\documentclass[a4paper,10pt]{book} \textwidth 10,5 cm %\textwidth 11,8 cm \textheight 17 cm %\documentclass[twocolumn,10pt]{book} %dvips -t landscape -O 0,-0.9 jean2.dvi \if@twoside \oddsidemargin 44pt \evensidemargin 82pt \marginparwidth 107pt \else \oddsidemargin -0.29cm \evensidemargin -0.29cm \marginparwidth 0cm \fi \marginparsep 0cm \topmargin 0.26cm \headheight 10pt \headsep 10pt \footskip 30pt %\topmargin 0.36cm %\baselineskip = 15pt %\textheight 24.5cm %\textwidth 17cm %\columnsep 30pt \columnseprule 0pt %\textwidth 25 cm %\textheight 18 cm %\def\@evenhead{\thepage\hfill{\footnotesize\textit{\leftmark}}} %\def\@oddhead{\footnotesize{\textit{\rightmark}}\hfill\thepage} %\usepackage{hp} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} %\usepackage[francais]{babel} \usepackage{latexsym} \usepackage{amssymb} \usepackage{makeidx} \usepackage{times} \usepackage[ps2pdf,breaklinks=true,colorlinks=true,linkcolor=red,citecolor=green]{hyperref}%\usepackage[colorlinks,pdftex]{hyperref} \title {Computer Algebra \\and \\ Mathematics\\ with\\ the {\tt HP40GS}} \author{Renée De Graeve\\ Version 3.0} %corrections MarK Howell \makeindex \begin{document} \newcommand{\asinh}{\,\,\mbox{asinh\,}} \newcommand{\atanh}{\,\,\mbox{atanh\,}} \maketitle {\bf \centerline{Acknowledgments}} \vspace{1cm} It was not believed possible to write an efficient program for computer algebra all on one's own. But one bright person by the name of Bernard Parisse didn't know that...and did it!!!\\ This is his program for computer algebra (called {\tt ERABLE}), built for the second time into an {\tt HP} calculator.\\ The development of this calculator has led Bernard Parisse to modify his program somewhat so that the computer algebra functions could be edited and cause the appropriate results to be displayed in the Equation Editor.\\ Explore all the capabilities of this calculator, as set out in the following pages.\\ \vspace{0.2cm} I would like to thank: \begin{itemize} \item Bernard Parisse for his invaluable counsel, his remarks on the text, his reviews, and for his ability to provide functions on demand both efficiently and graciously. \item Jean Tavenas for the concern shown towards the completion of the first version of this guide. %\item Jean Yves Avenard for taking on board our requests, and for writing the {\tt PROMPT} command in the very spirit of promptness and with no advance warning... (cf \ref{sec:prompt}). \end{itemize} \vspace{1cm} \vfill %\copyright\ 2000, Hewlett-Packard, http://www.hp.com/calculators\\ \copyright\ 2005, Renée De Graeve \\ The reproduction, distribution and/or the modification of this document is authorised according to the terms of the {\tt GNU Free Documentation License}, Version 1.1 or later, published by the Free Software Foundation, with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts.\\ A copy of this license exists under the section entitled :\\ {\tt GNU Free Documentation License} (chapter \ref{sec:fdl}, p. \pageref{sec:fdl}). \newpage {\bf \centerline{Preface}} \vspace{0.2cm} The {\tt HP40G} marked a new stage in the democratisation of the use of symbolic calculation, on the one hand by its competitive price, and on the other hand by making it possible to execute, step-by-step, the principle algorithms taught in mathematics at secondary schools and in the first years of University. But it was still necessary to add adequate documentation, preferably written by a teacher of mathematics. That is what you find in this guide, written by Renée De Graeve. %Lecturer at the University of Grenoble I and founder of the Grenoble IREM. It contains, naturally, a complete reference of the functions for symbolic calculation, but also demonstrates, using examples taken from study for both certificate and diploma, how to take smart advantage of the calculating power of the {\tt HP40GS}. The guide ends with two chapters dedicated to programming: the first for learning to program, and the second to illustrate the application of algorithmic language to arithmetic programs as taught in French tertiary schools. This guide is an update for the {\tt HP40GS} calculator, with some documentation improvements in addition to the {\tt HP40GS} software and speed improvements. { \flushright Bernard Parisse\\ \flushright Lecturer at the University of Grenoble I\\ } \vfill \chapter{To Begin} \section{General Information} \subsection{Starting the Calculator} Press the {\tt ON} button.\\ The {\tt HOME} screen is displayed.\\ As you work, you can use the {\tt ON} button to cancel the current operation. In other words, it has the same function as {\tt CANCEL}.\\ To turn the calculator off, press {\tt SHIFT} plus {\tt ON (OFF)}.\\ If the calculator fails to respond after several presses of the {\tt ON ( CANCEL)} button, you can press {\tt ON} and {\tt F3} simultaneously to reinitialise the calculator. \subsection{What You See} >From top to bottom:\\ 1.The {\tt HOME} screen 1.a The calculator's status 1.b A horizontal line 1.c A menu bar of commands\\ 2. The keyboard\\ \noindent 1. The {\tt HOME} screen : 1.a The calculator's status gives the {\tt HOME} screen's current mode: \begin{itemize} \item {\tt RAD}, {\tt DEG}, or {\tt GRD} when you are working with radians, degrees, or gradians. \item {\tt \{FUNCTION\}} to indicate the name of the {\tt Aplet} currently selected here, the {\tt Function Aplet}. \item $\blacktriangle$ to indicate that you can use the up-arrow to move back through the history. \end{itemize} 1.b Horizontal line :\\ - Above the horizontal line is found an overview of the calculations carried out in the {\tt HOME} screen.\\ {\sc Principle} : On the screen, the requested calculation is written on the left, and the result is written on the right.\\ - Beneath the horizontal line is the command line.\\ Using the up-arrow, you can move back through the history and, using {\tt COPY} on the menu bar, copy a command or a preceding result into the command line.\\ 1.c Menu Bar:\\ The commands on the menu bar are accessible through the six unlabelled grey keys, which we refer to here as :\\ {\tt F1 F2 F3 F4 F5 F6}.\\ The menu bar can contain items consisting of groups of commands; they are marked by a folder icon.\\ To activate a command on the menu bar, it's enough to press the corresponding {\tt Fi} key.\\ In the {\tt HOME} screen, the menu bar contains two commands:\\ - ${\tt STO \triangleright}$ which enables you to store a value in a variable, and\\ - {\tt CAS}, which enables you to open the Equation Editor to perform computer algebra.\\ 2. The keyboard :\\ You have already been introduced to:\\ The {\tt ON} key for starting the calculator or for cancelling the current operation, and {\tt SHIFT} {\tt ON} for turning the calculator off.\\ Other keys you must know include: \begin{itemize} \item The four arrow keys (left, right, up, down), which enable you to move the cursor while you're in the Equation Editor, in a menu, and so on, \item The {\tt SHIFT} key, which enables you to access two functions with one key, \item The {\tt ALPHA} key for typing text in upper case, and the {\tt SHIFT ALPHA} keys for typing text in lower case. To remain in alphabetic mode, you must press and hold the {\tt ALPHA} key. \item ${\tt X\ T\ \theta}$ which enable you to type ${\tt X,\ T,\ \theta, N}$ directly, depending on the active aplet. \item The {\tt ENTER} key, which is used to confirm a command. \end{itemize} \section{Notations} The four arrow keys are represented by four triangles:\\ $$\vartriangle\ \lhd \ \rhd \ \triangledown $$ \index{\tt $ \vartriangle\ \lhd \ \rhd \ \triangledown$} The ${\tt STO\triangleright}$ on the {\tt HOME} menu is represented in the program by:\\ $$ {\tt STO\triangleright \mbox{ or }\triangleright} \mbox{ or }->$$\index{$\triangleright$}\index{STO$\triangleright$}\index{->} In the Equation Editor, the position of the cursor is represented by: $$\blacktriangleleft$$\index{$\blacktriangleleft$} \section{Online Help}\index{HELP} With this calculator, you're able to get practical and efficient online help in either French or English (cf \ref{sec:cfg}).\\ You are shown an alphabetised list of computer algebra functions. As with the drop-down menus, you are able to access the functions by pressing the corresponding letter-keys without having to worry about pressing {\tt ALPHA}.\\ The help consists of a succinct description of the command, as well as an example and its result. Each example can be tested with {\tt ECHO} (on the menu) and can be either used as is or modified. You can view help for related commands by means of {\tt SEE1 SEE2}... on the menu. You have to use {\tt EXIT} to return, without change, in the Equation Editor.\\ For more details, refer to the description of {\tt SHIFT 2 (SYNTAX)}, sections \ref{sec:syntaxe} and \ref{sec:syntax}. \chapter{Aplets} \section{{\tt APLET} Key} The {\tt APLET} key gives you access to the list of available {\tt Aplets}.\\ This calculator in effect enables you to work with {\tt Aplets}.\\ But what is an {\tt Aplet} ?\\ An {\tt Aplet} is a program stored in the calculator, which enables you to easily obtain three views of a mathematical object (a symbolic view, a numeric view, and a graphic view) and all this is pre-programmed !\\ The various {\tt Aplets} enable you to work with mathematical objects such as: functions, sequences, statistical data, and so on...\\ Certain {\tt Aplets} illustrate particular classroom lessons. \section{The Various aplets} While you're in {\tt HOME}, you can see the name of the selected aplet by looking at the status line.\\ Here are the possible choices for the {\tt APLET} key: \begin{itemize} \item {\tt Sequence}\\ This {\tt Aplet} enables you to define sequences having the names:\\ {\tt U1, U2 .. U9, U0}\\ One defines {\tt U1(N)}:\\ - To be a function of {\tt N},\\ - To be a function of {\tt U1(N-1)},\\ - To be a function of {\tt U1(N-1)} and {\tt U1(N-2)}.\\ You define (for example):\\ {\tt U1(N) = N*N+1}\\ and then the values of {\tt U1(1)} and {\tt U1(2)} are automatically calculated and displayed.\\ To display the values of {\tt U1(N)}, choose {\tt U1}, then press {\tt NUM}.\\ You can find other examples that use the {\tt Sequence Aplet} in the following section, such as the calculation of the {\tt GCD} of two numbers (cf \ref{sec:pgcd}), and the calculation of the coefficients of Bézout's Identity (cf \ref{sec:bezout}). \item {\tt Function}\\ This {\tt Aplet} enables you to define functions having the names:\\ {\tt F1(X), F2(X) .. F9(X), F0(X)}\\ One defines {\tt F1(X)} :\\ - To be an expression of a function of {\tt X}:\\ For example, the formula:\\ {\tt F1(X) = X*LN(X)} defines the function: \\ $f1(x) = x.\ln(x)$\\ - To use Booleans ({\tt X(0} etc.) if the function is defined in parts :\\ For example, a formula of the form:\\ {\tt F1(X) = X((X(0)+2(X((X>0)} defines the function:\\ $f_1(x) = x$ if $x< 0$ and\\ $f_1(x) = 2 \cdot x$ if $x \geq 0$ \item {\tt Parametric} for tracing curves in parametric coordinates. \item {\tt Polar} for tracing curves in polar coordinates. \item {\tt Solve} for solving numeric equations. \item {\tt Statistics} for working with statistics. \item {\tt Inference} for working with inferential statistics. \item {\tt Quad Explorer} for exploring quadratic functions. \item {\tt Trig Explorer} for exploring trigonometric functions. \end{itemize} \section{Examples using the {\tt Sequence Aplet}} \subsection{Notation in Base $b$} Given $a$ and $b$, produce the series $q_n$ ($ n \geq 1$) and $r_n$ ($n \geq 2$) from the quotients and the remainders of the division of $q_i$ by $b$, defined by:\\ $q_1=a$\\ $q_1=b \cdot q_2+r_2 \ (0 \leq r_2 < b)$\\ $q_2=b \cdot q_3+r_3 \ (0 \leq r_3 < b)$\\ ......\\ $q_{n-1}=b \cdot q_n+r_n \ (0 \leq r_n < b)$\\ Note that if $r_{n+1}=0$, the number $r_n r_{n-1} .....r_3r_2$ is notated in base $b$ of $a$, while one assumes $2 \leq b \leq 10$.\\ Put into {\tt B} the value of the base, for example:\\ ${\tt 7\ STO\triangleright\ B}$ \\ Put into {\tt A} the number to write in base {\tt B} (for example, ${\tt 1789\ STO\triangleright\ A}$ )\\ Define the following two series:\\ {\tt U1(1)=A}\\ {\tt U1(2)=FLOOR(A/B)}\\ {\tt U1(N)=FLOOR(U1(N-1)/B)}\\ and\\ {\tt U2(1)=0}\\ {\tt U2(2)=A MOD B}\\ {\tt U2(N)=U1(N-1) MOD B }\\ Therefore, $q_n$={\tt U1(N)} and $r_n $={\tt U2(N)}.\\ We obtain :\\ {\tt U2(2)=4 U2(3)=3 U2(4)=1 U2(5)=5 U2(6)=0}\\ and so the notation in base 7 of 1789 is: 5134. \subsection{Calculating the GCD} \label{sec:pgcd} This is an application of Euclid's Algorithm on the {\tt HP40GS}.\\ Here is the description of this algorithm:\\ If one performs the successive Euclidean divisions: \begin{eqnarray*} A = & B \times Q_1+R_1 & 0 \leq R_1 < B \\ B = & R_1 \times Q_2+R_2 & 0 \leq R_2 < R_1 \\ R_1 = & R_2 \times Q_3+R_3 & 0 \leq R_3 < R_2 \\ .......\\ R_{n-2} = & R_{n-1} \times Q_n+R_n & 0 \leq R_n < R_{n-1} \end{eqnarray*} then after a finite number of steps (in excess of $B$), there exists a whole number $n$ such that: $R_n = 0.$ \\ We have then:\\ $GCD (A,B) = GCD (B, R_1) =...$\\ $GCD (R_{n-1}, R_n) = GCD (R_{n-1},0) = R_{n-1}$\\ Using a sequence, one then writes the sequences of remainders.\\ With the {\tt HP40GS}, you use the {\tt Sequence Aplet} ({\tt APLET} key, then select {\tt Sequence}, then {\tt START} on the menu bar).\\ To determine the GCD(78,56), you define the sequence:\\ ${\tt U1(1)=78} $ ${\tt U1(2)=56} $ ${\tt U1(N)=U1(N-2)\ MOD \ U1(N-1)}$ Press {\tt NUM} to get the numerical list of {\tt U1(N)} that is, the list of the remainders of the successive divisions...