texttt {cauchyd\_icdf} }{720}{section*.116}%
\contentsline {subsection}{\numberline {9.4.15}The exponential distribution }{720}{subsection.9.4.15}%
\contentsline {subsubsection}{The probability density function for the exponential distribution: \texttt {exponential} \texttt {exponentiald} }{720}{section*.117}%
\contentsline {subsubsection}{The cumulative distribution function for the exponential distribution: \texttt {exponential\_cdf} \texttt {exponentiald\_cdf} }{721}{section*.118}%
\contentsline {subsubsection}{The inverse distribution function for the exponential distribution: \texttt {exponential\_icdf} \texttt {exponentiald\_icdf} }{721}{section*.119}%
\contentsline {subsection}{\numberline {9.4.16}The Weibull distribution }{722}{subsection.9.4.16}%
\contentsline {subsubsection}{The probability density function for the Weibull distribution: \texttt {weibull} \texttt {weibulld} }{722}{section*.120}%
\contentsline {subsubsection}{The cumulative distribution function for the Weibull distribution: \texttt {weibull\_cdf} \texttt {weibulld\_cdf} }{722}{section*.121}%
\contentsline {subsubsection}{The inverse distribution function for the Weibull distribution: \texttt {weibull\_icdf} \texttt {weibulld\_icdf} }{723}{section*.122}%
\contentsline {subsection}{\numberline {9.4.17}The Kolmogorov-Smirnov distribution: \texttt {kolmogorovd} }{724}{subsection.9.4.17}%
\contentsline {subsection}{\numberline {9.4.18}The Wilcoxon or Mann-Whitney distribution }{724}{subsection.9.4.18}%
\contentsline {subsubsection}{The Wilcoxon test polynomial: \texttt {wilcoxonp}}{724}{section*.123}%
\contentsline {subsubsection}{The Wilcoxon/Mann-Whitney statistic: \texttt {wilcoxons} }{725}{section*.124}%
\contentsline {subsubsection}{The Wilcoxon or Mann-Whitney test: \texttt {wilcoxont} }{725}{section*.125}%
\contentsline {subsection}{\numberline {9.4.19}Moment generating functions for probability distributions: \texttt {mgf} }{726}{subsection.9.4.19}%
\contentsline {subsection}{\numberline {9.4.20}Cumulative distribution functions: \texttt {cdf} }{727}{subsection.9.4.20}%
\contentsline {subsection}{\numberline {9.4.21}Inverse distribution functions: \texttt {icdf}}{728}{subsection.9.4.21}%
\contentsline {subsection}{\numberline {9.4.22}Kernel density estimation: \texttt {kernel\_density} \texttt {kde} }{728}{subsection.9.4.22}%
\contentsline {subsection}{\numberline {9.4.23}Distribution fitting by maximum likelihood: \texttt {fitdistr} }{731}{subsection.9.4.23}%
\contentsline {subsection}{\numberline {9.4.24}Markov chains: \texttt {markov}}{733}{subsection.9.4.24}%
\contentsline {subsection}{\numberline {9.4.25}Generating a random walks: \texttt {randmarkov} }{733}{subsection.9.4.25}%
\contentsline {section}{\numberline {9.5}Hypothesis testing}{734}{section.9.5}%
\contentsline {subsection}{\numberline {9.5.1}General}{734}{subsection.9.5.1}%
\contentsline {subsection}{\numberline {9.5.2}Testing the mean with the Z test: \texttt {normalt} }{735}{subsection.9.5.2}%
\contentsline {subsection}{\numberline {9.5.3}Testing the mean with the T test: \texttt {studentt} }{736}{subsection.9.5.3}%
\contentsline {subsection}{\numberline {9.5.4}Testing a distribution with the $\chi ^2$ distribution: \texttt {chisquaret}}{737}{subsection.9.5.4}%
\contentsline {subsection}{\numberline {9.5.5}Testing a distribution with the Kolmogorov-Smirnov distribution: \texttt {kolmogorovt} }{739}{subsection.9.5.5}%
\contentsline {chapter}{\numberline {10}Numerical computations}{741}{chapter.10}%
\contentsline {section}{\numberline {10.1}Floating point representation.}{741}{section.10.1}%
\contentsline {subsection}{\numberline {10.1.1}Digits}{741}{subsection.10.1.1}%
\contentsline {subsection}{\numberline {10.1.2}Representation by hardware floats}{742}{subsection.10.1.2}%
\contentsline {subsection}{\numberline {10.1.3}Examples of representations of normalized floats}{742}{subsection.10.1.3}%
\contentsline {subsection}{\numberline {10.1.4}Difference between the representation of (3.1-3) and of 0.1}{743}{subsection.10.1.4}%
\contentsline {section}{\numberline {10.2}Approx. evaluation: \texttt {evalf} \texttt {approx} \texttt {Digits} }{744}{section.10.2}%
\contentsline {section}{\numberline {10.3}Numerical algorithms}{745}{section.10.3}%
\contentsline {subsection}{\numberline {10.3.1}Approximate solution of an equation: \texttt {newton} }{745}{subsection.10.3.1}%
\contentsline {subsection}{\numberline {10.3.2}Approximate computation of the derivative number: \texttt {nDeriv} }{745}{subsection.10.3.2}%
\contentsline {subsection}{\numberline {10.3.3}Approximate computation of integrals: \texttt {romberg} \texttt {nInt} }{746}{subsection.10.3.3}%
\contentsline {subsection}{\numberline {10.3.4}Approximate integral with an adaptive Gaussian quadrature at 15 points: \texttt {gaussquad} }{747}{subsection.10.3.4}%
\contentsline {subsection}{\numberline {10.3.5}Approximate solution of y'=f(t,y): \texttt {odesolve} }{748}{subsection.10.3.5}%
\contentsline {subsection}{\numberline {10.3.6}Approximate solution of the system v'=f(t,v): \texttt {odesolve} }{749}{subsection.10.3.6}%
\contentsline {subsection}{\numberline {10.3.7}Approximate solution of a nonlinear second-order boundary value problem: \texttt {bvpsolve} }{751}{subsection.10.3.7}%
\contentsline {section}{\numberline {10.4}Solve equations with \texttt {fsolve} \texttt {nSolve} \texttt {cfsolve} }{753}{section.10.4}%
\contentsline {subsection}{\numberline {10.4.1}\texttt {fsolve} with the option \texttt {bisection\_solver} }{755}{subsection.10.4.1}%
\contentsline {subsection}{\numberline {10.4.2}\texttt {fsolve} with the option \texttt {brent\_solver} }{755}{subsection.10.4.2}%
\contentsline {subsection}{\numberline {10.4.3}\texttt {fsolve} with the option \texttt {falsepos\_solver} }{755}{subsection.10.4.3}%
\contentsline {subsection}{\numberline {10.4.4}\texttt {fsolve} with the option \texttt {newton\_solver} }{755}{subsection.10.4.4}%
\contentsline {subsection}{\numberline {10.4.5}\texttt {fsolve} with the option \texttt {secant\_solver} }{756}{subsection.10.4.5}%
\contentsline {subsection}{\numberline {10.4.6}\texttt {fsolve} with the option \texttt {steffenson\_solver} }{756}{subsection.10.4.6}%
\contentsline {section}{\numberline {10.5}Solve systems with \texttt {fsolve} and \texttt {cfsolve}}{757}{section.10.5}%