\\ The final non-zero remainder is 2, so the $GCD(78,56)=2$.\\ {\sc Remark}:\\ In {\tt HOME}, you can use the variables {\tt A} and {\tt B} to store the two numbers, and then make {\tt U1(1)=A} and {\tt U1(2)=B}.\\ It's also important to note that {\tt A MOD 0 = A}. \subsection{Calculating the Coefficients of Bézout's Identity}\label{sec:bezout} Euclid's Algorithm enables you to find a pair $U$, $V$ such that: $ A \times U + B \times V= PGCD(A,B) $\\ Using the idea of sequence:\\ Define "the sequence of remainder" $R_n$ and two sequences $U_n$ and $V_n$, such that at each step one has:\\ $R_n=U_n \times A+V_n \times B$. Seeing that one has:\\ $R_n=R_{n-2}-Q_n \times R_{n-1}$ , $U_n$ and $V_n$ serve to satisfy the same recurrence relation (where $Q_n$ is the whole-number quotient of $R_{n-2}$ divided by $R_{n-1}$). One then has (from the beginning): $R_1=A \ R_2=B$ $U_1=1 \ U_2=0$ since $A=1 \times A+0 \times B$ $V_1=0 \ V_2=1$ since $B=0 \times A+1 \times B$ With the {\tt HP40GS}, using the {\tt Sequence Aplet}, you then define the sequence of remainders {\tt U1} and the sequences {\tt U2} and {\tt U3} such that for all {\tt N} one has: \\ {\tt U1(N)=A*U2(N)+B*U3(N)}.\\ For this, you need the sequence of quotients, which you put into {\tt U4}.\\ The sequences {\tt U1}, {\tt U2} and {\tt U3} satisfy the same recurrence relation:\\ $U_n=U_{n-2}-Q_n \times U_{n-1} $ with $Q_n={\tt U4(N)=FLOOR(U1(N-2)/U1(N-1))}$ On définit donc : ${\tt U1(1)=A} $ ${\tt U1(2)=B} $ ${\tt U1(N)=U1(N-2)-U4(N)* U1(N-1)}$ ${\tt U2(1)=1} $ ${\tt U2(2)=0} $ ${\tt U2(N)=U2(N-2)-U4(N)* U2(N-1)}$ ${\tt U3(1)=0}$ ${\tt U3(2)=1 }$ ${\tt U3(N)=U3(N-2)-U4(N)* U3(N-1)}$ ${\tt U4(1)=0} $ ${\tt U4(2)=0} $ ${\tt U4(N)=FLOOR(U1(N-2)/U1(N-1))}$ It's important to note that you use {\tt U4(N)} only for {\tt N > 2}; you have therefore defined the two first values (which are useless!) as zero.\\ {\tt NUM} then displays the values of these various sequences, and on the line of the final non-zero remainder you can read the GCD and the coefficients of Bézout's Identity. \section{The {\tt SYMB, NUM, PLOT} and {\tt MODES} Keys} In general, an {\tt Aplet} can be viewed in three different ways:\\ A symbolic view, which corresponds to the {\tt SYMB} key\\ A numeric view, which corresponds to the {\tt NUM} key\\ A graphic view, which corresponds to the {\tt PLOT} key\\ When these keys are SHIFTed ({\tt SETUP}), this corresponds to choosing the various available parameters (choosing the parameters of the graphic window, the step size for the table etc...).\\ The {\tt SHIFT HOME (MODES)} key pressed in {\tt HOME} screen or in {\tt Aplets}, corresponds to the choice between Radians, Degrees or Grads. %%%% HP40GS BEGIN CAS %%%%% %%%% CAS INTRO %%%% \chapter{{\tt CAS} user's guide} \section{Terms and Conditions} {\bf Use of the CAS Software requires from the user an appropriate mathematical knowledge. There is no warranty for the CAS Software, to the extent permitted by applicable law. Except when otherwise stated in writing the copyright holder provides the CAS Software As Is without warranty of any kind, either expressed or implied, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. The entire risk as to the quality and performance of the CAS Software is with you. Should the CAS Software prove defective, you assume the cost of all necessary servicing, repair or correction. In no event unless required by applicable law will any copyright holder be liable to you for damages, including any general, special, incidental or consequential damages arising out of the use or inability to use the CAS Software (including but not limited to loss of data or data being rendered inaccurate or losses sustained by you or third parties or a failure of the CAS Software to operate with any other programs), even if such holder or other party has been advised of the possibility of such damages. If required by applicable law the maximum amount payable for damages by the copyright holder shall not exceed the royalty amount paid by Hewlett-Packard to the copyright holder for the CAS Software } Note that a large part of the HP40GS CAS is (c) Bernard Parisse, Institut Fourier, Université de Grenoble I, France. This part is also licensed under the LGPL License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.\\ For more details, see\\ \verb|http://www-fourier.ujf-grenoble.fr/~parisse| \section{Introduction} The word {\tt CAS} is a shortcut for Computer Algebra System. This chapter gives a brief overview on the HP40GS CAS. \subsection{What is {\tt CAS}?}\label{sec:exarg} The {\tt CAS} enables you to perform exact or symbolic calculations. Make sure you understand the difference between: \begin{itemize} \item exact or symbolic calculations, which are performed by means of the {\tt CAS} functions. You work in {\tt exact mode}, with infinite precision, and \item numeric calculations, which are performed by means of the {\tt MATH} key's {\tt MTH} menu, either in the {\tt HOME} screen or in {\tt Aplets} or programs. You work in {\tt approximate mode}, with a precision of $10^{-12}$. \end{itemize} Example: \\ If you're working in the {\tt HOME} screen:\\ {\tt 1/2+1/6} returns 0.6666666666667\\ whereas in {\tt CAS}:\\ {\tt 1/2+1/6} will return {\tt 2/3}. Each mode has advantages and drawbacks, in exact mode there is no round-off error, but some computations require much more time and memory than in numeric mode. For example computing the factorial of 300 is slightly faster in numeric mode than in exact mode. \subsection{How do you perform a symbolic calculation?} The {\tt HP40GS} has been designed to use computer algebra functions from a special environment: the Equation Editor. You can, however, perform computer algebra in the {\tt HOME} screen, as long as you take certain precautions (cf \ref{sec:cashome}), and it is mandatory if you want do symbolic linear algebra, since the Equation Editor does not handle vectors and matrices. To open the Equation Editor, press {\tt CAS} on the menu bar of the {\tt HOME} screen.\\ To leave the Equation Editor, press {\tt HOME} to return to the {\tt HOME} screen. Section \ref{sec:eqw} describes how to use the {\tt CAS} functions from the Equation Editor, section \ref{sec:cashome} how to use the {\tt CAS} functions from the {\tt HOME} screen and chapter \ref{sec:cas} describes the {\tt CAS} functions. \subsection{Variables} When you use the symbolic calculation functions, you are working with symbolic variables (variables that do not contain a permanent value). In the {\tt HOME} screen, a variable of this kind must have a name like {\tt S1..S5,s1.. s5,n1..n5}, but not {\tt X} which is assigned to a real value (by default {\tt X} is assigned to 0) and to store for symbolic expressions, the variables are {\tt E0,E1..E9}. In the Equation Editor, all variables may, or may not be, assigned: for example {\tt X} is not assigned to a real value by default, computing {\tt X+X} will return {\tt 2X}. Moreover, Equation Editor variables may have long names, like {\tt XY} or {\tt ABC}, unlike in {\tt HOME} where implied multiplication is assumed (for example {\tt ABC} becomes {\tt A*B*C} in {\tt HOME}). For these reasons,variables used in the Equation Editor cannot be used in {\tt HOME}, and vice versa. Using the command {\tt PUSH}, you can transfer expressions from the {\tt HOME} screen history to the {\tt CAS} history (cf section \ref{sec:push}) and using the command {\tt POP}, you can also transfer expressions from the {\tt CAS} history to the {\tt HOME} screen history (cf section \ref{sec:pop}). \subsection{The current variable} In the Equation Editor, the current variable is the name of the symbolic variable contained in {\tt VX}, it is almost always {\tt X}.\\ The current variable is always {\tt S1} in the {\tt HOME} screen.\\ Some {\tt CAS} functions depend on a current variable, for example, the function {\tt DERVX} calculates the derivative with respect to the current variable. Hence in the Equation Editor,\\ {\tt DERVX(2*X+Y)} returns 2 if {\tt VX=X}, but\\ {\tt DERVX(2*X+Y)} returns 1 if {\tt VX=Y} and\\ in the {\tt HOME} screen,\\ {\tt DERVX(2*S1+S2)=2}, but\\ {\tt DERIV(2*S1+S2,S2)=1}. %%%% EQW CHAPTER %%%%%% \section{Using the CAS in the Equation Editor} \label{sec:eqw} The Equation Editor enables you to type expressions that you want to simplify, factor, differentiate, integrate, and so on, and then work them through as if on paper. The {\tt CAS} key on the {\tt HOME} menu bar takes you into the Equation Editor, and the {\tt HOME} key takes you back out. This section tells you how to write an expression in the Equation Editor using the menus (section \ref{sec:eqwmenu}) and the keyboard (section \ref{sec:eqwkbd}), how to select a sub-expression (section \ref{sec:eqwsub}), how to call the {\tt CAS} functions on a sub-expression (section \ref{sec:eqwsub}) and how to store values in the Equation Editor variables (section \ref{sec:eqwvar}). Chapter \ref{sec:cas} explains all the symbolic calculation functions contained in the various menus, together with examples of use.\\ You can consult the online help with {\tt SHIFT 2 (SYNTAX)} (cf \ref{sec:syntaxe}), to get help for the other available functions, and you can use {\tt SHIFT {\tt MATH} (CMDS)} (cf \ref{sec:cmds}), to type them. \subsection{The Equation editor menu}\label{sec:eqwmenu} The editor is supplied with a menu bar of menus: \begin{enumerate} \item The {\tt TOOL} menu contains the commands: \begin{itemize} \item {\tt Cursor mode} \index{Cursor mode} enables you to go into cursor mode (cf \ref{sec:curseur}). \item {\tt Edit expr.} \index{Edit expr.} enables you to edit (modify) the highlighted expression. \item {\tt Change font} \index{Change font} enables you to choose to type using large or small characters (you can make this choice at any time). \item {\tt Cut} \index{Cut} copies the selection into the buffer, then erases the selection. \item {\tt Copy} \index{Copy} copies the selection into the buffer. \item {\tt Paste} \index{Paste} copies the buffer to the location of the cursor. The buffer contains, either whatever {\tt Copy} or {\tt Cut} has selected the last time, or the highlighted level when you hit {\tt COPY} in the CAS history toolbar. \end{itemize} \item The ALGB menu contains functions that enable you to perform algebra: factoring, expansion, simplification, substitution... \item The DIFF menu contains functions that enable you to perform differential calculus: differentiation, integration, series expansion, limits... \item The REWRI menu contains functions that enable you to rewrite an expression in another form. \item The SOLV menu contains functions that enable you to solve equations, linear systems, and differential equations. \item The TRIG menu contains functions that enable you to transform trigonometric expressions. \end{enumerate} \subsection{The keyboard in the Equation Editor} \label{sec:eqwkbd} The keys mentioned in this section have different functions depending on whether they are pressed in the {\tt Equation Editor} or in the {\tt HOME} screen. For the functionality of these keys outside the Equation Editor, refer to section \ref{sec:thome}, or consult the User's Guide. \subsubsection{{\tt MATH} key}\label{sec:tmath} The {\tt MATH} key, if pressed in the Equation Editor, displays the functions used in symbolic calculation. These functions are contained in menus: \begin{itemize} \item The five preceding menus (cf \ref{sec:eqwmenu}) :\\ {\tt Algebra\,(ALGB)},~ {\tt Diff\&Int\,(DIFF)}, ~ {\tt Rewrite\,(REWRI)},\\ {\tt Solve\,(SOLV)},~{\tt Trig.\,(TRIG)}. \item The {\tt Complex} menu containing functions that enable you to work with complex numbers \item The {\tt Constant} menu containing${\tt e\ i \ \infty \ \ pi}$. \item The {\tt Hyperb.} menu containing hyperbolic functions \item The {\tt Integer} menu containing functions that enable you to perform integer arithmetic. \item The {\tt Modular} menu containing functions that enable you to perform calculations in $Z/pZ$ or $Z/pZ[X]$, $p$ being the value contained in the variable {\tt MODULO}. \item The {\tt Polynom.} menu containing functions that enable you to perform calculations with polynomials. \item The {\tt Tests} menu containing:\\ {\tt ASSUME}\index{ASSUME} {\tt UNASSUME }\index{UNASSUME} (to make hypotheses about the parameters, and to modify the variable {\tt REALASSUME}\index{REALASSUME} cf \ref{sec:realassume})\\ ${\tt > \ \geq \ < \ \leq \ == \ \neq}$ {\tt AND \ OR \ NOT}\\ {\tt IFTE}\index{IFTE} (to write an algebraic function having the same result as an IF THEN ELSE) \end{itemize} Refer to section \ref{sec:math}, for the list of the functions contained in the various menus. \subsubsection{{\tt SHIFT {\tt MATH} (CMDS)} keys}\label{sec:cmds} This key combination opens the list of all the {\tt CAS} commands available in the {\tt Equation Editor}.\\ In this way, functions that are not presented elsewhere can be called from this menu, so you don't have to type them in {\tt ALPHA} mode. \subsubsection{{\tt VARS} key}\label{sec:vars} This key, if pressed while you're in the {\tt Equation Editor}, displays the names of the variables defined in {\tt CAS}.\\ Take special note of {\tt namVX}, which contains the name of the current variable.\\ To see the contents of a variable, all you have to do is highlight its name and press {\tt F2} for {\tt VIEW} on the menu bar.\\ To change the contents of a variable, highlight its name and press {\tt F3} for {\tt EDIT} on the menu bar.