\contentsline {subsection}{\numberline {10.5.1}\texttt {fsolve} with the option \texttt {dnewton\_solver} }{758}{subsection.10.5.1}%
\contentsline {subsection}{\numberline {10.5.2}\texttt {fsolve} with the option \texttt {hybrid\_solver} }{758}{subsection.10.5.2}%
\contentsline {subsection}{\numberline {10.5.3}\texttt {fsolve} with the option \texttt {hybrids\_solver} }{759}{subsection.10.5.3}%
\contentsline {subsection}{\numberline {10.5.4}\texttt {fsolve} with the option \texttt {newtonj\_solver} }{759}{subsection.10.5.4}%
\contentsline {subsection}{\numberline {10.5.5}\texttt {fsolve} with the option \texttt {hybridj\_solver} }{759}{subsection.10.5.5}%
\contentsline {subsection}{\numberline {10.5.6}\texttt {fsolve} with the option \texttt {hybridsj\_solver} }{759}{subsection.10.5.6}%
\contentsline {section}{\numberline {10.6}Numeric roots of a polynomial: \texttt {proot} }{759}{section.10.6}%
\contentsline {section}{\numberline {10.7}Numeric factorization of a matrix: \texttt {cholesky} \texttt {qr} \texttt {lu} \texttt {svd}}{760}{section.10.7}%
\contentsline {chapter}{\numberline {11}Unit objects and physical constants}{761}{chapter.11}%
\contentsline {section}{\numberline {11.1}Unit objects}{761}{section.11.1}%
\contentsline {subsection}{\numberline {11.1.1}Notation of unit objects }{761}{subsection.11.1.1}%
\contentsline {subsection}{\numberline {11.1.2}Computing with units}{763}{subsection.11.1.2}%
\contentsline {subsection}{\numberline {11.1.3}Convert units into MKSA units: \texttt {mksa} }{764}{subsection.11.1.3}%
\contentsline {subsection}{\numberline {11.1.4}Convert units: \texttt {convert} \texttt {=>} }{764}{subsection.11.1.4}%
\contentsline {subsection}{\numberline {11.1.5}Convert between Celsius and Fahrenheit: \texttt {Celsius2Fahrenheit} \texttt {Fahrenheit2Celsius} }{765}{subsection.11.1.5}%
\contentsline {subsection}{\numberline {11.1.6}Factorize a unit: \texttt {ufactor} }{766}{subsection.11.1.6}%
\contentsline {subsection}{\numberline {11.1.7}Simplify a unit: \texttt {usimplify} }{766}{subsection.11.1.7}%
\contentsline {section}{\numberline {11.2}Constants}{767}{section.11.2}%
\contentsline {subsection}{\numberline {11.2.1}Notation of physical constants }{767}{subsection.11.2.1}%
\contentsline {chapter}{\numberline {12}Programming }{769}{chapter.12}%
\contentsline {section}{\numberline {12.1}Functions, programs and scripts}{769}{section.12.1}%
\contentsline {subsection}{\numberline {12.1.1}The program editor}{769}{subsection.12.1.1}%
\contentsline {subsection}{\numberline {12.1.2}Functions: \texttt {function} \texttt {endfunction} \texttt {\{} \texttt {\}} \texttt {local} \texttt {return} }{769}{subsection.12.1.2}%
\contentsline {subsection}{\numberline {12.1.3}Local variables}{770}{subsection.12.1.3}%
\contentsline {subsection}{\numberline {12.1.4}Default values of the parameters}{771}{subsection.12.1.4}%
\contentsline {subsection}{\numberline {12.1.5}Programs}{771}{subsection.12.1.5}%
\contentsline {subsection}{\numberline {12.1.6}Scripts}{772}{subsection.12.1.6}%
\contentsline {subsection}{\numberline {12.1.7}Code blocks }{772}{subsection.12.1.7}%
\contentsline {section}{\numberline {12.2}Basic instructions}{772}{section.12.2}%
\contentsline {subsection}{\numberline {12.2.1}Comments: \texttt {//} }{772}{subsection.12.2.1}%
\contentsline {subsection}{\numberline {12.2.2}Input: \texttt {input} \texttt {Input} \texttt {InputStr} \texttt {textinput} \texttt {output} \texttt {Output} }{772}{subsection.12.2.2}%
\contentsline {subsection}{\numberline {12.2.3}Reading a single keystroke: \texttt {getKey} }{774}{subsection.12.2.3}%
\contentsline {subsection}{\numberline {12.2.4}Checking conditions: \texttt {assert} }{774}{subsection.12.2.4}%
\contentsline {subsection}{\numberline {12.2.5}Checking the type of the argument: \texttt {type} \texttt {subtype} \texttt {compare} \texttt {getType} }{775}{subsection.12.2.5}%
\contentsline {subsection}{\numberline {12.2.6}Printing: \texttt {print} \texttt {Disp} \texttt {ClrIO} }{777}{subsection.12.2.6}%
\contentsline {subsection}{\numberline {12.2.7}Displaying exponents: \texttt {printpow} }{778}{subsection.12.2.7}%
\contentsline {subsection}{\numberline {12.2.8}Infixed assignments: \texttt {=>} \texttt {:=} \texttt {=<} }{779}{subsection.12.2.8}%
\contentsline {subsection}{\numberline {12.2.9}Assignment by copying: \texttt {copy} }{780}{subsection.12.2.9}%
\contentsline {subsection}{\numberline {12.2.10}The difference between \texttt {:=} and \texttt {=<}}{781}{subsection.12.2.10}%
\contentsline {section}{\numberline {12.3}Control structures}{782}{section.12.3}%
\contentsline {subsection}{\numberline {12.3.1}\texttt {if} statements: \texttt {if} \texttt {then} \texttt {else} \texttt {end} \texttt {elif} }{782}{subsection.12.3.1}%
\contentsline {subsection}{\numberline {12.3.2}The switch statement: \texttt {switch} \texttt {case} \texttt {default} }{784}{subsection.12.3.2}%
\contentsline {subsection}{\numberline {12.3.3}The for loop: \texttt {for} \texttt {from} \texttt {to} \texttt {step} \texttt {do} \texttt {end\_for} }{785}{subsection.12.3.3}%
\contentsline {paragraph}{The first form:}{785}{section*.126}%
\contentsline {paragraph}{The second form:}{785}{section*.127}%
\contentsline {paragraph}{The third form:}{786}{section*.128}%
\contentsline {subsection}{\numberline {12.3.4}The repeat loop: \texttt {repeat} \texttt {until} }{787}{subsection.12.3.4}%
\contentsline {subsection}{\numberline {12.3.5}The while loop: \texttt {while} }{787}{subsection.12.3.5}%
\contentsline {subsection}{\numberline {12.3.6}Breaking out a loop: \texttt {break} }{788}{subsection.12.3.6}%
\contentsline {subsection}{\numberline {12.3.7}Going to the next iteration of a loop: \texttt {continue}}{788}{subsection.12.3.7}%
\contentsline {subsection}{\numberline {12.3.8}Changing the order of execution: \texttt {goto} \texttt {label} }{789}{subsection.12.3.8}%
\contentsline {section}{\numberline {12.4}Other useful instructions}{789}{section.12.4}%
\contentsline {subsection}{\numberline {12.4.1}Define a function with a variable number of arguments: \texttt {args} }{789}{subsection.12.4.1}%
\contentsline {subsection}{\numberline {12.4.2}Assignments in a program}{790}{subsection.12.4.2}%
\contentsline {subsection}{\numberline {12.4.3}Writing variable values to a file: \texttt {write} }{791}{subsection.12.4.3}%
\contentsline {subsection}{\numberline {12.4.4}Writing output to a file: \texttt {fopen} \texttt {fclose} \texttt {fprint} }{792}{subsection.12.4.4}%
\contentsline {subsection}{\numberline {12.4.5}Using strings as names: \texttt {\#} }{793}{subsection.12.4.5}%
\contentsline {subsection}{\numberline {12.4.6}Using strings as commands: \texttt {expr} }{794}{subsection.12.4.6}%
\contentsline {subsection}{\numberline {12.4.7}Converting an expression to a string: \texttt {string} }{796}{subsection.12.4.7}%
\contentsline {subsection}{\numberline {12.4.8}Converting a real number into a string: \texttt {format} }{796}{subsection.12.4.8}%
\contentsline {subsection}{\numberline {12.4.9}Working with the graphics screen: \texttt {DispG} \texttt {DispHome} \texttt {ClrGraph} \texttt {ClrDraw} }{797}{subsection.12.4.9}%
\contentsline {subsection}{\numberline {12.4.10}Pausing a program: \texttt {Pause} \texttt {WAIT} }{798}{subsection.12.4.10}%
\contentsline {subsection}{\numberline {12.4.11}Dealing with errors: \texttt {try} \texttt {catch} \texttt {throw} \texttt {error} \texttt {ERROR} }{799}{subsection.12.4.11}%
\contentsline {section}{\numberline {12.5}Debugging}{801}{section.12.5}%
\contentsline {subsection}{\numberline {12.5.1}Starting the debugger: \texttt {debug} \texttt {sst} \texttt {in} \texttt {sst\_in} \texttt {cont} \texttt {kill} \texttt {break} \texttt {breakpoint} \texttt {halt} \texttt {rmbrk} \texttt {rmbreakpoint} \texttt {watch} \texttt {rmwtch} }{801}{subsection.12.5.1}%
\contentsline {chapter}{\numberline {13}Two-dimensional Graphics}{805}{chapter.13}%
\contentsline {section}{\numberline {13.1}Introduction}{805}{section.13.1}%
\contentsline {subsection}{\numberline {13.1.1}Points, vectors and complex numbers}{805}{subsection.13.1.1}%
\contentsline {section}{\numberline {13.2}Basic commands}{806}{section.13.2}%
\contentsline {subsection}{\numberline {13.2.1}Clear the \texttt {DispG} screen: \texttt {erase} }{806}{subsection.13.2.1}%
\contentsline {subsection}{\numberline {13.2.2}Toggle the axes: \texttt {switch\_axes} }{806}{subsection.13.2.2}%
\contentsline {subsection}{\numberline {13.2.3}Draw unit vectors in the plane: \texttt {Ox\_2d\_unit\_vector} \texttt {Oy\_2d\_unit\_vector} \texttt {frame\_2d} }{807}{subsection.13.2.3}%
\contentsline {subsection}{\numberline {13.2.4}Draw dotted paper: \texttt {dot\_paper} }{807}{subsection.13.2.4}%
\contentsline {subsection}{\numberline {13.2.5}Draw lined paper: \texttt {line\_paper}}{808}{subsection.13.2.5}%
\contentsline {subsection}{\numberline {13.2.6}Draw grid paper: \texttt {grid\_paper} }{809}{subsection.13.2.6}%
\contentsline {subsection}{\numberline {13.2.7}Draw triangular paper: \texttt {triangle\_paper} }{809}{subsection.13.2.7}%
\contentsline {section}{\numberline {13.3}Display features of graphics}{810}{section.13.3}%
\contentsline {subsection}{\numberline {13.3.1}Graphic features}{810}{subsection.13.3.1}%
\contentsline {subsection}{\numberline {13.3.2}Parameters for changing features}{810}{subsection.13.3.2}%
\contentsline {subsubsection}{Parameters for local features }{811}{section*.129}%
\contentsline {subsubsection}{Parameters for global features}{813}{section*.130}%
\contentsline {subsection}{\numberline {13.3.3}Commands for global display features}{815}{subsection.13.3.3}%
\contentsline {subsubsection}{Add a legend: \texttt {legend} }{815}{section*.131}%
\contentsline {subsubsection}{Change various features:\texttt {display} \texttt {color} }{815}{section*.132}%
\contentsline {section}{\numberline {13.4}Define geometric objects without drawing them: \texttt {nodisp}}{817}{section.13.4}%
\contentsline {section}{\numberline {13.5}Geometric demonstrations: \texttt {assume}}{819}{section.13.5}%
\contentsline {section}{\numberline {13.6}Points in the plane}{820}{section.13.6}%
\contentsline {subsection}{\numberline {13.6.1}Points and complex numbers}{820}{subsection.13.6.1}%
\contentsline {subsection}{\numberline {13.6.2}The point in the plane: \texttt {point} }{820}{subsection.13.6.2}%
\contentsline {subsection}{\numberline {13.6.3}The difference and sum of two points in the plane:\texttt {+} \texttt {-} }{822}{subsection.13.6.3}%
\contentsline {subsection}{\numberline {13.6.4}Define random points in the plane: \texttt {point2d} }{823}{subsection.13.6.4}%
\contentsline {subsection}{\numberline {13.6.5}Points in polar coordinates: \texttt {polar\_point} \texttt {point\_polar} }{823}{subsection.13.6.5}%
\contentsline {subsection}{\numberline {13.6.6}Find a point of intersection of two objects in the plane: \texttt {single\_inter} \texttt {line\_inter} }{825}{subsection.13.6.6}%
\contentsline {subsection}{\numberline {13.6.7}Find the points of intersection of two geometric objects in the plane: \texttt {inter} }{826}{subsection.13.6.7}%
\contentsline {subsection}{\numberline {13.6.8}Find the orthocenter of a triangle in the plane: \texttt {orthocenter} }{827}{subsection.13.6.8}%
\contentsline {subsection}{\numberline {13.6.9}Find the midpoint of a segment in the plane: \texttt {midpoint} }{828}{subsection.13.6.9}%
\contentsline {subsection}{\numberline {13.6.10}The barycenter in the plane: \texttt {barycenter} }{828}{subsection.13.6.10}%
\contentsline {subsection}{\numberline {13.6.11}The isobarycenter of $n$ points in the plane: \texttt {isobarycenter} }{829}{subsection.13.6.11}%
\contentsline {subsection}{\numberline {13.6.12}The center of a circle in the plane: \texttt {center} }{829}{subsection.13.6.12}%
\contentsline {subsection}{\numberline {13.6.13}The vertices of a polygon in the plane: \texttt {vertices} \texttt {vertices\_abc} }{830}{subsection.13.6.13}%
\contentsline {subsection}{\numberline {13.6.14}The vertices of a polygon in the plane, closed: \texttt {vertices\_abca} }{831}{subsection.13.6.14}%
\contentsline {subsection}{\numberline {13.6.15}A point on a geometric object in the plane: \texttt {element} }{831}{subsection.13.6.15}%
\contentsline {section}{\numberline {13.7}Lines in plane geometry}{832}{section.13.7}%
\contentsline {subsection}{\numberline {13.7.1}Lines and directed lines in the plane: \texttt {line} }{832}{subsection.13.7.1}%
\contentsline {subsection}{\numberline {13.7.2}Half-lines in the plane: \texttt {half\_line} }{834}{subsection.13.7.2}%
\contentsline {subsection}{\numberline {13.7.3}Line segments in the plane: \texttt {segment} \texttt {Line} }{834}{subsection.13.7.3}%