\\ Note also on the menu bar:\\ {\tt PURGE}, which enables you to destroy an existing variable,\\ {\tt RENAME}, which enables you to change the name of an existing variable,\\ {\tt NEW}, which enables you to define a new variable: just enter the contents ({\tt object}), then the ({\tt name}).\\ For more details, refer to section \ref{sec:eqwvar}. \subsubsection{{\tt SHIFT 2 (SYNTAX)} keys}\label{sec:syntaxe}\index{HELP} While you're in the Equation Editor, the key combination {\tt SHIFT 2 (SYNTAX)} opens the {\tt CAS HELP ON} menu.\\ To get help in French, choose Français on the {\tt CFG} menu, which enables you to change your configuration (cf \ref{sec:cfg}).\\ If there is no {\tt CAS} function selected in the editor, the menu shows a list of functions available in the {\tt Equation Editor}. Just highlight a function and press {\tt OK} to see the help for that function.\\ If there is a {\tt CAS} function selected in the editor, for example: \\ {\tt FACTOR(45)}, the {\tt CAS HELP ON} menu directly opens the help topic for {\tt FACTOR}. The help consists of a short description of the command, as well as an example and its result. Each example can be copied into the {\tt Equation Editor} by means of {\tt ECHO} on the menu bar, where it can be used as is or modified.\\ Note that in the help examples, {\tt VX=X} is used as the current variable. If that is not the case for you, the example will be automatically transformed, taking your value of {\tt VX} into account , when you transfer it with {\tt ECHO}.\\ You can also go directly to see the help of a command pointed to by {\tt See:} with {\tt SEE1, SEE2...} on the menu bar. \subsubsection{{\tt SYMB} key}\label{sec:history} Pressing the {\tt SYMB} key in the Equation Editor enables you to access the {\tt CAS} history.\\ The history of the calculations performed in {\tt CAS} differs from the history of the calculations performed in {\tt HOME}.\\ As in the {\tt HOME} screen history, the requested calculations are written on the left, and the results are written on the right. Using the up-arrow, you can move back through the history.\\ Press {\tt COPY} to copy the current CAS history level in the buffer in order to paste it later with {\tt TOOL->Paste} in the Equation Editor.\\ Press {\tt ENTER}, or {\tt ECHO} on the menu bar, to replace the current Equation Editor selection by the current CAS history level.\\ Press {\tt ON} to leave the CAS history without any changes. \subsubsection{{\tt SHIFT SYMB} or {\tt SHIFT HOME} keys}\label{sec:config} While you are working in the Equation Editor, the key combination:\\ {\tt SHIFT SYMB (SETUP)} or {\tt SHIFT HOME (MODES)} open the {\tt CAS MODES} screen to define the CAS configuration and it is similar to {\tt CFG} menu (the first item in the {\tt ALGB} menu on the menu bar, cf \ref{sec:cfg}).\\ This enables you to specify:\\ - The name of the variable contained in {\tt VX}, by typing its name next to {\tt Indep var.},\\ - The value of {\tt MODULO}, by typing its value next to {\tt Modulo},\\ - Whether you want to work in {\tt exact mode} (or in {\tt approximate mode}, if you've chosen {\tt Approx} with {\tt CHK} on the menu bar)\\ - Whether you want to work in real mode (or in complex mode, if you've chosen {\tt Complex} with {\tt CHK} on the menu bar)\\ - Whether you want to work in {\tt Direct} mode (or in {\tt Step by Step} mode, if you've chosen {\tt Step/Step} with {\tt CHK} on the menu bar)\\ - Whether you want polynomials to be written in decreasing order according to exponent (or increasing order, if you've chosen {\tt Incr Pow} with {\tt CHK} on the menu bar)\\ - Whether you want numerical factors suppressed (or not, if you've chosen {\tt Num.Factor} with {\tt CHK} on the menu bar)\\ - Whether you want to work in {\tt non-rigorous} mode (or in {\tt rigorous} mode, if you've chosen {\tt Rigorous} with {\tt CHK} on the menu bar so as not to neglect the absolute values!)\\ - Whether you want to simplify non-rational expressions if you've chosen {\tt Simp Non-Rational} (or not, if you clear the selection by pressing {\tt CHK} on the menu bar).\\ Use {\tt OK} or {\tt ENTER} to confirm your choices. \subsubsection{{\tt SHIFT ,} key}\label{sec:undo} While you are working in the Equation Editor, the key combination:\\ {\tt SHIFT ,(MEMORY)} plays the role of an "undo" key.\\ This is very useful when you've made a mistake, because it enables you to cancel the last command. \subsubsection{{\tt PLOT} key} When you press {\tt PLOT} in the Equation Editor, a dialog box asks you if you want to graph a function, a parametric curve, or a polar curve.\\ Depending on what you choose, the highlighted expression is copied into the appropriate aplet, to the location that you have specified as the destination.\\ {\sc Note} : This supposes that the current variable is also the variable of the function or curve you want to graph: when the expression is copied, it is evaluated, and the current variable (variable contained in {\tt VX}) is changed to {\tt X, T}, or ${\tt \theta}$, depending on the type of plot.\\ {\sc Note} : If the function depends on a parameter, it is preferable to give the parameter a value before pressing {\tt PLOT}. If, however, you want the parameterised expression to be copied with its parameter, then the name of the parameter must consist of a single letter other than {\tt X, T}, or ${\tt \theta}$, so that there is no confusion.\\ If the highlighted expression has real values :\\ the {\tt Function Aplet} or {\tt Polar Aplet} can be chosen, and the graph will be of {\tt Function} or {\tt Polar} type.\\ If the highlighted expression has complex values :\\ the {\tt Parametric Aplet} must be chosen, and the graph will be of {\tt Parametric} type.\\ If you choose: \begin{itemize} \item the {\tt Function Aplet}, the highlighted expression is copied into the chosen function {\tt Fi}, and the current variable is changed to {\tt X} during the copy. \item the {\tt Parametric Aplet}, the real part and the imaginary part of the highlighted expression are copied into the chosen functions {\tt Xi, Yi}, and the current variable is changed to {\tt T} during the copy. \item the {\tt Polar Aplet}, the highlighted expression is copied into the chosen function {\tt Ri}, and the current variable is changed to ${\tt \theta}$ during the copy. \end{itemize} \subsubsection{{\tt NUM} key} Pressing the {\tt NUM} key in the Equation Editor causes the highlighted expression to be replaced by a numeric approximation.\\ {\tt NUM} puts the calculator into approximate mode. \subsubsection{{\tt SHIFT NUM} key} Pressing the {\tt SHIFT NUM} key in the Equation Editor causes the highlighted expression to be replaced by a rationnal number.\\ {\tt SHIFT NUM} puts the calculator into exact mode. \subsubsection{{\tt VIEWS} key} Pressing the {\tt VIEWS} key in the Equation Editor enables you to move the cursor with the $\rhd$ and $\lhd$ arrows to see the entire highlighted expression.\\ Then press {\tt OK} on the menu bar to return in the Equation Editor. \subsubsection{Short-cut keys} Note: In the Equation Editor, the following short-cut keys are available on the keyboard:\\ {\tt SHIFT 0} for ${\tt \infty}$\\ {\tt SHIFT 1} for ${\tt i}$\\ {\tt SHIFT 2} to open the {\tt CAS HELP ON}\\ {\tt SHIFT 3} for ${\tt \pi}$\\ {\tt SHIFT 5} for ${\tt <}$\\ {\tt SHIFT 6} for ${\tt >}$\\ {\tt SHIFT 8} for ${\tt \leq}$\\ {\tt SHIFT 9} for ${\tt \geq}$\\ {\tt SHIFT NUM} to put the calculator into exact mode\\ {\tt NUM} to put the calculator into approximate mode. \subsection{Expressions, subexpressions, selection} \label{sec:eqwsub} When you type expressions in the Equation Editor, the operator that you are typing always carries over to the adjacent or selected expression.\\ You don't have to preoccupy yourself with where the parentheses go... just select!\\ You need to view a mathematical expression as a binary tree, and the four arrow keys as enabling you move through the tree in a natural fashion:\\ The right and left arrow keys enable you to move from one branch to another,\\ The up and down arrow keys enable you to go up and down a particular tree,\\ The SHIFT-up and SHIFT-down arrow keys enable you to make multiple selection (cf page \pageref{sec:exemple3} example 3). \subsubsection{How do you select?}\label{sec:exemple2} There are two ways of going into selection mode : \begin{itemize} \item The up-arrow $\vartriangle$ takes you into selection mode and selects the element adjacent to the cursor.\\ Example: $$1+2+3+4 \ \vartriangle$$ selects 4, then $\vartriangle$ selects the entire tree 1+2+3+4.\\ \item The right-arrow $\rhd$ takes you into selection mode and selects the branch adjacent to the cursor.\\ Pressing $\rhd$ again augments the selection, adding the next branch to the right.\\ Example:\\ $$1+2+3+4 \ \rhd$$ selects 3 + 4, then $\rhd$ selects 2+3+4, then $\rhd$ selects 1+2+3+4. \item {\sc Note}: If you are typing a templated function with multiple arguments (such as $\sum$, $\int$, SUBST, or the like), the right arrow $\rhd$ enables you to move through the template by changing the location of the cursor. In effect, the left and right arrow, keys $\lhd$ and $\rhd$ enable you to move from one argument to another. In this case, you always use the up arrow $\vartriangle$ to select (cf \ref{sec:sum}).\\ \end{itemize} Examples of the way this Equation Editor works:\\ Press {\tt CAS} on the menu bar to open the Equation Editor, then type the expressions in the examples.\\ \begin{itemize} \item Example 1\\ Typing: $${\tt 2\ +\ X\ \times \ 3 \ -\ X }$$ returns: $${\tt 2+X \cdot 3-X}$$ Press $\rhd$ $\rhd$ $\rhd$ to select the expression,\\ then the {\tt ENTER} key to produce: $${\tt 2+2 \cdot X}$$ Typing: $${\tt 2\ +\ X \ \rhd\ \times \ 3 \ -\ X}$$ returns: $${\tt (2+X) \cdot 3-X}$$ Press $\rhd$ $\rhd$ to select the expression,\\ then the {\tt ENTER} key to produce: $${\tt 6+2 \cdot X}$$ Typing: $${\tt 2\ +\ X\ \rhd\ \times \ 3\ \vartriangle\ -\ X }$$ returns: $${\tt (2+X) \cdot (3-X)}$$ Press $\rhd$ $\rhd$ $\rhd$ to select the expression,\\ then the {\tt ENTER} key to produce: $${\tt-(X^2-X-6)}$$ \item Example 2 \\ To express:\\ $${\tt X^2-3 \cdot X+1}$$ type: $${\tt X \ x^y \ 2 \ \rhd \ - \ 3 \ X \ + \ 1 }$$ To express: $${\tt -X^2-3 \cdot X+1}$$ type: $${\tt (-)\ X \ x^y \ 2 \ \rhd \rhd\ - \ 3 \ X \ + \ 1 }$$ In effect, you must select ${\tt -X^2}$ before typing the rest. \item Example 3\label{sec:exemple3}\\ To express: $$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$$ Here, the tree ends in a $+$, and there are four branches; each of these branches ends in a $\div$ and has two leaves. Press {\tt CAS} on the menu bar to open the Equation Editor, then type the first branch: $${\tt1 \div 2} $$ then select this branch with $$ \rhd$$ Then, type $$+$$ and the second branch: $${\tt 1 \div 3}$$ then select this branch with $$ \rhd$$ Then, type $$+$$ and the third branch: $${\tt 1 \div 4}$$ then select this branch with $$ \rhd$$ Then, type $$+$$ and the fourth branch: $${\tt 1 \div 5}$$ then select this branch with $$ \rhd$$ At this point, the desired expression : $${\tt \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}}$$ is in the Equation Editor, and ${\tt \frac{1}{5}}$ is selected. To move back through the tree and select: $${\tt\frac{1}{3}+\frac{1}{4}}$$ type $$ \lhd \ \lhd$$ to select the ${\tt \frac{1}{3}}$, then press $${\tt SHIFT \rhd}$$ which enables you to select two contiguous branches, the one already selected and the one to the right of it, like this: $${\tt \frac{1}{3}+\frac{1}{4}}$$ Advantage : If you want, you can calculate the selected part by pressing {\tt ENTER}.\\ This produces: $${\tt \frac{1}{2}+\frac{7}{12}+\frac{1}{5}}$$ with the $\frac{7}{12}$ selected.\\ If you want to perform the partial calculation $$\frac{1}{2}+\frac{1}{5}$$ you must first perform a permutation so that the ${\tt \frac{1}{2}}$ and the ${\tt \frac{1}{5}}$ are side by side. To do this, type: $${\tt SHIFT \lhd} $$ which exchanges the selected element with its neighbour to the left. This produces: $$ {\tt \frac{7}{12}+\frac{1}{2}+\frac{1}{5}}$$ with the ${\tt \frac{7}{12}}$ still selected. Press : $${\tt \rhd SHIFT \rhd}$$ to select : $${\tt \frac{1}{2}+\frac{1}{5}}$$ Pressing {\tt ENTER} then produces the result.\\ \item Summing up: ${\tt SHIFT \rhd}$ enables you to select the selected element and its neighbour to the right. ${\tt SHIFT \lhd}$ enables you to exchange the selected element with its neighbour to the left. The selected element remains selected, even if you move it. \end{itemize} \subsubsection{How to modify an expression} If you're typing an expression, the {\tt DEL} key enables you to erase what you've typed. If you're selecting, you can:\\ - Cancel the selection without deleting the expression, by typing : $${\tt DEL}$$ The cursor moves to the end of the deselected portion.\\ - Replace the selection with an expression, just by typing the desired expression\\ - Transform the selected expression by applying a {\tt CAS} function to it: you call the function via one of the {\tt CAS} menu options.\\ - Delete the selected expression by typing: $${\tt ALPHA\ SHIFT\ DEL \ \ (ALPHA\ CLEAR)}$$ - Delete a selected unary operator "the top of the tree" by typing: $${\tt SHIFT\ DEL \ \ (CLEAR)} $$ For example, to replace {\tt SIN(expr)} with {\tt COS(expr)}, select {\tt SIN(expr)}, then press {\tt SHIFT\ DEL} and press {\tt COS}.