\contentsline {subsection}{\numberline {13.7.4}Vectors in the plane: \texttt {segment} \texttt {vector} }{835}{subsection.13.7.4}%
\contentsline {subsection}{\numberline {13.7.5}Parallel lines in the plane: \texttt {parallel} }{836}{subsection.13.7.5}%
\contentsline {subsection}{\numberline {13.7.6}Perpendicular lines in the plane: \texttt {perpendicular} }{837}{subsection.13.7.6}%
\contentsline {subsection}{\numberline {13.7.7}Tangents to curves in the plane: \texttt {tangent} }{838}{subsection.13.7.7}%
\contentsline {subsection}{\numberline {13.7.8}The median of a triangle in the plane: \texttt {median\_line} }{839}{subsection.13.7.8}%
\contentsline {subsection}{\numberline {13.7.9}The altitude of a triangle: \texttt {altitude} }{839}{subsection.13.7.9}%
\contentsline {subsection}{\numberline {13.7.10}The perpendicular bisector of a segment in the plane: \texttt {perpen\_bisector} }{840}{subsection.13.7.10}%
\contentsline {subsection}{\numberline {13.7.11}The angle bisector: \texttt {bisector} }{840}{subsection.13.7.11}%
\contentsline {subsection}{\numberline {13.7.12}The exterior angle bisector: \texttt {exbisector} }{841}{subsection.13.7.12}%
\contentsline {section}{\numberline {13.8}Triangles in the plane}{841}{section.13.8}%
\contentsline {subsection}{\numberline {13.8.1}Arbitrary triangles in the plane: \texttt {triangle} }{841}{subsection.13.8.1}%
\contentsline {subsection}{\numberline {13.8.2}Isosceles triangles in the plane: \texttt {isosceles\_triangle} }{842}{subsection.13.8.2}%
\contentsline {subsection}{\numberline {13.8.3}Right triangles in the plane: \texttt {right\_triangle} }{843}{subsection.13.8.3}%
\contentsline {subsection}{\numberline {13.8.4}Equilateral triangles in the plane: \texttt {equilateral\_triangle} }{844}{subsection.13.8.4}%
\contentsline {section}{\numberline {13.9}Quadrilaterals in the plane}{845}{section.13.9}%
\contentsline {subsection}{\numberline {13.9.1}Squares in the plane: \texttt {square} }{846}{subsection.13.9.1}%
\contentsline {subsection}{\numberline {13.9.2}Rhombuses in the plane: \texttt {rhombus} }{847}{subsection.13.9.2}%
\contentsline {subsection}{\numberline {13.9.3}Rectangles in the plane: \texttt {rectangle} }{848}{subsection.13.9.3}%
\contentsline {subsection}{\numberline {13.9.4}Parallelograms in the plane: \texttt {parallelogram} }{849}{subsection.13.9.4}%
\contentsline {subsection}{\numberline {13.9.5}Arbitrary quadrilaterals in the plane: \texttt {quadrilateral} }{850}{subsection.13.9.5}%
\contentsline {section}{\numberline {13.10}Other polygons in the plane}{851}{section.13.10}%
\contentsline {subsection}{\numberline {13.10.1}Regular hexagons in the plane: \texttt {hexagon} }{851}{subsection.13.10.1}%
\contentsline {subsection}{\numberline {13.10.2}Regular polygons in the plane: \texttt {isopolygon} }{852}{subsection.13.10.2}%
\contentsline {subsection}{\numberline {13.10.3}General polygons in the plane: \texttt {polygon} }{853}{subsection.13.10.3}%
\contentsline {subsection}{\numberline {13.10.4}Polygonal lines in the plane: \texttt {open\_polygon} }{854}{subsection.13.10.4}%
\contentsline {subsection}{\numberline {13.10.5}Convex hulls: \texttt {convexhull} }{855}{subsection.13.10.5}%
\contentsline {section}{\numberline {13.11}Circles}{855}{section.13.11}%
\contentsline {subsection}{\numberline {13.11.1}Circles and arcs in the plane: \texttt {circle} }{855}{subsection.13.11.1}%
\contentsline {subsection}{\numberline {13.11.2}Circular arcs: \texttt {arc} }{858}{subsection.13.11.2}%
\contentsline {subsection}{\numberline {13.11.3}Circles (TI compatibility): \texttt {Circle}}{859}{subsection.13.11.3}%
\contentsline {subsection}{\numberline {13.11.4}Inscribed circles: \texttt {incircle} }{859}{subsection.13.11.4}%
\contentsline {subsection}{\numberline {13.11.5}Circumscribed circles: \texttt {circumcircle} }{860}{subsection.13.11.5}%
\contentsline {subsection}{\numberline {13.11.6}Excircles: \texttt {excircle} }{860}{subsection.13.11.6}%
\contentsline {subsection}{\numberline {13.11.7}The power of a point relative to a circle: \texttt {powerpc} }{861}{subsection.13.11.7}%
\contentsline {subsection}{\numberline {13.11.8}The radical axis of two circles: \texttt {radical\_axis} }{861}{subsection.13.11.8}%
\contentsline {section}{\numberline {13.12}Other conic sections}{862}{section.13.12}%
\contentsline {subsection}{\numberline {13.12.1}The ellipse in the plane: \texttt {ellipse} }{862}{subsection.13.12.1}%
\contentsline {subsection}{\numberline {13.12.2}The hyperbola in the plane: \texttt {hyperbola} }{863}{subsection.13.12.2}%
\contentsline {subsection}{\numberline {13.12.3}The parabola in the plane: \texttt {parabola} }{864}{subsection.13.12.3}%
\contentsline {section}{\numberline {13.13}Coordinates in the plane}{866}{section.13.13}%
\contentsline {subsection}{\numberline {13.13.1}The affix of a point or vector: \texttt {affix} }{866}{subsection.13.13.1}%
\contentsline {subsection}{\numberline {13.13.2}The abscissa of a point or vector in the plane: \texttt {abscissa} }{867}{subsection.13.13.2}%
\contentsline {subsection}{\numberline {13.13.3}The ordinate of a point or vector in the plane: \texttt {ordinate} }{867}{subsection.13.13.3}%
\contentsline {subsection}{\numberline {13.13.4}The coordinates of a point, vector or line in the plane: \texttt {coordinates} }{868}{subsection.13.13.4}%
\contentsline {subsection}{\numberline {13.13.5}The rectangular coordinates of a point: \texttt {rectangular\_coordinates} }{870}{subsection.13.13.5}%
\contentsline {subsection}{\numberline {13.13.6}The polar coordinates of a point: \texttt {polar\_coordinates} }{871}{subsection.13.13.6}%
\contentsline {subsection}{\numberline {13.13.7}The Cartesian equation of a geometric object in the plane: \texttt {equation} }{871}{subsection.13.13.7}%
\contentsline {subsection}{\numberline {13.13.8}The parametric equation of a geometric object in the plane: \texttt {parameq} }{872}{subsection.13.13.8}%
\contentsline {section}{\numberline {13.14}Measurements}{872}{section.13.14}%
\contentsline {subsection}{\numberline {13.14.1}Measurement and display: \texttt {distanceat} \texttt {distanceatraw} \texttt {angleat} \texttt {angleatraw} \texttt {areaat} \texttt {areaatraw} \texttt {perimeterat} \texttt {perimeteratraw} \texttt {slopeat} \texttt {slopeatraw} \texttt {extract\_measure} }{872}{subsection.13.14.1}%
\contentsline {subsection}{\numberline {13.14.2}The distance between objects in the plane: \texttt {distance} }{874}{subsection.13.14.2}%
\contentsline {subsection}{\numberline {13.14.3}The length squared of a segment in the plane: \texttt {distance2} }{876}{subsection.13.14.3}%
\contentsline {subsection}{\numberline {13.14.4}The measure of an angle in the plane: \texttt {angle} }{876}{subsection.13.14.4}%
\contentsline {subsection}{\numberline {13.14.5}The graphical representation of the area of a polygon: \texttt {plotarea} \texttt {areaplot} }{878}{subsection.13.14.5}%
\contentsline {subsection}{\numberline {13.14.6}The area of a polygon: \texttt {area} }{879}{subsection.13.14.6}%
\contentsline {subsection}{\numberline {13.14.7}The perimeter of a polygon: \texttt {perimeter} }{879}{subsection.13.14.7}%
\contentsline {subsection}{\numberline {13.14.8}The slope of a line: \texttt {slope} }{881}{subsection.13.14.8}%
\contentsline {subsection}{\numberline {13.14.9}The radius of a circle: \texttt {radius} }{882}{subsection.13.14.9}%
\contentsline {subsection}{\numberline {13.14.10}The length of a vector: \texttt {abs} }{882}{subsection.13.14.10}%
\contentsline {subsection}{\numberline {13.14.11}The angle of a vector: \texttt {arg} }{882}{subsection.13.14.11}%
\contentsline {subsection}{\numberline {13.14.12}Normalize a complex number: \texttt {normalize} }{883}{subsection.13.14.12}%
\contentsline {section}{\numberline {13.15}Transformations}{883}{section.13.15}%
\contentsline {subsection}{\numberline {13.15.1}General remarks}{883}{subsection.13.15.1}%
\contentsline {subsection}{\numberline {13.15.2}Translations in the plane: \texttt {translation} }{883}{subsection.13.15.2}%
\contentsline {subsection}{\numberline {13.15.3}Reflections in the plane: \texttt {reflection} }{884}{subsection.13.15.3}%
\contentsline {subsection}{\numberline {13.15.4}Rotation in the plane: \texttt {rotation} }{885}{subsection.13.15.4}%
\contentsline {subsection}{\numberline {13.15.5}Homothety in the plane: \texttt {homothety} }{887}{subsection.13.15.5}%
\contentsline {subsection}{\numberline {13.15.6}Similarity in the plane: \texttt {similarity} }{888}{subsection.13.15.6}%
\contentsline {subsection}{\numberline {13.15.7}Inversion in the plane: \texttt {inversion} }{890}{subsection.13.15.7}%
\contentsline {subsection}{\numberline {13.15.8}Orthogonal projection in the plane: \texttt {projection} }{891}{subsection.13.15.8}%
\contentsline {section}{\numberline {13.16}Properties}{893}{section.13.16}%
\contentsline {subsection}{\numberline {13.16.1}Check if a point is on an object in the plane: \texttt {is\_element} }{893}{subsection.13.16.1}%
\contentsline {subsection}{\numberline {13.16.2}Check if three points are collinear in the plane: \texttt {is\_collinear} }{893}{subsection.13.16.2}%
\contentsline {subsection}{\numberline {13.16.3}Check if four points are concyclic in the plane: \texttt {is\_concyclic} }{894}{subsection.13.16.3}%
\contentsline {subsection}{\numberline {13.16.4}Check if a point is in a polygon or circle: \texttt {is\_inside} }{895}{subsection.13.16.4}%
\contentsline {subsection}{\numberline {13.16.5}Check if an object is an equilateral triangle in the plane: \texttt {is\_equilateral} }{895}{subsection.13.16.5}%
\contentsline {subsection}{\numberline {13.16.6}Check if an object in the plane is an isosceles triangle: \texttt {is\_isosceles} }{896}{subsection.13.16.6}%
\contentsline {subsection}{\numberline {13.16.7}Check if an object in the plane is a right triangle or a rectangle: \texttt {is\_rectangle} }{898}{subsection.13.16.7}%
\contentsline {subsection}{\numberline {13.16.8}Check if an object in the plane is a square: \texttt {is\_square} }{899}{subsection.13.16.8}%
\contentsline {subsection}{\numberline {13.16.9}Check if an object in the plane is a rhombus: \texttt {is\_rhombus} }{900}{subsection.13.16.9}%
\contentsline {subsection}{\numberline {13.16.10}Check if an object in the plane is a parallelogram: \texttt {is\_parallelogram} }{901}{subsection.13.16.10}%
\contentsline {subsection}{\numberline {13.16.11}Check it two lines in the plane are parallel: \texttt {is\_parallel} }{902}{subsection.13.16.11}%
\contentsline {subsection}{\numberline {13.16.12}Check if two lines in the plane are perpendicular: \texttt {is\_perpendicular} }{902}{subsection.13.16.12}%
\contentsline {subsection}{\numberline {13.16.13}Check if two circles in the plane are orthogonal: \texttt {is\_orthogonal} }{903}{subsection.13.16.13}%
\contentsline {subsection}{\numberline {13.16.14}Check if elements are conjugates: \texttt {is\_conjugate} }{904}{subsection.13.16.14}%
\contentsline {subsection}{\numberline {13.16.15}Check if four points form a harmonic division: \texttt {is\_harmonic} }{905}{subsection.13.16.15}%
\contentsline {subsection}{\numberline {13.16.16}Check if lines are in a bundle: \texttt {is\_harmonic\_line\_bundle} }{905}{subsection.13.16.16}%
\contentsline {subsection}{\numberline {13.16.17}Check if circles are in a bundle: \texttt {is\_harmonic\_circle\_bundle} }{906}{subsection.13.16.17}%
\contentsline {section}{\numberline {13.17}Harmonic division}{906}{section.13.17}%
\contentsline {subsection}{\numberline {13.17.1}Find a point dividing a segment in the harminic ratio $k$: \texttt {division\_point} }{906}{subsection.13.17.1}%
\contentsline {subsection}{\numberline {13.17.2}The cross ratio of four collinear points: \texttt {cross\_ratio} }{907}{subsection.13.17.2}%
\contentsline {subsection}{\numberline {13.17.3}Harmonic division: \texttt {harmonic\_division} }{908}{subsection.13.17.3}%
\contentsline {subsection}{\numberline {13.17.4}The harmonic conjugate: \texttt {harmonic\_conjugate} }{909}{subsection.13.17.4}%
\contentsline {subsection}{\numberline {13.17.5}Pole and polar: \texttt {pole} \texttt {polar} }{910}{subsection.13.17.5}%
\contentsline {subsection}{\numberline {13.17.6}The polar reciprocal: \texttt {reciprocation} }{911}{subsection.13.17.6}%
\contentsline {section}{\numberline {13.18}Loci and envelopes}{912}{section.13.18}%
\contentsline {subsection}{\numberline {13.18.1}Loci: \texttt {locus} }{912}{subsection.13.18.1}%
\contentsline {subsubsection}{The locus of points depending on points on a curve.}{912}{section*.133}%
\contentsline {subsubsection}{The envelope of a family of lines which depend on points on a curve.}{913}{section*.134}%