\\ - Delete a binary infix operator and one of its argument by selecting the argument you want delete and typing: $${\tt SHIFT\ DEL \ \ (CLEAR)}$$ For example : if you have the expression 1+2, you select 1, then {\tt SHIFT DEL} deletes 1+ and it leaves only 2.\\ So, to delete {\tt F(X)=} of the expression {\tt F(X)=}${\tt X^2-X+1}$ you select {\tt F(X)}, then press {\tt SHIFT DEL}, and this produces ${\tt X=X^2-X+1}$ (you have deleted the unary operatora {\tt F}), then {\tt SHIFT DEL}, produces ${\tt X^2-X+1}$ (you have deleted {\tt X} and the operator {\tt =}).\\ - Delete a binary operator by editing the expression, you select : $${\tt Edit\ expr.}$$ from the {\tt TOOL} menu on the menu bar, and then make the correction.\\ - Copy an element from the CAS history. You access to the CAS history by pressing {\tt SYMB}. In the history, pressing {\tt ENTER} or selecting {\tt ECHO} on the menu bar inserts the copy where the cursor is, or in place of the selection. You can also use {\tt COPY}, on the history menu bar, to copy the selection in the buffer, or the commands {\tt Cut, Copy and Paste} from the {\tt TOOL} menu on the Equation Editor menu bar to delete, copy and paste expressions as you would with any text editor (cf.\ref{sec:eqwmenu}). \subsubsection{The cursor mode}\label{sec:curseur} The cursor mode enables you to select a large expression quickly. To go into cursor mode, select:\\ Cursor mode in the {\tt TOOL} menu,\\ then use the arrow keys to include your selection in a box (when you release the arrow key, the expression pointed to by the cursor is enclosed).\\ Then, press {\tt ENTER} to select the contents of the box. \subsubsection{To see everything} By selecting {\tt Change font} from the {\tt TOOL} menu on the menu bar, you can increase or decrease the font size of the expression. This enables you to view a large expression in its entirety when you need to.\\ If this is insufficient to see the whole expression, then you'll need to go into cursor mode:\\ select {\tt Cursor mode} in the {\tt TOOL} menu, then use the arrow key $\rhd$, and then use {\tt ENTER} to quit cursor mode,\\ or you can also press:\\ the {\tt VIEWS} key, then use the arrow key $\rhd$, and press {\tt OK} on the menu bar to quit {\tt VIEWS} screen. \subsection{Accessing the {\tt CAS} functions} \label{sec:eqwfun} While you are in the Equation Editor, you can access all {\tt CAS} functions, and you can access them in various ways. General principle:\\ When you have written an expression in the Equation Editor, all you have to do is press {\tt ENTER} to evaluate the selection (or the entire expression, if nothing is selected). \subsubsection{How to type $\int$ and $\sum$}\label{sec:sum} $\sum$ is found on the keyboard, all you have to do is type: $${\tt SHIFT +\ (\sum)}$$ The symbol $\int$ is also found on the keyboard, it can be produced by typing: $${\tt SHIFT \ d/dX \ (\int) }$$ The symbols ${\tt \int}$ and ${\tt \sum}$ are treated as prefix functions with multiple arguments.\\ ${\tt \int}$ and ${\tt \sum}$ are automatically placed before the selected element, if there is one (hence "prefix functions").\\ You can move the cursor with the arrow keys: $${\tt \rhd \ \lhd} $$ Enter the expressions according to the rules of selection explained earlier, but you must first go into selection mode by pressing $\vartriangle$.\\ {\sc Note}: Do not use the index $i$ to define a summation, because $i$ designates the complex-number solution of $x^2+1=0$.\\ $\sum $ performs exact calculations if its argument has a dicrete primitive else $\sum $ performs approximate calculations, even in exact mode. For example in exact, or numeric mode : $$\sum_{k=0}^4 \frac{1}{k!}=2.70833333334$$ whereas in exact mode : $$1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}=\frac{65}{24}$$ the symbol \ $!$ \ is obtained by typing ${\tt SHIFT\ \times}$.\\ Note that $\sum $ can symbolically calculate summations of rational fractions and hypergeometric series that allow a discrete primitive.\\ Example :\\ If you type: $${\tt \sum_{K=1}^4 \frac{1}{K \cdot (K+1)}}$$ then select the entire expression and press {\tt ENTER}, you obtain: $${\tt \frac{4}{5}}$$ If you type: $${\tt \sum_{K=1}^\infty \frac{1}{K \cdot (K+1)}}$$ then select the entire expression and press {\tt ENTER}, you obtain: $${\tt 1}$$ \subsubsection{How to call infix functions}\label{sec:and} These functions are typed between their arguments, for example :\\ {\tt AND | MOD ,} are infix functions.\\ You can either:\\ - type them in {\tt Alpha} mode (using {\tt AND MOD}), then type the arguments, or\\ - call them by selecting a {\tt CAS} menu option or by pressing a key, provided you have already written and selected the first argument.\\ You move from one argument to the other by using the arrow keys $\rhd\ \lhd$.\\ The comma {\tt ,} enables you to write a complex number:\\ when you type ${\tt 1+2 \cdot i}$ or {\tt (1,2)}, the parentheses are automatically placed when you type the comma.\\ If you want to type (-1,2), you must of course select -1 before you type the comma. \subsubsection{How to call prefix functions}\label{sec:prefixe} These functions are typed before their arguments (which is the usual case). To call a prefix function:\\ you can type the first argument, select it, then call the function using a menu,\\ or, you can call the function using a menu or a keystroke in {\tt Alpha} mode, then type the arguments.\\ The following example illustrates the different ways of calling a prefix function.\\ Example:\\ Say you want to factor the expression $x^2-4$, then find its value for $x=4$. You know that {\tt FACTOR} is the function for factoring, and that this function is found in the {\tt ALGB} menu.\\ You also know that {\tt SUBST} is the function for substituting a value for a variable in an expression, and that this function is found in the {\tt ALGB} menu as well. {\bf First possibility: Function call, then arguments}\\ Press the {\tt F2} key to activate the {\tt ALGB} on the menu bar, then highlight {\tt FACTOR} and press {\tt ENTER}.\\ ${\tt FACTOR(\ \blacktriangleleft )}$ is displayed in the editor, with the cursor between the parentheses.\\ Type your expression, using the rules of selection given earlier: $${\tt X \ X^y\ 2\ \rhd - \ 4 \rhd \ \rhd \ \rhd} $$ The following is now selected: $${\tt FACTOR (X^2-4 )}$$ Pressing {\tt ENTER} then produces the result: $${\tt (X+2) \cdot (X-2)}$$ The result is selected, and replaces the command.\\ You do not see this, but after each {\tt ENTER}, the display is written to the history, so in this case, ${\tt \ FACTOR (X^2-4) \ }$ and the answer ${\tt \ (X+2) \cdot (X-2) \ } $ are written to the history (press {\tt SYMB} to see the history and then press {\tt ON} to return to the Equation Editor).\\ At this point, you can erase the preceding result with {\tt ALPHA SHIFT DEL (CLEAR)}, because the result is selected.\\ Press the key that activates {\tt ALGB} on the menu bar, then highlight {\tt SUBST} and press {\tt ENTER}. $${\tt SUBST (\blacktriangleleft ,\bullet)}$$ is displayed in the editor, with the cursor between the parentheses at the l ocation of the first argument.\\ Type your expression, using the rules of selection given earlier:\\ {\sc Note}: Here, {\tt SUBST} has two arguments, so you must go into selection mode using $\vartriangle$ : $${\tt X \ X^y\ 2\ \vartriangle \ \vartriangle\ - \ 4 \rhd X=4\ \rhd \ \rhd} $$ The following is now selected: $${\tt SUBST (X^2-4,X=4 )}$$ Pressing {\tt ENTER} then produces the result: $${\tt 4^2-4}$$ The result is selected, and replaces the command. Pressing {\tt ENTER} then yields the simplified result: $${\tt 12}$$ Naturally, ${\tt \ SUBST (X^2-4,X=4)\ }$, ${\tt \ 4^2-4 \ }$ and ${\tt \ 12}$ are all written to the history.\\ {\sc Remark}:\\ When you call a {\tt CAS} function that has arguments, you can type it in {\tt Alpha} mode with its parentheses. {\bf Second possibility: Arguments, then function call}\\ First of all, type the expression and select it using the rules of selection given earlier.\\ In this case, type: $${\tt X \ X^y\ 2\ \rhd - \ 4 \rhd \ \rhd } $$ Then call {\tt FACTOR}:\\ press the key {\tt F2} that activates {\tt ALGB} on the menu bar, then highlight {\tt FACTOR} and press {\tt ENTER}.\\ This produces: $${\tt FACTOR (X^2-4 )}$$ Pressing {\tt ENTER} then yields the result: $${\tt (X+2) \cdot (X-2)}$$ The result is selected, and replaces the command.\\ Naturally, ${\tt\ FACTOR (X^2-4)\ }$ and ${\tt \ (X+2) \cdot (X-2)\ }$ are both written to the history.\\ Recall now that because your result is selected, you can apply another command to it.\\ At this point, then, call {\tt SUBST}: press the key {\tt F2} that activates {\tt ALGB} on the menu bar, then highlight {\tt SUBST} and press {\tt ENTER}.\\ $${\tt SUBST ((X+2) \cdot (X-2) , \blacktriangleleft )}$$ is displayed in the editor, with your expression as the first argument between the parentheses, and with the cursor at the location of the second argument.\\ All you have to do then is type:\\ ${\tt X=4}$ then, $\rhd \ \rhd$, followed by {\tt ENTER}.\\ This produces: $${\tt (4+2) \cdot (4-2)}$$ Pressing {\tt ENTER} then yields: $${\tt 12}$$ Naturally, ${\tt SUBST (X^2-4,X=4 )}$, ${\tt \ (4+2) \cdot (4-2) \ }$ and ${\tt \ 12}$ are all written to the history.\\ {\sc Remark}:\\ If you call a {\tt CAS} function while you're writing an expression, whatever is currently selected is copied into the function's first or "main" argument. If nothing is selected, the cursor is placed at the appropriate location for completing the arguments. \subsection{Equation editor variables} \label{sec:eqwvar} You can store objects in variables, then access an object by using the name of its variable.\\ {\sc Notes}:\\ 1- Variables used in {\tt CAS} cannot be used in {\tt HOME}, and vice versa.\\ 2- In {\tt HOME} or in the program editor, use ${\tt STO\triangleright}$ (represented here as ${\tt STO\triangleright}$ or ${\tt \triangleright}$ or ->) to store an object in a variable.\\ 3- In {\tt CAS}, use the {\tt STORE} command (cf. \ref{sec:store}) to store a value in a variable.\\ 4- The {\tt VARS} key displays a menu that contains all the available variables.\\ Pressing this key while you are in {\tt HOME} displays the names of the variables defined in {\tt HOME} and in the {\tt Aplets}.\\ Pressing this key while you are in the Equation Editor displays the names of the variables defined in {\tt CAS}. \subsubsection{To store an object in a {\tt CAS} variable : {\tt STORE}}\index{STORE}\label{sec:store} In {\tt CAS}, it is necessary to use the {\tt STORE} command to store an object in a variable, or to use the {\tt VARS} key in the Equation Editor (and then choose {\tt NEW} or {\tt EDIT} on the menu bar; cf. \ref{sec:vars}).\\ All you need to provide is the name of the variable.\\ {\tt STORE} is found in the {\tt ALGB} menu on the Equation Editor menu bar.\\ Example:\\ Type: $${\tt STORE (X^2-4,ABC)}$$ Or, type: $${\tt X^2-4}$$ then select it and call {\tt STORE}, \\ then type {\tt ABC}, then, {\tt ENTER} confirms the definition of the variable {\tt ABC}.\\ To destroy the variable, use the {\tt VARS} key in the Equation Editor (then choose {\tt PURGE} on the menu bar; cf. \ref{sec:vars}), or invoke the {\tt UNASSIGN} \index{UNASSIGN} command on the {\tt ALGB} menu by typing (for example):\\ $${\tt UNASSIGN(ABC)}$$ \subsubsection{Predefined {\tt CAS} variables}\label{sec:realassume} {\tt VX} contains the name of the current symbolic variable.\\ Generally this is {\tt X}, so you should not use {\tt X} as the name of a numeric variable, or erase the contents of {\tt X} with the {\tt UNASSIGN} command in the {\tt ALGB} menu after having done a symbolic calculation by typing, for example, {\tt UNASSIGN(X)}. {\tt EPS} contains the value of epsilon used in the {\tt EPSX0} command (cf. \ref{sec:epsx0}). {\tt MODULO} contains the value of $p$ for performing symbolic calculations in $Z/pZ$ or in $Z/pZ[X]$. You can change the value of $p$ either with the {\tt MODSTO} command in the {\tt MODULAR} menu, (by typing, for example, {\tt MODSTO(13)}\index{MODSTO} to give $p$ a value of 13), or use {\tt SHIFT SYMB} to open the {\tt CAS MODES} screen and change {\tt Modulo}. {\tt PERIOD} must contain the period of a function before you can find its Fourier coefficients (cf. \ref{sec:fourier}). {\tt PRIMIT} contains the primitive of the last integrated function. {\tt REALASSUME}\index{REALASSUME} contains a list of the names of the symbolic variables that are considered reals. If you've chosen the {\tt Cmplx vars} option on the {\tt CFG} configuration menu, these are by default:\\ {\tt X, Y, t, S1, S2}, as well as any integration variables that are in use.\\ Of course, if you've chosen the {\tt Real vars} option on the {\tt CFG} configuration menu, all symbolic variables are considered reals (cf. \ref{sec:cfg}). You can also use an assumption to define a variable such as {\tt X>1}.\\ In a case like this, you use the {\tt ASSUME(X>1)}\index{ASSUME} command to make {\tt REALASSUME} contain {\tt X>1}. The command {\tt UNASSUME(X)}\index{UNASSUME} destroys all the assumptions we've previously made about {\tt X}. To see or to modify all these variables, as well as those that you've defined in {\tt CAS}, press {\tt VARS} in the Equation Editor, then on {\tt EDIT} on the menu bar (cf. \ref{sec:vars}). %%% CAS FROM HOME %%% \section{{\tt CAS} in the {\tt HOME} screen}\label{sec:cashome} You can use many computer algebra functions directly in the {\tt HOME} screen, as long as you take certain precautions. CAS functions that take matrices arguments work only from {\tt HOME}. For the description of these functions see chapter \ref{sec:mtrcas}. \subsection{{\tt HOME} variables} You can not use the same variable names in {\tt HOME} as in the Equation Editor. For example {\tt X} can not be used as a symbolic variable, since the {\tt A..Z} variables names are numeric variables, and always contain a real value (0 by default).