\contentsline {subsection}{\numberline {13.18.2}Envelopes: \texttt {envelope} }{915}{subsection.13.18.2}%
\contentsline {subsection}{\numberline {13.18.3}The trace of a geometric object: \texttt {trace}}{916}{subsection.13.18.3}%
\contentsline {chapter}{\numberline {14}Three-dimensional Graphics}{917}{chapter.14}%
\contentsline {section}{\numberline {14.1}Introduction}{917}{section.14.1}%
\contentsline {section}{\numberline {14.2}Change the view}{918}{section.14.2}%
\contentsline {section}{\numberline {14.3}The axes}{918}{section.14.3}%
\contentsline {subsection}{\numberline {14.3.1}Draw unit vectors: \texttt {Ox\_3d\_unit\_vector} \texttt {Oy\_3d\_unit\_vector} \texttt {Oz\_3d\_unit\_vector} \texttt {frame\_3d}}{918}{subsection.14.3.1}%
\contentsline {section}{\numberline {14.4}Points in space}{919}{section.14.4}%
\contentsline {subsection}{\numberline {14.4.1}Define a point in three-dimensions: \texttt {point}}{919}{subsection.14.4.1}%
\contentsline {subsection}{\numberline {14.4.2}Define a random point in three-dimensions: \texttt {point3d}}{920}{subsection.14.4.2}%
\contentsline {subsection}{\numberline {14.4.3}Find an intersection point of two objects in space: \texttt {single\_inter} \texttt {line\_inter}}{920}{subsection.14.4.3}%
\contentsline {subsection}{\numberline {14.4.4}Find the intersection points of two objects in space: \texttt {inter}}{922}{subsection.14.4.4}%
\contentsline {subsection}{\numberline {14.4.5}Find the midpoint of a segment in space: \texttt {midpoint}}{923}{subsection.14.4.5}%
\contentsline {subsection}{\numberline {14.4.6}Find the isobarycenter of a set of points in space: \texttt {isobarycenter}}{923}{subsection.14.4.6}%
\contentsline {subsection}{\numberline {14.4.7}Find the barycenter of a set of points in space: \texttt {barycenter}}{923}{subsection.14.4.7}%
\contentsline {section}{\numberline {14.5}Lines in space}{924}{section.14.5}%
\contentsline {subsection}{\numberline {14.5.1}Lines and directed lines in space: \texttt {line}}{924}{subsection.14.5.1}%
\contentsline {subsection}{\numberline {14.5.2}Half lines in space: \texttt {half\_line}}{925}{subsection.14.5.2}%
\contentsline {subsection}{\numberline {14.5.3}Segments in space: \texttt {segment}}{926}{subsection.14.5.3}%
\contentsline {subsection}{\numberline {14.5.4}Vectors in space: \texttt {vector}}{926}{subsection.14.5.4}%
\contentsline {subsection}{\numberline {14.5.5}Parallel lines and planes in space: \texttt {parallel}}{928}{subsection.14.5.5}%
\contentsline {subsection}{\numberline {14.5.6}Perpendicular lines and planes in space: \texttt {perpendicular}}{930}{subsection.14.5.6}%
\contentsline {subsection}{\numberline {14.5.7}Planes orthogonal to lines and lines orthogonal to planes in space: \texttt {orthogonal}}{931}{subsection.14.5.7}%
\contentsline {subsection}{\numberline {14.5.8}Common perpendiculars to lines in space: \texttt {common\_perpendicular}}{932}{subsection.14.5.8}%
\contentsline {section}{\numberline {14.6}Planes in space}{933}{section.14.6}%
\contentsline {subsection}{\numberline {14.6.1}Planes in space: \texttt {plane}}{933}{subsection.14.6.1}%
\contentsline {subsection}{\numberline {14.6.2}The bisector plane in space: \texttt {perpen\_bisector}}{934}{subsection.14.6.2}%
\contentsline {subsection}{\numberline {14.6.3}Tangent planes in space: \texttt {tangent}}{934}{subsection.14.6.3}%
\contentsline {section}{\numberline {14.7}Triangles in space}{936}{section.14.7}%
\contentsline {subsection}{\numberline {14.7.1}Draw a triangle in space: \texttt {triangle}}{936}{subsection.14.7.1}%
\contentsline {subsection}{\numberline {14.7.2}Isosceles triangles in space: \texttt {isosceles\_triangle}}{936}{subsection.14.7.2}%
\contentsline {subsection}{\numberline {14.7.3}Right triangles in space: \texttt {right\_triangle}}{938}{subsection.14.7.3}%
\contentsline {subsection}{\numberline {14.7.4}Equilateral triangles in space: \texttt {equilateral\_triangle}}{940}{subsection.14.7.4}%
\contentsline {section}{\numberline {14.8}Quadrilaterals in space}{940}{section.14.8}%
\contentsline {subsection}{\numberline {14.8.1}Squares in space: \texttt {square}}{941}{subsection.14.8.1}%
\contentsline {subsection}{\numberline {14.8.2}Rhombuses in space: \texttt {rhombus}}{941}{subsection.14.8.2}%
\contentsline {subsection}{\numberline {14.8.3}Rectangles in space: \texttt {rectangle}}{943}{subsection.14.8.3}%
\contentsline {subsection}{\numberline {14.8.4}Parallelograms in space: \texttt {parallelogram}}{945}{subsection.14.8.4}%
\contentsline {subsection}{\numberline {14.8.5}Arbitrary quadrilaterals in space: \texttt {quadrilateral}}{945}{subsection.14.8.5}%
\contentsline {section}{\numberline {14.9}Polygons in space}{946}{section.14.9}%
\contentsline {subsection}{\numberline {14.9.1}Hexagons in space: \texttt {hexagon}}{946}{subsection.14.9.1}%
\contentsline {subsection}{\numberline {14.9.2}Regular polygons in space: \texttt {isopolygon}}{947}{subsection.14.9.2}%
\contentsline {subsection}{\numberline {14.9.3}General polygons in space: \texttt {polygon}}{948}{subsection.14.9.3}%
\contentsline {subsection}{\numberline {14.9.4}Polygonal lines in space: \texttt {open\_polygon}}{948}{subsection.14.9.4}%
\contentsline {section}{\numberline {14.10}Circles in space: \texttt {circle}}{949}{section.14.10}%
\contentsline {section}{\numberline {14.11}Conics in space}{950}{section.14.11}%
\contentsline {subsection}{\numberline {14.11.1}Ellipses in space: \texttt {ellipse}}{950}{subsection.14.11.1}%
\contentsline {subsection}{\numberline {14.11.2}Hyperbolas in space: \texttt {hyperbola}}{951}{subsection.14.11.2}%
\contentsline {subsection}{\numberline {14.11.3}Parabolas in space: \texttt {parabola}}{951}{subsection.14.11.3}%
\contentsline {section}{\numberline {14.12}Three-dimensional coordinates}{952}{section.14.12}%
\contentsline {subsection}{\numberline {14.12.1}The abscissa of a three-dimensional point: \texttt {abscissa}}{952}{subsection.14.12.1}%
\contentsline {subsection}{\numberline {14.12.2}The ordinate of a three-dimensional point: \texttt {ordinate}}{952}{subsection.14.12.2}%
\contentsline {subsection}{\numberline {14.12.3}The cote of a three-dimensional point: \texttt {cote}}{953}{subsection.14.12.3}%
\contentsline {subsection}{\numberline {14.12.4}The coordinates of a point, vector or line in space: \texttt {coordinates}}{953}{subsection.14.12.4}%
\contentsline {subsection}{\numberline {14.12.5}The Cartesian equation of an object in space: \texttt {equation}}{954}{subsection.14.12.5}%
\contentsline {subsection}{\numberline {14.12.6}The parametric equation of an object in space: \texttt {parameq}}{954}{subsection.14.12.6}%
\contentsline {subsection}{\numberline {14.12.7}The length of a segment in space: \texttt {distance}}{955}{subsection.14.12.7}%
\contentsline {subsection}{\numberline {14.12.8}The length squared of a segment in space: \texttt {distance2}}{955}{subsection.14.12.8}%
\contentsline {subsection}{\numberline {14.12.9}The measure of an angle in space: \texttt {angle}}{956}{subsection.14.12.9}%
\contentsline {section}{\numberline {14.13}Properties}{956}{section.14.13}%
\contentsline {subsection}{\numberline {14.13.1}Check if an object in space is on another object: \texttt {is\_element}}{956}{subsection.14.13.1}%
\contentsline {subsection}{\numberline {14.13.2}Check if points and/or lines in space are coplanar: \texttt {is\_coplanar}}{957}{subsection.14.13.2}%
\contentsline {subsection}{\numberline {14.13.3}Check if lines and/or planes in space are parallel: \texttt {is\_parallel}}{958}{subsection.14.13.3}%
\contentsline {subsection}{\numberline {14.13.4}Check if lines and/or planes in space are perpendicular: \texttt {is\_perpendicular}}{958}{subsection.14.13.4}%
\contentsline {subsection}{\numberline {14.13.5}Check if two lines or two spheres in space are orthogonal: \texttt {is\_orthogonal}}{959}{subsection.14.13.5}%
\contentsline {subsection}{\numberline {14.13.6}Check if three points in space are collinear: \texttt {is\_collinear}}{960}{subsection.14.13.6}%
\contentsline {subsection}{\numberline {14.13.7}Check if four points in space are cocyclic: \texttt {is\_concyclic}}{960}{subsection.14.13.7}%
\contentsline {subsection}{\numberline {14.13.8}Check if five points in space are cospherical: \texttt {is\_cospherical}}{961}{subsection.14.13.8}%
\contentsline {subsection}{\numberline {14.13.9}Check if an object in space is an equilateral triangle: \texttt {is\_equilateral}}{961}{subsection.14.13.9}%
\contentsline {subsection}{\numberline {14.13.10}Check if an object in space is an isosceles triangle: \texttt {is\_isosceles}}{962}{subsection.14.13.10}%
\contentsline {subsection}{\numberline {14.13.11}Check if an object in space is a right triangle or a rectangle: \texttt {is\_rectangle}}{962}{subsection.14.13.11}%
\contentsline {subsection}{\numberline {14.13.12}Check if an object in space is a square: \texttt {is\_square}}{963}{subsection.14.13.12}%
\contentsline {subsection}{\numberline {14.13.13}Check if an object in space is a rhombus: \texttt {is\_rhombus}}{964}{subsection.14.13.13}%
\contentsline {subsection}{\numberline {14.13.14}Check if an object in space is a parallelogram: \texttt {is\_parallelogram}}{964}{subsection.14.13.14}%
\contentsline {section}{\numberline {14.14}Transformations in space}{965}{section.14.14}%
\contentsline {subsection}{\numberline {14.14.1}General remarks}{965}{subsection.14.14.1}%
\contentsline {subsection}{\numberline {14.14.2}Translation in space: \texttt {translation}}{966}{subsection.14.14.2}%
\contentsline {subsection}{\numberline {14.14.3}Reflection in space with respect to a plane, line or point: \texttt {reflection} \texttt {symmetry}}{966}{subsection.14.14.3}%
\contentsline {subsection}{\numberline {14.14.4}Rotation in space: \texttt {rotation}}{967}{subsection.14.14.4}%
\contentsline {subsection}{\numberline {14.14.5}Homothety in space: \texttt {homothety}}{967}{subsection.14.14.5}%
\contentsline {subsection}{\numberline {14.14.6}Similarity in space: \texttt {similarity}}{968}{subsection.14.14.6}%
\contentsline {subsection}{\numberline {14.14.7}Inversion in space: \texttt {inversion}}{968}{subsection.14.14.7}%
\contentsline {subsection}{\numberline {14.14.8}Orthogonal projection in space: \texttt {projection}}{969}{subsection.14.14.8}%
\contentsline {section}{\numberline {14.15}Surfaces}{970}{section.14.15}%
\contentsline {subsection}{\numberline {14.15.1}Cones: \texttt {cone}}{970}{subsection.14.15.1}%
\contentsline {subsection}{\numberline {14.15.2}Half-cones: \texttt {half\_cone}}{970}{subsection.14.15.2}%
\contentsline {subsection}{\numberline {14.15.3}Cylinders: \texttt {cylinder}}{971}{subsection.14.15.3}%
\contentsline {subsection}{\numberline {14.15.4}Spheres: \texttt {sphere}}{971}{subsection.14.15.4}%
\contentsline {subsection}{\numberline {14.15.5}The graph of a function of two variables: \texttt {funcplot}}{972}{subsection.14.15.5}%
\contentsline {subsection}{\numberline {14.15.6}The graph of parametric equations in space: \texttt {paramplot}}{972}{subsection.14.15.6}%
\contentsline {section}{\numberline {14.16}Solids}{973}{section.14.16}%
\contentsline {subsection}{\numberline {14.16.1}Cubes: \texttt {cube}}{973}{subsection.14.16.1}%
\contentsline {subsection}{\numberline {14.16.2}Tetrahedrons: \texttt {tetrahedron} \texttt {pyramid}}{975}{subsection.14.16.2}%
\contentsline {subsection}{\numberline {14.16.3}Parallelepipeds: \texttt {parallelepiped}}{976}{subsection.14.16.3}%
\contentsline {subsection}{\numberline {14.16.4}Prisms: \texttt {prism}}{977}{subsection.14.16.4}%
\contentsline {subsection}{\numberline {14.16.5}Polyhedra: \texttt {polyhedron}}{977}{subsection.14.16.5}%
\contentsline {subsection}{\numberline {14.16.6}Vertices: \texttt {vertices}}{978}{subsection.14.16.6}%
\contentsline {subsection}{\numberline {14.16.7}Faces: \texttt {faces}}{978}{subsection.14.16.7}%
\contentsline {subsection}{\numberline {14.16.8}Edges: \texttt {line\_segments}}{979}{subsection.14.16.8}%
\contentsline {section}{\numberline {14.17}Platonic solids}{979}{section.14.17}%
\contentsline {subsection}{\numberline {14.17.1}Centered tetrahedra: \texttt {centered\_tetrahedron}}{980}{subsection.14.17.1}%
\contentsline {subsection}{\numberline {14.17.2}Centered cubes: \texttt {centered\_cube}}{980}{subsection.14.17.2}%
\contentsline {subsection}{\numberline {14.17.3}Octahedra: \texttt {octahedron}}{981}{subsection.14.17.3}%
\contentsline {subsection}{\numberline {14.17.4}Dodecahedra: \texttt {dodecahedron}}{982}{subsection.14.17.4}%
\contentsline {subsection}{\numberline {14.17.5}Icosahedra: \texttt {icosahedron}}{982}{subsection.14.17.5}%
\contentsline {chapter}{\numberline {15}Multimedia}{985}{chapter.15}%