\\ The names of the symbolic {\tt HOME} variables are :\\ {\tt S1..S5, s1..s5, n1..n5}.\\ Unlike with Equation Editor variables, it is not possible to assign a value to a {\tt HOME} symbolic variable: note that {\tt s1..s5} and {\tt n1..n5} are sometimes returned by {\tt QUAD} and {\tt ISOLATE} respectively instead of a sign (it should be replaced by $\pm 1$ to get both solutions in {\tt QUAD}, for instance, {\tt QUAD(S1*S1-9,S1)} returns {\tt s1*6/2}) or instead of an integer (e.g. for isolation inside a trigonometric function, for instance, {\tt ISOLATE(SIN(S1),S1)} returns {\tt 3.14..+n1}).\\ %note that {\tt s1..s5} and {\tt n1..n5}are sometimes returned by {\tt QUAD} and {\tt ISOLATE} respectivelyinstead of a sign (it should be replaced by $\pm 1$) or instead of an integer (for instance, {\tt QUAD(S1*S1-9,S1)} returns {\tt s1*6/2} and {\tt ISOLATE(SIN(S1),S1)} returns {\tt 3.14..+n1}).\\ If you want to store symbolic expressions in {\tt HOME} you must use the symbolic expression variables {\tt E0..E9}. If you want to store lists of symbolic expressions in {\tt HOME} you must use the variables {\tt L0..L9}. \subsection{Storing an object in a {\tt HOME} variable : ${\tt STO\triangleright}$}\index{STO} ${\tt STO\triangleright}$ enables you to store an object in a {\tt HOME} variable. ${\tt STO\triangleright}$ is obtained by pressing {\tt F1} on the {\tt HOME} menu bar or on program editor menu bar,. A variable can store only one kind of object, this kind is determined by the variable name (see above).\\ For example, type:\\ ${\tt S1^2+2 \times S1\ STO\triangleright E1}$, then,\\ ${\tt DERIV(E1,S1)}$ \\ you get : \\ ${\tt 2*S1+2}$. \subsection{Some tricks and examples.} CAS functions may be accessed from {\tt HOME}, either from the {\tt CAS} submenu of the {\tt MATH} key's menu bar ({\tt F4} key), or directly (type the function name in alpha mode). Note that: \begin{itemize} \item Certain calculations will be performed in approximate mode because numbers are interpreted as reals instead of integers in {\tt HOME}. To do exact calculations, you should use the command {\tt XQ}, it converts an approximate argument into an exact argument. For example, if you work with {\tt Radians}: $${\tt ARG(XQ(1+i))= \frac{\pi}{4}}$$ $${\tt ARG(1+i)=0.7853...}$$ $${\tt FACTOR(XQ(45))=3^2*5}$$ $${\tt FACTOR(45)=45}$$ \item The symbolic {\tt HOME} variable {\tt S1} serves as the current variable for the relevant {\tt CAS} functions in {\tt HOME}. For example : $${\tt DERVX(S1^2+2 \times S1)= 2*S1+2}$$ The result ${\tt 2*S1+2}$ does not depend on the Equation Editor variable {\tt VX}). Note that a few {\tt CAS} function can not work inside {\tt HOME} because they require to change the current variable. \item Remember that you must use :\\ {\tt S1,S2,..S5,s1,s2,..s5,n1,n2,..n5} for symbolic variables and {\tt E0,E1,..E9} to store symbolic expressions. For example, if you type : $${\tt S1^2-4\times S2\ STO\triangleright E1}$$ then you have : $${\tt DERVX(E1)=S1*2}$$ $${\tt DERIV(E1,S2)=-4}$$ $${\tt INTVX(E1)=1/3*S1^3-4*(S2*S1)}$$ \item If you want to work with symbolic matrices, you must store them in {\tt L0, L1..L9}, because symbolic matrices are interpreted in the same these matrices are interpreted in the same way as lists of lists (whereas numeric matrices are stored in {\tt M0, M1..M9}). The CAS linear algebra instructions accept list of lists as input.\\ For example, if you type in {\tt HOME}: $${\tt XQ(\{ \{ S2+1, 1 \}, \{ \sqrt2,1 \} \}) \ STO\triangleright\ L1}$$ then you have : $${\tt TRAN(L1)=\{\{S2+1,\sqrt2\},\{1,1\}\}}$$ Some numeric linear algebra instructions do not work on list of lists, but will after a conversion by {\tt AXL}, like:\\ ${\tt DET(AXL(L1)) ) \ STO\triangleright\ E1}$\\ you get :\\ ${\tt S2-(-1+\sqrt2)}$ Section \ref{sec:mtrcas} describes CAS instructions that work only from HOME, including some CAS symbolic linear algebra instructions. For more information you can also use the on-line help ({\tt HELP} or e.g. {\tt HELPWITH TRAN}) or look at the HP49G, HP49G+ or HP48GII CAS documentation available from:\\ \verb|http://www-fourier.ujf-grenoble.fr/~parisse|\\ \end{itemize} \subsection{Send expressions from {\tt HOME} into the {\tt CAS} history : {\tt PUSH}}\index{PUSH}\label{sec:push} In the {\tt HOME} screen, you can use the {\tt PUSH} command to send expressions into the {\tt CAS} history.\\ In the {\tt HOME} screen, you type:\\ {\tt PUSH(S1+1)}\\ and {\tt S1+1} is written to the {\tt CAS} history. \subsection{Retrieve expressions written to the {\tt CAS} history in {\tt HOME} : {\tt POP}}\index{POP}\label{sec:pop} In the {\tt HOME} screen, you can use the {\tt POP} command to retrieve the last expression written to the {\tt CAS} history.\\ In the {\tt HOME} screen, you type:\\ {\tt POP}\\ and, for example, {\tt S1+1} is written to the {\tt HOME} screen history and {\tt S1+1} is removed from the {\tt CAS} history. \subsection{The keyboard in the {\tt HOME} screen} \label{sec:thome} \subsubsection{{\tt MATH} key}\label{sec:tmathh} This opens the menu of mathematical functions.\\ This key, if pressed in the {\tt HOME} screen, open a window containing the mathematical (numeric) functions grouped by theme, since the MTH option on the menu bar ({\tt F1} key) is selected by default.\\ If you choose {\tt CAS} on this window's menu bar ({\tt F4} key), you'll find the same menus as when you press the {\tt MATH} key in the Equation Editor: this gives you access to all available computer algebra functions, grouped by theme, from the {\tt HOME} screen. \subsubsection{{\tt SHIFT F6} keys} The key combination {\tt SHIFT F6(CAS on the menu bar)} opens {\tt CAS MODES} screen, which enables you change the {\tt CAS} configuration from the {\tt HOME} screen (cf \ref{sec:config}). \subsubsection{{\tt SHIFT 2 (SYNTAX)} keys}\label{sec:syntax}\index{HELP}\index{HELPWITH} The key combination {\tt SHIFT 2 (SYNTAX)} places {\tt HELPWITH} in the command line. All you need to do then is complete the line with the name of the command or {\tt CAS} function for which you want help. You can enter the name of a {\tt CAS} function with {\tt MATH} and {\tt CAS} on this window's menu bar ({\tt F4} key), but take care to omit the parentheses.\\ For example: {\tt HELPWITH DERVX} opens the {\tt CAS} help topic on {\tt DERVX}.\\ If you want general {\tt CAS} help in the {\tt HOME} screen, press {\tt HELP}, then {\tt ENTER}. This gives you help on the {\tt CAS} functions available in the {\tt HOME} screen.\\ To get help in French, choose Français in the {\tt CFG} menu, which enables you to change your configuration (cf \ref{sec:cfg}).\\ Each example can be copied into the {\tt HOME} screen history by means of {\tt ECHO} on the menu bar, where it can be used as is or modified, and naturally, the variable {\tt X} of the example is replaced automatically by {\tt S1}.\\ In addition, the examples are sometimes modified automatically to change in {\tt HOME} screen, reals to integers using the {\tt XQ} function.\\ For example, you type in the {\tt HOME} screen: $${\tt HELPWITH\ PROPFRAC}$$ or $${\tt HELP} \mbox{ then you select } {\tt PROPFRAC} \mbox{ in } {\tt CAS\ HELP\ ON}$$ %and select {\tt PROPFRAC} on the {\tt CAS HELP ON}\\ You obtain the example for {\tt PROPFRAC} : $${\tt PROPFRAC(XQ(\frac{43}{12}))=3+\frac{7}{12}}$$ whereas if, in the equation editor, you select {\tt PROPFRAC} on the {\tt CAS HELP ON}, the example is : $${\tt PROPFRAC(\frac{43}{12})}$$ but typing in {\tt HOME} : $${\tt PROPFRAC(\frac{43}{12})}$$ gives $${\tt 3.5833..}$$ \chapter{The {\tt CAS} functions only valid in HOME screen}\label{sec:mtrcas} These functions have as arguments matrix or vectors and return matrix or vectors. \section{Linear systems} In this paragraph, we call "augmented matrix" of the system $A \cdot X=B$ (or matrix "representing" the system $A \cdot X=B$), the matrix obtained augmenting the matrix $A$ to the right with the column vector $B$. \subsection{Resolution of a linear system of equations : {\tt REF}}\index{REF} {\tt REF} solves a linear system of equations written in matrix form: $${\tt A\cdot X=B}$$ The argument of {\tt REF} is the augmented matrix of the system (the matrix obtained augmenting matrix {\tt A} to the right with the column vector {\tt B}).\\ The result is a matrix {\tt [A1,B1]} : {\tt A1} has zeros under its principal diagonal, and the solutions of: $${\tt A1\cdot X=B1}$$ are the same as: $${\tt A\cdot X=B}$$ For example, solve the system: $$\left \{\begin{array}{lcr} 3 \cdot x + y & = &-2 \\3 \cdot x +2 \cdot y & =& 2 \end{array}\right.$$\\ Typing: $${\tt REF([[3, 1, -2] [3, 2, 2]])}$$ gives: $$\left [ \begin{array}{rrr} 1 & \frac{1}{3} & \frac{-2}{3}\\0 & 1 & 4 \end{array} \right]$$ So, $y=4$ and $x=-2/3-4/3=-2$ \subsection{Resolution of a linear system of equations : {\tt rref}}\index{rref} {\tt rref} solves a linear system of equations written in matrix form: $${\tt A\cdot X=B}$$ TThe argument of {\tt rref} is the augmented matrix of the system (the matrix obtained augmenting matrix {\tt A} to the right with the column vector {\tt B}).\\ The result is a list containing the list of pivot elements used by the command and a matrix {\tt [A1,B1]} : {\tt A1} has zeros both above and under its principal diagonal, and the solutions of: $${\tt A1\cdot X=B1}$$ are the same as: $${\tt A\cdot X=B}$$ It is interesting to use {\tt rref} in step-by-step mode.\\ For example, to solve the system: $$\left \{ \begin{array}{lcr} 3 \cdot x + y & = &-2 \\3 \cdot x +2 \cdot y & =& 2 \end{array}\right.$$ Typing $${\tt rref([[3, 1, -2][3, 2, 2]])}$$ gives : $${\tt \left\{\{1\ 1\}\left [ \begin{array}{rrr}3 & 0 & -6\\0 & 1 & 4 \end{array} \right]\right\}}$$ Typing $${\tt rref([[1, S1][S1, 1]])}$$ gives : $${\tt \left\{\{S1^2-1 ,1\}\left [ \begin{array}{rr} S1^2-1 & 0\\0 & -(S1^2-1) \end{array} \right]\right\}}$$ \section{Polynomials} \subsection{List of factors : {\tt FACTORS}}\index{FACTORS} {\tt FACTORS} has either a polynomial or a list of polynomials as argument.\\ {\tt FACTORS} returns a list containing the factors of the polynomial and their exponents.\\ Typing: $${\tt FACTORS(S1^2-2*S1+1)}$$ gives: $${\tt \{S1-1,2.\}}$$ Typing: $${\tt FACTORS(S1^4-2*S1^2+1)}$$ gives: $${\tt \{S1-1,2\ ,\ S1+1 ,2\}}$$ Typing: $${\tt FACTORS(\{S1^3-2*S1^2+1,S1^2-S1\})}$$ gives: $${\tt \{\{S1-1,1\ ,\ 2*S1+-1+\sqrt5 ,1\ , \ 2*S1-(1+\sqrt5) ,1\ ,\ 4, -1\},}$$ $${\tt \{S1, 1\ ,\ S1-1,1\}\}}$$ \subsection{Rational function given by its roots and poles :{\tt FCOEF}}\index{FCOEF} {\tt FCOEF} has as argument a vector whose components are the roots and poles of a rational function $F[x]$, followed by their multiplicity.\\ {\tt FCOEF} returns the rational function $F[x]$.\\ Typing: $${\tt FCOEF([1,2,0,3,2,-1]) }$$ gives: $${\tt S1^3*(S1-1)^2/(S1-2)}$$ \subsection{Roots and poles ofa rational function : {\tt FROOTS}}\index{FROOTS} {\tt FROOTS} has a rational function $F[x]$ as argument.\\ {\tt FROOTS} returns a vector whose components are the roots and the poles of $F[x]$, followed by their multiplicity.\\ Typing: $${\tt FROOTS(\frac{S1^5-2 \cdot S1^4+S1^3}{S1-2}) }$$ gives: $${\tt [2,-1,0,3,1,2]}$$ The result means that: $2$ is a pole of order 1, $0$ is a triple root, and $1$ is a double root of $F[x]=\frac{x^5-2 \cdot x^4+x^3}{x-2}$ since $(x-1)^2 \cdot x^3=x^5-2 \cdot x^4+x^3$. \subsection{Gr{\oe}bner basis : {\tt GBASIS}}\index{GBASIS} {\tt GBASIS} has two arguments : a vector of polynomials with several variables and the vector of names of these variables.\\ {\tt GBASIS} returns a Gr{\oe}bner basis of the polynomial ideal built by the polynomials which are in the first argument.\\ We choose to write the polynomial variables with the order of the last argument and to write the polynomials in decreasing power. If $I$ is an ideal and if $(G_k)_{k \in K}$ is a Gr{\oe}bner basis of this ideal $I$ then if $F$ is a non zero polynomial in $I$, the greatest power term of $F$ is divisible by the greatest power term of a $G_k$. \\ Property : If you do an euclidian division of $F$ by its correspondant $G_k$, then if you continue with the remainder of this division, we have at end, a null remainder.\\ Typing : $${\tt GBASIS([2\cdot S1\cdot S2-S2^2,S1^2-2\cdot S1\cdot S2],[S1,S2])}$$ gives : $${\tt [S1^2-S2^2,2\cdot S1\cdot S2-S2^2,S2^3]}$$ \subsection{Gr{\oe}bner reduction : {\tt GREDUCE}}\index{GREDUCE} {\tt GREDUCE} has three arguments : a polynomial with several variables, a vector made with polynomials which are a Gr{\oe}bner basis and a vector of names of these variables.\\ {\tt GREDUCE} returns the reduction (+ a constant) of the polynomial given in the first argument with respect to the Gr{\oe}bner basis given in the second argument.\\ Typing : $${\tt GREDUCE(S1*S2-1,[S1^2-S2^2,2*S1*S2-S2^2,S2^3],[S1,S2])}$$ gives : $${\tt (S2^2-2)/2}$$ that is to say ${\tt Y^2-2}$ is the reduce polynomial of ${\tt 2\cdot X\cdot Y-2}$ (${\tt Y^2-2}$ is the remainder of ${\tt 2\cdot (X\cdot Y-1)}$ by ${\tt G_2=2\cdot X\cdot Y-Y^2}$). \subsection{Lagrange's polynomial : {\tt LAGRANGE}}\index{LAGRANGE} {\tt LAGRANGE} has as argument a matrix with two rows and $n$ columns:\\ the first row corresponds to the abscissa values $x_i$, and the second row corresponds to ordinate values $y_i$ ($i=1..n$).