\contentsline {section}{\numberline {15.1}Audio Tools}{985}{section.15.1}%
\contentsline {subsection}{\numberline {15.1.1}Creating audio clips: \texttt {createwav} }{986}{subsection.15.1.1}%
\contentsline {subsection}{\numberline {15.1.2}Reading \texttt {wav} files from disk: \texttt {readwav} }{987}{subsection.15.1.2}%
\contentsline {subsection}{\numberline {15.1.3}Writing \texttt {wav} files to disk: \texttt {writewav} }{987}{subsection.15.1.3}%
\contentsline {subsection}{\numberline {15.1.4}Audio playback: \texttt {playsnd}}{987}{subsection.15.1.4}%
\contentsline {subsection}{\numberline {15.1.5}Averaging channel data: \texttt {stereo2mono} }{988}{subsection.15.1.5}%
\contentsline {subsection}{\numberline {15.1.6}Audio clip properties: \texttt {channels} \texttt {bit\_depth} \texttt {samplerate} \texttt {duration} }{988}{subsection.15.1.6}%
\contentsline {subsection}{\numberline {15.1.7}Extracting samples from audio clips: \texttt {channel\_data} }{988}{subsection.15.1.7}%
\contentsline {subsection}{\numberline {15.1.8}Changing the sampling rate: \texttt {resample} }{989}{subsection.15.1.8}%
\contentsline {subsection}{\numberline {15.1.9}Visualizing waveforms: \texttt {plotwav} }{989}{subsection.15.1.9}%
\contentsline {subsection}{\numberline {15.1.10}Visualizing power spectra: \texttt {plotspectrum} }{990}{subsection.15.1.10}%
\contentsline {subsection}{\numberline {15.1.11}Reading a \texttt {wav} file: \texttt {readwav}}{991}{subsection.15.1.11}%
\contentsline {subsection}{\numberline {15.1.12}Writing a \texttt {wav} file: \texttt {writewav}}{992}{subsection.15.1.12}%
\contentsline {subsection}{\numberline {15.1.13}Listening to a digital sound: \texttt {playsnd}}{992}{subsection.15.1.13}%
\contentsline {subsection}{\numberline {15.1.14}Preparing digital sound data: \texttt {soundsec} }{992}{subsection.15.1.14}%
\contentsline {section}{\numberline {15.2}Signal Processing}{993}{section.15.2}%
\contentsline {subsection}{\numberline {15.2.1}Boxcar function: \texttt {boxcar}}{993}{subsection.15.2.1}%
\contentsline {subsection}{\numberline {15.2.2}Rectangle function: \texttt {rect}}{993}{subsection.15.2.2}%
\contentsline {subsection}{\numberline {15.2.3}Triangle function: \texttt {tri}}{994}{subsection.15.2.3}%
\contentsline {subsection}{\numberline {15.2.4}Cardinal sine function: \texttt {sinc}}{994}{subsection.15.2.4}%
\contentsline {subsection}{\numberline {15.2.5}Cross-correlation of two signals: \texttt {cross\_correlation}}{995}{subsection.15.2.5}%
\contentsline {subsection}{\numberline {15.2.6}Auto-correlation of a signal: \texttt {auto\_correlation}}{995}{subsection.15.2.6}%
\contentsline {subsection}{\numberline {15.2.7}Convolution of two signals or functions: \texttt {convolution} }{996}{subsection.15.2.7}%
\contentsline {subsection}{\numberline {15.2.8}Low-pass filtering: \texttt {lowpass}}{998}{subsection.15.2.8}%
\contentsline {subsection}{\numberline {15.2.9}High-pass filtering: \texttt {highpass}}{998}{subsection.15.2.9}%
\contentsline {subsection}{\numberline {15.2.10}Apply a moving average filter to a signal: \texttt {moving\_average}}{998}{subsection.15.2.10}%
\contentsline {subsection}{\numberline {15.2.11}Perform thresholding operations on an array: \texttt {threshold}}{999}{subsection.15.2.11}%
\contentsline {subsection}{\numberline {15.2.12}Bartlett-Hann window function: \texttt {bartlett\_hann\_window}}{1001}{subsection.15.2.12}%
\contentsline {subsection}{\numberline {15.2.13}Blackman-Harris window function: \texttt {blackman\_harris\_window}}{1002}{subsection.15.2.13}%
\contentsline {subsection}{\numberline {15.2.14}Blackman window function: \texttt {blackman\_window}}{1002}{subsection.15.2.14}%
\contentsline {subsection}{\numberline {15.2.15}Bohman window function: \texttt {bohman\_window}}{1002}{subsection.15.2.15}%
\contentsline {subsection}{\numberline {15.2.16}Cosine window function: \texttt {cosine\_window}}{1003}{subsection.15.2.16}%
\contentsline {subsection}{\numberline {15.2.17}Gaussian window function: \texttt {gaussian\_window}}{1003}{subsection.15.2.17}%
\contentsline {subsection}{\numberline {15.2.18}Hamming window function: \texttt {hamming\_window}}{1003}{subsection.15.2.18}%
\contentsline {subsection}{\numberline {15.2.19}Hann-Poisson window function: \texttt {hann\_poisson\_window}}{1004}{subsection.15.2.19}%
\contentsline {subsection}{\numberline {15.2.20}Hann window function: \texttt {hann\_window}}{1004}{subsection.15.2.20}%
\contentsline {subsection}{\numberline {15.2.21}Parzen window function: \texttt {parzen\_window}}{1004}{subsection.15.2.21}%
\contentsline {subsection}{\numberline {15.2.22}Poisson window function: \texttt {poisson\_window}}{1005}{subsection.15.2.22}%
\contentsline {subsection}{\numberline {15.2.23}Riemann window function: \texttt {riemann\_window}}{1005}{subsection.15.2.23}%
\contentsline {subsection}{\numberline {15.2.24}Triangular window function: \texttt {triangle\_window}}{1005}{subsection.15.2.24}%
\contentsline {subsection}{\numberline {15.2.25}Tukey window function: \texttt {tukey\_window}}{1006}{subsection.15.2.25}%
\contentsline {subsection}{\numberline {15.2.26}Welch window function: \texttt {welch\_window}}{1006}{subsection.15.2.26}%
\contentsline {subsection}{\numberline {15.2.27}An example: static noise removal by spectral subtraction}{1006}{subsection.15.2.27}%
\contentsline {section}{\numberline {15.3}Images}{1008}{section.15.3}%
\contentsline {subsection}{\numberline {15.3.1}Image structure in \texttt {Xcas}}{1008}{subsection.15.3.1}%
\contentsline {subsection}{\numberline {15.3.2}Reading images: \texttt {readrgb}}{1009}{subsection.15.3.2}%
\contentsline {subsection}{\numberline {15.3.3}Viewing images}{1009}{subsection.15.3.3}%
\contentsline {subsection}{\numberline {15.3.4}Creating or recreating images: \texttt {writergb}}{1009}{subsection.15.3.4}%
\contentsline {subsubsection}{Writing images given in \texttt {Xcas} format}{1009}{section*.135}%
\contentsline {subsubsection}{Creating images}{1011}{section*.136}%
\contentsline {chapter}{\numberline {16}Using \texttt {giac} inside a program}{1013}{chapter.16}%
\contentsline {section}{\numberline {16.1}Using \texttt {giac} inside a \texttt {C++} program}{1013}{section.16.1}%
\contentsline {section}{\numberline {16.2}Defining new \texttt {giac} functions}{1014}{section.16.2}%