\\ {\tt LAGRANGE} returns the polynomial $P$ of degree $n-1$, so that $P(x_i)=y_i$.\\ Typing: $${\tt LAGRANGE([[1,3],[0,1]])}$$ gives: $${\tt \frac{S1-1}{2}}$$ in fact $\frac{x-1}{2}=0 $ for $x=1$ and $\frac{x-1}{2}=1$ for $x=3$ \subsection{GCD of a list : {\tt LGCD}}\index{LGCD} {\tt LGCD} denotes the gcd (greatest common divisor) of a list of whole numbers or of a list of polynomials.\\ {\tt LGCD} returns the {\tt GCD} of all whole numbers of the list or the {\tt GCD} of all polynomials of the list.\\ Typing: $${\tt LGCD(\{125,45,35 \})}$$ gives: $${\tt 5}$$ Typing: $${\tt LGCD(\{S1^2+2*S1+1,S1^3+1 ,S1^2-1,S1^2+S1 \})}$$ gives : $${\tt S1+1}$$ \subsection{resultant of two polynomials: {\tt RESULTANT}}\index{RESULTANT} {\tt RESULTANT} has two polynomials as arguments.\\ {\tt RESULTANT} returns the resultant of the two polynomials. The resultant of two polynomials is non zero if and only if the polynomials are prime together. It is the determinant of their Sylvester matrix $S$.\\ The Sylvester matrix $S$ of two polynomials $A[X]=\sum_{i=0}^{i=n} a_iX^i$ and $B[X]=\sum_{i=0}^{i=m} a_iX^i$ is a square matrix with $m+n$ rows and columns; its first $m$ rows are made from the coefficients of $A[X]$: $$\left(\begin{array}{ccccccc} s_{11}=a_n & s_{12}=a_{n-1}& \cdots & s_{1(n+1)}=a_0 & 0 & \cdots & 0\\ s_{21}=0 & s_{22}=a_{n}& \cdots & s_{2(n+1)}=a_1 & s_{2(n+2)}=a_0 & \cdots & 0\\ \vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots\\ s_{m1}=0 & s_{m2}=0& \cdots & s_{m(n+1)}=a_{m-1} & s_{m(n+2)}=a_{m-2} & \cdots&a_0 \end{array}\right)$$ and the following $n$ rows are made in the same way from the coefficients of $B[X]$: $$\left(\begin{array}{ccccccc} s_{(m+1)1}=b_m & s_{(m+1)2}=b_{m-1}& \cdots & s_{(m+1)(m+1)}=b_0 & 0 & \cdots & 0\\ \vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots\\ \end{array}\right)$$ For example, typing: $${\tt RESULTANT(S1^3-S2*S1+S3,3*S1^2-S2)}$$ gives: $${\tt -(4*S2^3-27*S3^2)}$$ \subsection{Division by the PGCD :{\tt SIMP2}}\index{SIMP2} {\tt SIMP2} has two whole numbers (or two lists of whole numbers with the same length) as arguments. These two whole numbers are considered as representing a fraction.\\ {\tt SIMP2} returns the simplified fraction, represented as a list of two whole numbers.\\ Typing: $${\tt SIMP2(18,15) }$$ gives: $${\tt \{6,5\}}$$ Typing: $${\tt SIMP2(\{18,28\},\{15,21\}) }$$ gives: $${\tt \{6,5,4,3\}}$$ {\tt SIMP2} has two polynomials (or two lists of polynomials with the same length) as arguments. These two polynomials are considered as representing a rational fraction.\\ {\tt SIMP2} returns the simplified rational fraction, represented as a list of two polynomials.\\ Typing: $${\tt SIMP2(S1^3-1,S1^2-1) }$$ gives: $${\tt \{S1^2+S1+1,S1+1\}}$$ \subsection{Sturm's sequences : {\tt STURM}}\index{STURM} {\tt STURM} has a polynomial $P$ as argument.\\ {\tt STURM} returns a list containing the Sturm's sequences of $P$ and their multiplicities.\\ The Sturm sequence $R_1,R_2,...$ can be obtained from a square-free factor $F$ of $P$ as follows:\\ $R_1$ is the opposite of the remainder of the euclidean division of $F$ by $F'$;\\ then, $R_2$ is the opposite of the remainder of the euclidean division of $F'$ by $R_1$,\\ and so on, until $R_k=0$.\\ For example, type: $${\tt STURM(S1^3+1)}$$ You obtain: $${\tt \{[1],\ -1,\ [S1^3+1,\ -3\cdot S1^2 -1],\ 1.\}}$$ The first element of the list denotes that the denominator of $P$ (that is, the element with $-1$ power), is 1. \subsection{Zeros of a polynomial : {\tt ZEROS}}\index{ZEROS} {\tt ZEROS} has two arguments: a polynomial $P$ and a variable name.\\ {\tt ZEROS} returns a list containing the zeros of $P$ with respect to the given variable, {\sc without} their multiplicity.\\ Type: $${\tt ZEROS(S1^4-1,S1)}$$ gives:\\ - in real mode (change the mode with {\tt SHIFT F6} and {\tt UNCHK Complex}) $${\tt \{-1,1\}}$$ - in complex mode (change the mode with {\tt SHIFT F6} and {\tt CHK Complex}) $${\tt \{ i,-1, -i ,1\}}$$ \section{Rational fractions} \subsection{Numerator and denominator of a fraction : {\tt FXND}}\index{FXND} {\tt FXND} has a rational fraction as argument, and returns a list containing the simplified numerator and denominator of this fraction.\\ Typing: $${\tt FXND(3/6)}$$ gives : $${\tt \{1,2\}}$$ %bug ne simplifie pas Type: $${\tt FXND((S1^2-1)/(S1-1)) }$$ You obtain: %$${\tt \{S1+1,1\}}$$ $${\tt \{S1^2-1,S1-1\}}$$ \section{Modular calculations} \subsection{Resolution of a linear system of equations : {\tt RREFMOD}}\index{RREFMOD} {\tt RREFMOD} solves a system of linear equations $AX=B$ in $Z/pZ$.\\ The argument is the matrix $A$ augmented with the vector $B$ as its rightmost column. The result is a matrix composed of $A1$ and $B1$, where $A1$ has zeros both above and under its principal diagonal, and the system $A1X=B1$ is equivalent to $AX=B$.\\ To solve, in $Z/13Z$ : $$\left\{ \begin{array}{lcr}\ \ x\ +\ \ 2 \cdot y & = &9 \\3 \cdot x+10 \cdot y & =& 0 \end{array}\right. $$ Typing : $${\tt MODSTO(13)}$$ $${\tt RREFMOD([[1, 2, 9][3,10,0]])}$$ gives : $${\tt [[2, 0, 6],[0, 4, -1 ]]}$$ that is, ${\tt 2.X=6} $ and ${\tt 4.Y=-1}$ or, which is the same, ${\tt X=3 \ \ Y=3}$ (since $-4*3=1 \ (\bmod 13)$). \section{Variables} \subsection{The symbolic variables of an expression : {\tt LNAME}}\index{LNAME} {\tt LNAME} has an expression as argument, and returns a vector whose components are the symbolic variable names the expression contains.\\ Typing : $${\tt LNAME(S1*S2*SIN(S1))}$$ gives : $${\tt [S1,S2]}$$ \subsection{The variables of an expression : {\tt LVAR}}\index{LVAR}\label{sec:lvar} {\tt LVAR} has an expression as argument, and returns a vector such that the original expression is a rational fraction with respect to the components of this vector.\\ Typing : $${\tt LVAR(S1*S2*SIN(S1))}$$ gives : $${\tt [SIN(S1),S1,S2]}$$ \section{Numeric calculus} \subsection{Omit the values smaller than EPS : {\tt EPSX0}}\index{EPSX0}\label{sec:epsx0_} {\tt EPSX0} has as argument an expression; in the expression, it replaces all numeric values whose magnitude is smaller than {\tt EPS} with zero and returns the result.\\ Type: $${\tt EPSX0(0.001+S1)}$$ You obtain (when {\tt EPS=0.01}) : $${\tt 0+S1}$$ You obtain (when {\tt EPS=0.0001}) : $${\tt .001+S1}$$ \subsection{Solve numerically a system : {\tt MSLV}}\index{MSLV} {\tt MSLV} solves numerically a system of non-polynomial equations.\\ {\tt MSLV} has three vectors as arguments: a vector containing the equations, a vector containing the system's variables, and a vector containing an initial guess for the solution.\\ {\tt MSLV} returns a vector containing an approximate solution of the given system of equations.\\ While the command is running, the first display line shows the last estimate $\overrightarrow V$, and the second line shows the modulo of $\Delta \overrightarrow V$\\ For example, type: $${\tt MSLV([SIN(S1)+S2,S1+SIN(S2)-1],[S1,S2],[0,0])}$$ You obtain: $${\tt [1.82384112611,\ -.968154636174]}$$ \subsubsection{List of values : {\tt TABVAL}}\index{TABVAL} {\tt TABVAL} has an expression and a list of numbers as arguments.\\ {\tt TABVAL} returns a list. This list holds two lists : the list of numbers given as input and the list of the value the expression where the variable {\tt S1} is replaced by the list of numbers given as input.\\ For example, typing: $${\tt TABVAL(S1^2+S1,\{1,2,3,XQ(\sqrt5\})}$$ gives: $${\tt \{\{1,2,3,\sqrt5\},\{2,6,12,5+\sqrt5\}\}}$$ \subsection{Time for evaluation of the answer : {\tt TEVAL}}\index{TEVAL} {\tt TEVAL} has a command as argument.\\ {\tt TEVAL} returns the list of the answer and the time for the evaluation of this answer (unit=second).\\ For example, typing: $${\tt TEVAL(FACTOR(S1^2-1))}$$ gives: $${\tt \{(S1-1)*(S1+1),0.2991\}}$$ \subsection{Numeric approximation : {\tt XNUM}}\index{XNUM} {\tt XNUM} has either an expression or an array as argument.\\ {\tt XNUM} enables approximate mode and returns the numeric approximation of its argument.\\ Type: $${\tt XNUM(\sqrt2)}$$ You obtain: $${\tt 1.41421356237}$$ \subsection{Exact value : {\tt XQ}}\index{XQ} {\tt XQ} has a real numeric expression as argument.\\ {\tt XQ} enables exact mode and returns either a rational or a real approximation of the expression.\\ Typing: $${\tt XQ(1.41421)}$$ gives: $${\tt \frac{66441}{46981}}$$ Typing: $${\tt XQ(1.414213562)}$$ gives: $${\tt \sqrt2}$$ \section{The lists} \subsection{Concatenation : {\tt AUGMENT}}\index{AUGMENT} {\tt AUGMENT} has as arguments two vectors, or two lists, or a list and an element.\\ {\tt AUGMENT} concatenates its arguments.\\ Typing: $${\tt AUGMENT(\{1,2\},3)}$$ gives: $${\tt \{ 1,2,3 \}}$$ Typing: $${\tt AUGMENT(\{1,2\},\{3,4\})}$$ gives: $${\tt \{ 1,2,3,4 \}}$$ \subsection{Map a function to a list : {\tt MAP}}\index{MAP} {\tt MAP} has two arguments: a list and a function defining an holomorphic operator.\\ {\tt MAP} applies the operator to the list, and returns the result.\\ Typing: $${\tt MAP({1,0,2},EXP(S1)}$$ gives: $${\tt \{EXP(1),EXP(0),EXP(2)\}}$$ \section{Quadratic forms} \subsection{Matrice to quadratic form : {\tt AXQ}}\index{AXQ} {\tt AXQ} has two arguments: a symmetric matrix $A$ representing a quadratic form $q$ and a vector whose components are the quadratic form's variables.\\ {\tt AXQ} returns the quadratic form $q$.\\ Typing : $${\tt AXQ([[0,1],[1,0]]\ ,\ [S1,S2])}$$ gives : $${\tt 2*S2*S1}$$ \subsection{Gauss decompsition : {\tt GAUSS}}\index{GAUSS} {\tt GAUSS} has two arguments: a quadratic form $q$ and a vector whose components are the quadratic form's variables.\\ {\tt GAUSS} returns $q$ expressed as a sum of squares.\\ Typing: $${\tt GAUSS(2 \cdot S1 \cdot S2\ ,\ [S1,S2])}$$ gives: $${\tt -2\cdot(\frac{S2-S1}{2})^2+\frac{1}{2}\cdot (S2+S1)^2}$$ \subsection{Matrix of a quadratic form : {\tt QXA}}\index{QXA} {\tt QXA} has two arguments: a quadratic form $q$ and a vector whose components are the form's variables.\\ {\tt QXA} returns the matrix $A$ associated with $q$.\\ Typing : $${\tt QXA(2*S1*S2\ ,\ [S1,S2])}$$ gives : $${\tt[[0,1],[1,0]]}$$ and with {\tt SHOW} $${\tt \left[\begin{array}{cc} 0& 1\\ 1&0 \end{array}\right]}$$ \subsection{Sylvester decomposition : {\tt SYLVESTER}}\index{SYLVESTER} {\tt SYLVESTER} has one argument: a symmetric matrix representing a quadratic form $q$.\\ {\tt SYLVESTER} returns a list of two elements: the diagonal elements of the diagonal matrix $B$ (obtained expressing $q$ as a sum of squares) and the matrix of change of base $Q$.\\ We have: $$ ^tQ \cdot B \cdot Q=A$$ Typing: $${\tt SYLVESTER([[0,1],[1,0]])}$$ gives : $${\tt \{[\frac{1}{2},-2]\ ,\ \left[\begin{array}{cc} 1& 1\\\frac{-1}{2} &\frac{1}{2} \end{array}\right]\}}$$ \section{Functions of multiple variables} \subsection{The rotor : {\tt CURL}}\index{CURL} Here $n=3$.\\ {\tt CURL} has two arguments: the value [$E1,E2,E3$] of a vectorial function (application from $R^3 $ in $R^3$) and a vector of $R^3$ denoting the variable names.\\ {\tt CURL} returns the rotor of [$E1,E2,E3$]. $${\tt CURL([E1,E2,E3],[S1,S2,S3])=[\frac{\partial E3}{\partial S2}-\frac{ \partial E2}{\partial S3},\frac{\partial E1}{\partial S3}-\frac{\partial E3} {\partial S1},\frac{\partial E2}{\partial S1}-\frac{\partial E1}{\partial S2}]}$$ Type: $${\tt CURL([S1*S3,-S2^2,2*S1*S2],[S1,S2,S3])}$$ You obtain: $${\tt [2*S1,S1-S2*2,0]}$$ \subsection{The gradient : {\tt DERIV}}\index{DERIV}\label{sec:deriv} {\tt DERIV} has two arguments: the value of an application $E1$ from $R^n $ in $R$ and a vector of $R^n$ denoting the variable names.\\ {\tt DERIV} returns the gradient of $E1$ ($[\frac{\partial E1}{\partial S1}, \frac{\partial E1}{\partial S2},\frac{\partial E1}{\partial S3}]$ if $n=3$).\\ Type: $${\tt DERIV(2*S1^2*S2-S1*S3^3,[S1,S2,S3])}$$ After simplification, you obtain: $${\tt [4 \cdot S2 \cdot S1-S3^3,2 \cdot S1^2,-(3 \cdot S1\cdot S3^2 )]}$$ \subsection{The divergence : {\tt DIV}}\index{DIV} {\tt DIV} has two arguments: the value [$E1,E2,E3$] of a vectorial function (application from $R^n $ in $R^n$) and a vector of $R^n$ denoting the variable names.\\ {\tt DIV} returns the divergence of [$E1,E2,E3$]. $${\tt DIV([E1,E2,E3],[S1,S2,S3])=\frac{\partial E1}{\partial S1}+\frac{ \partial E2}{\partial S2}+\frac{\partial E3}{\partial S3}} \mbox{ (here } n=3)$$\\ Typing: $${\tt DIV([S1* S3,-S2^2,2*S1*S2],[S1,S2,S3])}$$ gives: $${\tt S3-2*S2}$$ \subsection{The hessian : {\tt HESS}}\index{HESS} {\tt HESS} has two arguments: the value of an application $E1$ from $R^n $ in $R$ and a vector of $R^n$ denoting the variable names.\\ {\tt HESS} returns a list containing the hessian of $E1$, the gradient of $E1$ and the vector of the variable names.\\ Typing : $${\tt HESS(2*S1^2*S2-S1*S3\ ,\ [S1,S2,S3])}$$ gives : $$ \{ \left[\begin{array}{ccc} {\tt 2 \cdot2\cdot S2}& {\tt 2 \cdot2\cdot S1}&-1 \\ {\tt2\cdot2\cdot S1}&0&0 \\ -1&0&0 \end{array}\right]{\tt \ ,\ [4\cdot S1\cdot S2-S3,2\cdot S1^2,-S1]\ , \ [S1,S2,S3]\}}$$ Now, to obtain the critical points of $E1$, you must type: $${\tt SOLVE( [4.S1.S2-S3,2.S1^2,-S1]\ ,\ [S1,S2,S3])}$$ \subsection{The laplacian : {\tt LAPL}}\index{LAPL} {\tt LAPL} has two arguments: the value of an application $E1$ from $R^n $ in $R$ and a vector of $R^n$ denoting the variable names.\\ {\tt LAPL} returns the laplacian of $E1$ ($\frac{\partial^2 E1}{\partial S1^2}+ \frac{\partial^2 E1}{\partial S2^2}+\frac{\partial^2 E1}{\partial S3^2}$ if $n=3$).\\ Type: $${\tt LAPL(2*S1^2*S2-S1*S3^3\ ,\ [S1,S2,S3])}$$ You obtain: $${\tt 4*S2-6*S1*S3}$$ \section{Equations} \subsection{Members of an equation : {\tt EXLR}}\index{EXLR} {\tt EXLR} has an equation as argument (you have to type {\tt ==}).\\ {\tt EXLR} returns a list containing the left and right hand sides of the equation.\\ Type:\\ $${\tt EXLR(XQ(2\cdot S1+1==3/2))}$$ You obtain: $${\tt \{2\cdot S1+1 ,3/2\}}$$ \section{Matrices} \subsection{Matrix to list : {\tt AXL}}\index{AXL} When {\tt AXL} is given a matrix as argument, it returns the same matrix rewritten as a list of lists. Vice versa, {\tt AXL} transforms a list of lists into a matrix.\\ Typing: $${\tt AXL(XQ([[1,1/2],[3,4]]))}$$ gives: $${\tt \{\{1,1/2\},\{3,4\}\}}$$ Typing: $${\tt AXL(XQ(\{\{1,1/2\},\{3,4\}\}))}$$ gives: $${\tt [[1,1/2],[3,4]]}$$ But typing: $${\tt AXL([[1,1/2],[3,4]])}$$ gives: $${\tt \{\{1,0.5\},\{3,4\}\}}$$ and typing $${\tt AXL(\{\{1,1/2\},\{3,4\}\})}$$ gives: $${\tt [[1,0.5,[3,4]]}$$ \subsection{Symbolic matrix to numeric matrix : {\tt AXM}}\index{AXM} If given a symbolic matrix as argument, {\tt AXM} returns an equivalent (but approximate) numeric matrix.\\ %and vice versa.\\ Type: $${\tt AXM([[1/2,2],[3,4]])}$$ You obtain: $${\tt[[0.5,2],[3,4]]}$$ \subsection{Basis vectors : {\tt BASIS}}\index{BASIS} {\tt BASIS} has as argument a list of vectors, defining a vectorial subspace of $R^n$.\\ {\tt BASIS} returns a list containing the vectors of a basis of the vectorial subspace.\\ For example, type: $${\tt BASIS(\{[1,2,3],[1,1,1],[2,3,4]\})}$$ You obtain: $${\tt \{[1,0,-1],\ [0,1,2]\}}$$ \subsection{Cholesky matrix : {\tt CHOLESKY}}\index{CHOLESKY} {\tt CHOLESKY} has as argument a square matrix {\tt M}, positive by definition.\\ {\tt CHOLESKY} returns an upper triangular matrix {\tt P} so that:\\ ${\tt ^tP*P=M}$\\ Typing: $${\tt CHOLESKY([[1,1],[1,5]])}$$ gives : $${\tt \left[\begin{array}{rr}1 & 1\\0&2\end{array}\right]}$$ \subsection{Map an operator to a diagonalizable matrix : {\tt DIAGMAP}}\index{DIAGMAP} {\tt DIAGMAP} has two arguments: a diagonalizable matrix and an expression defining an holomorphic operator.\\ {\tt DIAGMAP} applies the operator to the matrix, and returns the result.\\ Typing: $${\tt DIAGMAP([[1,1],[0,2]], EXP(S1))}$$ gives: $${\tt \left[\begin{array}{cc}{\tt EXP}(1)& {\tt EXP}(2)-{\tt EXP}(1)\\0&{\tt EXP}(2)\end{array}\right]}$$ \subsection{Hadamard product : {\tt HADAMARD}}\index{HADAMARD} {\tt HADAMARD} has two matrices $A$ and $B$, with the same size, as arguments.\\ {\tt HADAMARD} returns the element-by-element product between $A$ and $B$.\\ Type: $${\tt HADAMARD(XQ([[ 1/2, 2],[3,4]]),XQ([[ 5, 6],[7,8]]))}$$ or $${\tt HADAMARD(XQ(\{\{ 1/2, 2\},\{3,4\}\}),XQ(\{\{ 5, 6\},\{7,8\}\})}$$ You obtain: $${\tt [[ 5/2, 12],[21,32]]}$$ but $${\tt HADAMARD([[ 1/2, 2],[3,4]],[[ 5, 6],[7,8]])}$$ or $${\tt HADAMARD(\{\{ 1/2, 2\},\{3,4\}\},\{\{ 5, 6\},\{7,8\}\}}$$ You obtain: $${\tt [[ 2.5, 12],[21,32]]}$$ \subsection{Hilbert matrix : {\tt HILBERT}}\index{HILBERT} {\tt HILBERT} has an integer $n$ as argument.\\ {\tt HILBERT} returns the square Hilbert matrix of order $n$ whose elements are given by:\\ $$a_{i,j}=\frac{1}{i+j-1}$$ Type: $${\tt HILBERT(4)}$$ You obtain, pressing {\tt ENTER}, then $\uparrow$, and then {\tt SHOW} : $${\tt \left[\begin{array}{cccc} 1&\frac{1}{2}&\frac{1}{3} &\frac{1}{4}\\ \frac{1}{2}&\frac{1}{3}&\frac{1}{4} &\frac{1}{5}\\\frac{1}{3}&\frac{1}{4} &\frac{1}{5} &\frac{1}{6}\\\frac{1}{4}&\frac{1}{5}&\frac{1}{6} &\frac{1}{7}\\ \end{array}\right] }$$ \subsection{Basis of an intersection : {\tt IBASIS}}\index{IBASIS} {\tt IBASIS} has as arguments two lists of vectors, defining two vectorial spaces.\\ {\tt IBASIS} returns a list containing the vectors of a basis of the intersection between these two vectorial spaces.\\ Typing: $${\tt IBASIS(\{[1,2]\},\{[2,4]\})}$$ gives: $${\tt \{[1,2]\}}$$ \subsection{basis of the image : {\tt IMAGE}}\index{IMAGE} {\tt IMAGE} has as argument a matrix representing a linear application $f$ in terms of the standard basis.\\ {\tt IMAGE} returns a list of vectors; they are a basis of the image of $f$.\\ Typing: $${\tt IMAGE([[1,2,3],[2,1,3],[3,1,4]])}$$ gives : $${\tt \{[1,0,-1/3],[0,1,5/3] \}}$$ \subsection{Jordan matrix : {\tt JORDAN}}\index{JORDAN} {\tt JORDAN} has a matrix $A$, of order $n$, as argument.\\ {\tt JORDAN} returns a list composed by the minimal polynomial $M $ of $A$, the characteristic polynomial $P $ of $A$, the list of the eigenvectors and characteristic vectors (each vector is preceded by its characteristic value), and the vector of the $n$ eigenvalues of $A$.\\ Type: $${\tt JORDAN([[4,1,-2],[1,2,-1],[2,1,0]])}$$ or $${\tt JORDAN(\{\{4,1,-2\},\{1,2,-1\},\{2,1,0\}\})}$$ You obtain pressing {\tt ENTER}, then $\uparrow$, and then {\tt SHOW} : \begin{eqnarray*} {\tt \{S1^3-6\cdot S1^2+12\cdot S1-8, S1^3-6\cdot S1^2+12\cdot S1-8,}\\ {\tt \{[1,0,0],[2,1,2],[1,0,1]\}, [2,2,2]\} } \end{eqnarray*} \subsection{Basis of the kernel : {\tt KER}}\index{KER} {\tt KER} has as argument a matrix representing a linear application $f$ in terms of the standard basis.\\ {\tt KER} returns a list of vectors; they are a basis of the kernel of $f$.\\ Typing: $${\tt KER([[1,2,3],[2,1,3],[3,1,4]])}$$ gives: $${\tt \{[1,1,-1] \}}$$ \subsection{Make a matrix : {\tt LCXM}}\index{LCXM} {\tt LCXM} has as arguments two integers, $n$ and $p$, and an expression of {\tt S1} (a row number) and og {\tt S2} (a column number) and yielding the value of $a_{i,j}$.\\ {\tt LCXM} returns a $n \cdot p$ matrix having coefficients $a_{i,j}$. Type: $${\tt LCXM(2,3, S1+S2 )}$$ You obtain: $${\tt [[1+1,1+2,1+3],[2+1,2+2,2+3]]}$$ and then after simplification : $${\tt [[2,3,4],[3,4,5]]}$$ \subsection{List of several elements : {\tt MAD}}\index{MAD} {\tt MAD } has a square matrix $A$, of order $n$, as argument.\\ {\tt MAD } returns a list containing the determinant of $A$, the inverse of $A$, a list containing the matrix coefficients of a polynomial $Q$, and of the characteristic poloynomial $P$ of $A$.\\ We have: $$P(x)=(-1)^n \cdot \det(A-x \cdot I)$$ The polynomial with matrix coefficients $P(A)-P(x) \cdot I$ is divisible by $A-x \cdot I$ (since its value is zero for $x=A$). Let $Q(x)$ be their quotient.\\ Since $P(A)=0$, we have $\ P(A)-P(x) \cdot I\ =\ -P(x) \cdot I\ =\ (A-x \cdot I) \cdot Q(x)$.\\ Therefore, $Q(x)$ is also the co-matrix of $A-x \cdot I$ and the following holds:\\ $Q(x)\ =\ I \cdot x^{n-1}+...+B_0 $, where $B_0$ is the co-matrix of $A$ (with the sign exchanged if $n$ is even!).\\ Type: $${\tt MAD([[ 1, 1/2],[1,3]])}$$ or $${\tt MAD(\{\{ 1, 1/2\},\{1,3\}\})}$$ You obtain: $${\tt \{5/2,[[6/5,-1/5],[-2/5,2/5]],\{[[1,0],[0,1]],[[-3,1/2],[1,-1]]\},S1^2-4*S1+5/2\}}$$ or pressing {\tt ENTER}, then $\uparrow$, and then {\tt SHOW} gives: $${\tt \left\{\frac{5}{2}, \left[ \begin{array}{cc} \frac{6}{5}&-\frac{-1}{5}\\ \frac{-2}{5}&\frac{2}{5}\\ \end{array} \right], \{\left[ \begin{array}{cc} 1&0\\0&1 \end{array} \right], \left[\begin{array}{cc} -3&1/2\\1&-1 \end{array}\right] \right\},S1^2-4 \cdot S1+\frac{5}{2}\}}$$ \subsection{Characteristic polynomial : {\tt PCAR}}\index{PCAR} {\tt PCAR} has a matrix $A$, of order $n$, as argument.\\ {\tt PCAR} returns the characteristic polynomial $P $ of $A$ ($P[x]=(-1)^n \cdot \det(A-x \cdot I))$ Type: $${\tt PCAR([[4,1,-2],[1,2,-1],[2,1,0]])}$$ or $${\tt PCAR(\{\{4,1,-2\},\{1,2,-1\},\{2,1,0\}\})}$$ You obtain: $${\tt S1^3-6*S1^2+12*S1-8}$$ \subsection{Minimal polynomial : {\tt PMINI}}\index{PMINI} {\tt PMINI} has a matrix $A$ as argument.\\ {\tt PMINI} returns another matrix, whose first "zero row" is ended by the minimal polynomial of $A$.\\ For example, type: $${\tt PMINI([[1,0],[0,1]])}$$ In step-by-step mode, you obtain:\\ {\tt L2=L2-L1}\\ $\left[\begin{array}{rrrrr}1&0&0&1&1\\1&0&0 & 1&S1\\1&0&0&1&S1^2\end{array}\right]$\\ {\tt L3=L3-L1}\\ $\left[\begin{array}{rrrrr}1&0&0&1&1\\0&0&0 & 0&S1-1\\1&0&0&1&S1^2\end{array}\right]$\\ {\tt Reduction result}\\ $\left[\begin{array}{rrrrr}1&0&0&1&1\\0&0&0 & 0&S1-1\\0&0&0&0&S1^2-1\end{array}\right]$\\ The row :\\ {\tt 0,0,0,0,S1-1} gives the minimal polynomial of $A$. So, the minimal polynomial of $A$: $\left[\begin{array}{rr}1&0\\0 & 1\end{array}\right]$ is: $${\tt S1-1}$$ \subsection{Q*R decomposition : {\tt qr}}\index{qr} {\tt qr} has a square matrix as argument.\\ {\tt qr} factors the matrix as Q*R, where Q is an orthogonal matrix and R is a triangular matrix.\\ Typing: $${\tt qr([[3, 5],[4,5]])}$$ gives: $${\tt \{\left[\begin{array}{rr}\frac{3}{5} & \frac{4}{5}\\\frac{4}{5}&-\frac{3}{5}\end{array}\right],\left[\begin{array}{rr}5 & 7\\0&1\end{array}\right]\}}$$ \subsection{System to matrix : {\tt SYST2MAT}}\index{SYST2MAT} {\tt SYST2MAT} has two arguments: a vector containing a system of linear equations ({\tt =} or {\tt ==} is not allowed and so {\tt =0} is omitted), and a vector whose elements are the system's variables.\\ {\tt SYST2MAT} rewrites the system in matrix notation, and returns the matrix.\\ For example, type:\\ $${\tt SYST2MAT([S1+S2,S1-S2-2],[S1,S2])}$$ You obtain: $${\tt \left[\begin{array}{rrr}1 & 1&0\\1&-1&-2\end{array}\right]}$$ \subsection{Transposed matrix : {\tt TRAN}}\index{TRAN} {\tt TRAN} has a matrix $A$ as argument.\\ {\tt TRAN} returns the input matrix $A$ transposed.\\ Typing : $${\tt TRAN(XQ([[ 1, 1/2],[3,4]]))}$$ gives: $${\tt [[ 1, 3],[1/2,4]]}$$ Typing : $${\tt TRAN(XQ(\{\{ 1, 1/2\},\{3,4\}\}))}$$ gives: $${\tt \{\{ 1, 3\},\{1/2,4\}\}}$$ But typing : $${\tt TRAN([[ 1, 1/2],[3,4]]))}$$ gives: $${\tt [[ 1, 3],[0.5,4]]}$$ \subsection{Vandermonde matrix : {\tt VANDERMONDE}}\index{VANDERMONDE} {\tt VANDERMONDE} has as argument a vector whose components are denoted by $x_i$ for $i=0..n-1$.\\ {\tt VANDERMONDE} returns the corresponding Vandermonde matrix (the $k$-th row of the matrix is the vector whose components are $x_i^{k}$ for $i=0..n-1$ and $k=0..n-1$).\\ Type: $${\tt VANDERMONDE([1,2,3])}$$ You obtain: $${\tt [[1,1,1],[1,2,4],[1,3,9]] }$$ Suppose that {\tt E1=6}, {\tt E2=2*S1+S1*S2} and then type: $${\tt VANDERMONDE([E1,E2,E3])}$$ You obtain pressing {\tt ENTER}, then $\uparrow$, and then {\tt SHOW}: $${\tt \left[\begin{array}{ccc} 1& 6&36\\ {\tt 1}&{\tt 2\cdot S1+S1\cdot S2}&{\tt (2\cdot S1+S1\cdot S2)^2}\\{\tt 1}&{\tt E3}&{\tt E3^2} \end{array}\right]}$$ %\section{Other matrix functions} \section{The permutations} Permutations are defined with the image list $\{P(1),P(2)...P(n)\}$.\\ For example, definining the permutation $P$ as $P =\{3,2,1\}$ means that:\\ $P(1)=3,\ P(2)=2,\ P(3)=1$.\\ A cycle is denoted with a list containing the images of an element through the cycle; for example, defining a cycle $C$ as $C=\{3,2,1\}$ means that:\\ $C(3)=2,\ C(2)=1,\ C(1)=3$.\\ Accordingly, a decomposition into cycles is denoted with a list of lists.\\ \subsection{Cycles to permutation : {\tt C2P}}\index{C2P} {\tt C2P} has a list of cycles as argument.\\ {\tt C2P} returns the permutation having the input list as decomposition into cycles (see also {\tt P2C}).\\ Typing: $${\tt C2P(\{\{1,3,5\},\{2,4\}\})}$$ gives: $${\tt\{3,4,5,2,1\}}$$ \subsection{Composition : {\tt CIRC}}\index{CIRC} {\tt CIRC} has two permutations as arguments.\\ {\tt CIRC} returns the permutation obtained by composition: ($1^{\mbox{st}}\mbox{argument} \circ 2^{\mbox{nd}} \mbox{argument}$).\\ Typing: $${\tt CIRC(\{3,4,5,2,1\},\{2,1,4,3,5\})}$$ gives : $${\tt \{4,3,2,5,1\}}$$ \subsection{Decomposition into cycles\tt P2C}\index{P2C} {\tt P2C} has a permutation as argument.\\ {\tt P2C} returns its decomposition into cycles and its signature.\\ Typing: $${\tt P2C(\{3,4,5,2,1\})}$$ gives : $${\tt \{\{\{1,3,5\},\{2,4\}\},-1\}}$$ \section{The isometries} \subsection{Isometry's characteristics elements : {\tt ISOM}}\index{ISOM} {\tt ISOM} has as argument a matrix representing a two or three-dimensional linear isometry.\\ {\tt ISOM} returns the list of the isometry's characteristics elements and either {\tt +1} (for direct isometries) or {\tt -1} (for indirect isometries).\\ For example, type: $${\tt ISOM([[0,0,1],[0,1,0],[1,0,0]])}$$ You obtain: $${\tt \{[1,0,-1], -1\}}$$ This result means that the isometry is a simmetry with respect to the plane $x\ -\ z\ =\ 0$.\\ If you type: $${\tt ISOM(\frac{\sqrt2}{2}\cdot \left[\begin{array}{rr}1 & -1\\1&1\end{array}\right])}$$ You obtain: $${\tt \{ACOS(\sqrt(1/2),1\}}$$ therefore, this isometry is a rotation of $ \displaystyle \frac{\pi}{4}$ radians. \subsection{Isometry's matrix : {\tt MKISOM}}\index{MKISOM} In a three-dimensional space, {\tt MKISOM} has the list of the characteristics elements of an isometry, and either {\tt +1} (denoting a direct isometry) or {\tt -1} (denoting an indirect isometry) as arguments,\\ In a two-dimensional space, {\tt MKISOM} has the characteristic element of an isometry (either an angle or a vector) and either {\tt +1} (denoting a direct isometry) or {\tt -1} (denoting an indirect isometry) as arguments.\\ {\tt MKISOM} returns the matrix representing the given isometry.\\ For example, type: $${\tt MKISOM(\{[-1,2,-1],\pi \},1)}$$ You simplify the answer typing :\\ {\tt XQ(} then $\uparrow$ and {\tt COPY}, then {\tt ) ENTER}\\ this gives the matrix of a rotation with axis $[-1,2,-1]$ and angle $\pi$ : $${\tt \left[\begin{array}{rrr}\frac{-2}{3} & \frac{-2}{3}&\frac{1}{3}\\\frac{-2}{3}&\frac{1}{3}&\frac{-2}{3}\\\frac{1}{3}&\frac{-2}{3}&\frac{-2}{3}\end{array}\right]}$$ For example, type: $${\tt MKISOM(\{\pi \},-1)}$$ You obtain the matrix of a symmetry with respect to the origin: $${\tt [[-1,0,0],[0,-1,0],[0,0,-1]]}$$ Type: $${\tt MKISOM(\{[1,1,1],\frac{\pi}{3} \},-1)}$$ You obtain (after simplification) the matrix of a rotation with axis $[1,1,1]$ and angle $\frac{\pi}{3}$ combined with a symmetry with respect to the plane $x+y+z=0$: $${\tt \left[\begin{array}{rrr}0 & -1&0\\0&0&-1\\-1&0&0\end{array}\right]}$$ Type: $${\tt MKISOM( \frac{\pi}{2},1)}$$ You obtain the matrix of a rotation of $\frac{\pi}{2}$ radians in two dimensions: $${\tt [[0, -1],[1,0]]}$$ (because ${\tt COS(1/2*\pi)=0}$ and ${\tt SIN(1/2*\pi)=1}$)\\ Type: $${\tt MKISOM([1,1],-1)}$$ You obtain the matrix of a symmetry with respect to $y=x$ in two dimensions: $${\tt [[0, -1],[-1,0]]}$$ %%% CAS FUNCTIONS %%% \chapter{{\tt CAS} Reference from the Equation Editor}\label{sec:cas} When you press on {\tt CAS} in the {\tt HOME} menu bar, you open the Equation Editor in wich you can do symbolic calculus. In this chapter, we describe the computer algebra functions as used from the Equation Editor (see section \ref{sec:cashome} and chapter \ref{sec:mtrcas} if you want to use {\tt CAS} functions from {\tt HOME}). \section{{\tt CAS} menu bar} The Equation Editor has a menu bar : it is the {\tt CAS} menu bar.\\ {\tt TOOL}, {\tt ALGB}, {\tt DIFF}, {\tt REWRI}, {\tt SOLV} and {\tt TRIG} compose the {\tt CAS} menu bar.\\ Only the {\tt TOOL} menu contains commands, the other menus contain the algebraic functions that can be written in {\tt Alpha} mode and have on its first line the sub-menu {\tt CFG} enable you to change your configuration. \subsection{Change your configuration : {\tt CFG}}\index{CFG}\label{sec:cfg} All the menus, except {\tt TOOL} display the state of your configuration and enable you to change it.\\ For example, say that you see the following on the first line of a menu: $${\tt CFG :\ R\ =\ X \ S}$$ This means that :\\ (1) you are in exact-real mode,\\ (2) {\tt X} is the current variable, and \\ (3) you are working in Step by step mode ({\tt S}).\\ If you highlight {\tt CFG} and press {\tt OK}, a menu is displayed with this at its head: $${\tt CFG :\ R\ =\ STEP\ \uparrow\ X \ 13\ ||}$$ This means that :\\ (1) you are in exact-real mode,\\ (2) Step by step mode is selected,\\ (3) polynomials are written with their terms in ascending order by exponent,\\ (4) {\tt X} is the current variable,\\ (5) modular calculations are carried out in {\tt Z/13Z} ($p = 13$), and\\ (6) you are working in Rigorous mode (that is, using absolute values).\\ You can change this configuration by selecting any of the following:\\ {\tt Quit cfg} (when you're finished making changes)\\ {\tt Complex} (or {\tt Real})\\ {\tt Approx} (or {\tt Exact})\\ {\tt Direct} (or {\tt Step/Step} if you want to work in Step by Step mode)\\ ${\tt 1 + x + x^2...}$ (or ${\tt ...x^2 + x +1}$) how polynomials will appear,\\ {\tt Sloppy} (or {\tt Rigorous}, if you want to work in absolute values)\\ {\tt Num. factor} (or {\tt Symb factor})\\ {\tt Cmplx vars} (or {\tt Real vars} if you want all symbolic variables to be treated as reals cf. \ref{sec:realassume})\\ {\tt English} (or {\tt Français} if you want the line-based help to be in French)\\ {\tt Default cfg} (configuration ${\tt R\ =\ STEP\ \downarrow\ X \ 13\ ||}$.\\ Press {\tt OK} to validate each of your choices.\\ Pressing {\tt CANCEL} takes you out of the {\tt CFG} menu (as does choosing {\tt quit cfg} and confirming it with {\tt OK}).\\ The name of the current variable contained in {\tt VX}, as well as the value of the variable {\tt MODULO}, can be changed by means of the {\tt SHIFT SYMB (SETUP)} keystroke, or by using the {\tt VARS} key (see \ref{sec:config} and \ref{sec:vars}).\\ {\sc Remarks} :\\ - You can also change your configuration with the {\tt SHIFT SYMB (SETUP)} (or {\tt SHIFT HOME (MODES)}) keystrokes pressed in the Equation Editor (see section \ref{sec:config}).\\ - In {\tt CAS}, angles are always expressed in {\tt Radians}.\\ When you are in {\tt HOME} screen, you can use the {\tt MODES} menu (the {\tt SHIFT HOME} keystrokes) to open the {\tt HOME MODES} screen and to change {\tt ANGLE MEASURE} or open the {\tt CAS MODES} screen with the {\tt SHIFT F6(CAS on the menu bar)} to change your CAS configuration. \subsection{{\tt Step/Step} Mode} You choose Step by Step mode ({\tt Step/Step}, abbreviated {\tt S}) when you want to see the details of the calculations.\\ The details of the calculations are displayed on the screen and you can view the next step by pressing {\tt OK}.\\ When the screen is not big enough to display all the information, directional arrows $\blacktriangledown$ and $\blacktriangle$ appear on the edge of the screen. You can then scroll the screen to see more information by using the arrow keys $\triangledown \ \vartriangle$.\\ If you do not need to see the details of the calculations, choose {\tt Direct} mode (abbreviated {\tt D}). \subsection{{\tt TOOL} menu} The functions contained in the {\tt TOOL} menu are described in section \ref{sec:eqwmenu}.\\ {\tt Cursor mode}\index{Cursor mode}\\ {\tt Edit expr.}\index{Edit expr.}\\ {\tt Change font}\index{Change font}\\ {\tt Cut}\index{Cut}\\ {\tt Copy}\index{Copy}\\ {\tt Paste}\index{Paste} \subsection{{\tt ALGB} menu} \noindent{\tt COLLECT}\index{COLLECT}\\ {\tt DEF}\index{DEF}\\ {\tt EXPAND}\index{EXPAND}\\ {\tt FACTOR}\index{FACTOR}\\ {\tt PARTFRAC}\index{PARTFRAC}\\ {\tt QUOTE}\index{QUOTE}\\ {\tt STORE}\index{STORE}\\ {\tt |}\index{$\mid$}\\ {\tt SUBST}\index{SUBST}\\ {\tt TEXPAND}\index{TEXPAND}\\ {\tt UNASSIGN}\index{UNASSIGN} \subsection{{\tt DIFF} menu} \noindent{\tt DERIV}\index{DERIV}\\ {\tt DERVX}\index{DERVX}\\ {\tt DIVPC}\index{DIVPC}\\ {\tt FOURIER}\index{FOURIER}\\ {\tt IBP}\index{IBP}\\ {\tt INTVX}\index{INTVX}\\ {\tt lim}\index{LIMIT}\\ {\tt PREVAL}\index{PREVAL}\\ {\tt RISCH}\index{RISCH}\\ {\tt SERIES}\index{SERIES}\\ %\tt SIGNTAB}\index{SIGNTAB}\\ {\tt TABVAR}\index{TABVAR}\\ {\tt TAYLOR0}\index{TAYLOR0}\\ {\tt TRUNC}\index{TRUNC} \subsection{{\tt REWRI} menu} \noindent{\tt DISTRIB}\index{DISTRIB}\\ {\tt EPSX0}\index{EPSX0}\\ {\tt EXPLN}\index{EXPLN}\\ {\tt EXP2POW}\index{EXP2POW}\\ {\tt FDISTRIB}\index{FDISTRIB}\\ {\tt LIN}\index{LIN}\\ {\tt LNCOLLECT}\index{LNCOLLECT}\\ {\tt POWEXPAND}\index{POWEXPAND}\\ {\tt SINCOS}\index{SINCOS}\\ {\tt SIMPLIFY}\index{SIMPLIFY}\\ {\tt XNUM}\index{XNUM}\\ {\tt XQ}\index{XQ} \subsection{{\tt SOLV} menu} \noindent{\tt DESOLVE}\index{DESOLVE}\\ {\tt ISOLATE}\index{ISOLATE}\\ {\tt LDEC}\index{LDEC}\\ {\tt LINSOLVE}\index{LINSOLVE}\\ {\tt SOLVE}\index{SOLVE}\\ {\tt SOLVEVX}\index{SOLVEVX} \subsection{{\tt TRIG} menu} \noindent{\tt ACOS2S}\index{ACOS2S}\\ {\tt ASIN2C}\index{ASIN2C}\\ {\tt ASIN2T}\index{ASIN2T}\\ {\tt ATAN2S}\index{ATAN2S}\\ {\tt HALFTAN}\index{HALFTAN}\\ {\tt SINCOS}\index{SINCOS}\\ {\tt TAN2CS2}\index{TAN2CS2}\\ {\tt TAN2SC}\index{TAN2SC}\\ {\tt TAN2SC2}\index{TAN2SC2}\\ {\tt TCOLLECT}\index{TCOLLECT}\\ {\tt TEXPAND}\index{TEXPAND}\\ {\tt TLIN}\index{TLIN}\\ {\tt TRIG}\index{TRIG}\\ {\tt TRIGCOS}\index{TRIGCOS}\\ {\tt TRIGSIN}\index{TRIGSIN}\\ {\tt TRIGTAN}\index{TRIGTAN} \subsection{{\tt The MATH} key}\label{sec:math} It contains, in addition to the previously described menus ({\tt Algebra}, {\tt Diff\&Int}, {\tt Rewrite}, {\tt Trig.}, {\tt Solve}) we have :\\ {\tt Complex} ({\tt i}\index{i} {\tt ABS}\index{ABS} {\tt ARG}\index{ARG} {\tt CONJ}\index{CONJ} {\tt DROITE}\index{DROITE} {\tt IM}\index{IM} {\tt -}\index{-} {\tt RE }\index{RE} {\tt SIGN}\index{SIGN})\\ {\tt Constant} (${\tt e\ i \ \infty \pi}$)\\ {\tt Hyperb.} ({\tt ACOSH ASINH ATANH COSH SINH TANH})\\ {\tt Integer} ({\tt DIVIS}\index{DIVIS} {\tt EULER}\index{EULER} {\tt FACTOR}\index{FACTOR} {\tt GCD}\index{GCD} {\tt IDIV2}\index{IDIV2} {\tt IEGCD}\index{IEGCD} {\tt IQUOT}\index{IQUOT} {\tt IREMAINDER}\index{IREMAINDER} {\tt ISPRIME?}\index{ISPRIME?} {\tt LCM}\index{LCM} {\tt MOD}\index{MOD} {\tt NEXTPRIME}\index{NEXTPRIME} {\tt PREVPRIME}\index{PREVPRIME})\\ {\tt Modular} ({\tt ADDTMOD}\index{ADDTMOD} {\tt DIVMOD}\index{DIVMOD} {\tt EXPANDMOD}\index{EXPANDMOD} {\tt FACTORMOD}\index{FACTORMOD} {\tt GCDMOD}\index{GCDMOD} {\tt INVMOD}\index{INVMOD} {\tt MODSTO}\index{MODSTO} {\tt MULTMOD}\index{MULTMOD} {\tt POWMOD}\index{POWMOD} {\tt SUBTMOD}\index{SUBTMOD})\\ {\tt Polynom.} ({\tt EGCD}\index{EGCD} {\tt FACTOR}\index{FACTOR} {\tt GCD}\index{GCD} {\tt HERMITE}\index{HERMITE} {\tt LCM}\index{LCM} {\tt LEGENDRE}\index{LEGENDRE} {\tt PARTFRAC}\index{PARTFRAC} {\tt PROPFRAC}\index{PROPFRAC} {\tt PTAYL}\index{PTAYL} {\tt QUOT}\index{QUOT} {\tt REMAINDER}\index{REMAINDER} {\tt TCHEBYCHEFF}\index{TCHEBYCHEFF})\\ {\tt Reals} ({\tt CEILING}\index{CEILING} {\tt FLOOR}\index{FLOOR} {\tt FRAC}\index{FRAC} {\tt INT}\index{INT} {\tt MAX}\index{MAX} {\tt MIN}\index{MIN})\\ {\tt Tests} ({\tt ASSUME}\index{ASSUME} {\tt UNASSUME}\index{UNASSUME} ${\tt > \ \geq \ < \ \leq \ == \ \neq}$ {\tt AND OR NOT} {\tt IFTE}\index{IFTE})\\ Refer to sections \ref{sec:eqwmenu} and \ref{sec:tmath} for descriptions of these menus. \section{Integers (and Gaussian Integers)} All the functions in this section are found in the {\tt MATH} key's {\tt Integer} menu, except {\tt IABCUV ICHINREM PA2B2}, which is in the {\tt SHIFT MATH (CMDS)} key's menu.\\ For certain functions, you can use Gaussian integers (numbers of the form $a+ib$, where $a$ and $b$ are integers) in the place of integers. \subsection{Unlimited precision} The calculator can manage integers with unlimited precision, such as the following: $$100!$$ \indent The symbol \ $!$ \ is obtained by typing ${\tt SHIFT\ \times}$\\ The decimal value of $100!$ is very large, but you can view it by using the {\tt VIEWS} key. \subsection{The divisors of a number : {\tt DIVIS}}\index{DIVIS} {\tt DIVIS} gives the divisors of a number.\\ Typing : $${\tt DIVIS(12) }$$ gives: $${\tt 12 \ OR\ 6\ OR\ 3\ OR\ 4\ OR\ 2\ OR\ 1 }$$ {\sc Note} : ${\tt DIVIS(0)=0 \ OR\ 1}$ \subsection{The Euler index : {\tt EULER}}\index{EULER} {\tt EULER} returns the Euler index for a whole number.\\ {\tt EULER(n)} is equal to the number of whole numbers less than $n$ and prime with $n$.\\ Typing: $${\tt EULER(21)}$$ gives: $${\tt 12}$$ In other words:\\ E=\{2,4,5,7,8,10,11,13,15,16,17,19\} is the set of whole numbers less than 21 and prime with 21. There are 12 members of the set, so Cardinal(E)=12. \subsection{Decomposition into prime factors : {\tt FACTOR }}\index{FACTOR} {\tt FACTOR} decomposes an integer into its prime factors.\\ Typing: $${\tt FACTOR(90)}$$ gives: $${\tt 2\cdot3^2\cdot5}$$ \subsection{The greatest common divisor : {\tt GCD}}\index{GCD} {\tt GCD} returns the greatest common divisor of two integers.\\ Typing: $${\tt GCD(18,15) }$$ gives: $${\tt 3}$$ In Step by step mode, typing: $${\tt GCD(78,24) }$$ gives: \\ ${\tt 78\ \bmod \ 24 \ = \ 6}$\\ ${\tt 24\ \bmod \ 6 \ = \ 0}$\\ {\tt Result 6}\\ Pressing {\tt ENTER} then causes {\tt 6} to be written to the Equation Editor. %Nouveau \subsection{Resolution of $au+bv=c$ : {\tt IABCUV}}\index{IABCUV} {\tt IABCUV(A,B,C)} returns {\tt U \ AND \ V} so that {\tt AU+BV=C}.\\ It is of course necessary that {\tt C} must be a multiple of {\tt GCD(A,B)} to obtain a solution.\\ Typing : $${\tt IABCUV(48,30,18) }$$ gives : $${\tt 6 \ AND\ -9}$$ %Nouveau \subsection{Chinese remainders :{\tt ICHINREM}}\index{ICHINREM} {\tt ICHINREM(A AND P,B AND Q)} returns {\tt C AND R}.\\ The numbers ${\tt X=C+k \cdot R}$ where ${\tt k}\in \mathbb Z$ are such that :\\ ${\tt X=A \bmod P}$ and ${\tt X=B \bmod Q}$.\\ A solution {\tt X} always exists when {\tt P} and {\tt Q} are mutually prime, ({\tt GCD(P,Q)=1}) and in this case, all the solutions are congruent modulo ${\tt R=P \cdot Q}$.\\ Typing : $${\tt ICHINREM(7 \ AND\ 10,12 \ AND\ 15)}$$ gives : $${\tt -3 \ AND\ 30}$$ that is to say that the solutions are such that :\\ ${\tt X=-3 \bmod 30}$\\ {\sc Example} : \\ Gives the solutions of : $${\tt \left \{ \begin{array}{rl} X=&3\ (\bmod\ 5)\\ X=&9\ (\bmod\ 13) \end{array}\right.}$$ Typing : $${\tt ICHINREM(3 \ AND\ 5,9 \ AND\ 13)}$$ gives : $${\tt -147 \ AND\ 65}$$ that is to say that the solutions are such that :\\ ${\tt X=-147 \bmod 65}$ %Nouveau \subsection{Quotient and remainder of two integers : {\tt IDIV2}}\index{IDIV2} {\tt IDIV2} returns the quotient {\tt AND} the remainder of the Euclidean division between two integers.\\ Typing: $${\tt IDIV2(148,5) }$$ gives: $${\tt 29\ AND\ 3}$$ In step-by-step mode, the calculator shows the division process like it is taught at school. The division is carried out as if in longhand, using the so-called "gallows" algorithm. \subsection{Bézout's Identity : {\tt IEGCD}}\index{IEGCD} {\tt IEGCD(A,B)} returns the value of Bézout's Identity for two integers.\\ In other words, {\tt IEGCD(A,B)} returns {\tt U AND V = D}, with {\tt U, V, D} such that:\\ {\tt AU+BV=D} and {\tt D=GCD(A,B)}.\\ Typing: $${\tt IEGCD(48,30) }$$ gives $${\tt 2 \ AND\ -3=6}$$ In other words: $2 \cdot 48+ (-3) \cdot 30 =6$ and gcd(48,30)=6\\ In Step by step mode, we get:\\ {\tt [z,u,v]:z=u*48+v*30}\\ {\tt [48,1,0]}\\ {\tt [30,0,1]*-1}\\ {\tt [18,1,-1]*-1 }\\ {\tt [12,-1,2]*-1}\\ {\tt [6,2,-3]*-2}\\ {\tt Result : [6,2,-3]}\\ puis {\tt ENTER} ou {\tt OK},\\ $${\tt 2 \ AND\ -3=6}$$ to be written to the Equation Editor. \subsection{The integer Euclidean quotient : {\tt IQUOT}}\index{IQUOT}\label{sec:iquot} {\tt IQUOT} returns the integer quotient of the Euclidean division of two integers.\\ Typing: $${\tt IQUOT(148,5)}$$ gives: $${\tt 29}$$ In Step by step mode, the division is carried out as if in longhand: $$\begin{array}{rcl} 148 & | &\ 5\\ 48 & |&---\\ 3 &| &29 \end{array}$$ Press {\tt OK} to execute the division step by step, then press {\tt ENTER} to write the result {\tt 29} to the Equation Editor. \subsection{The integer remainder : {\tt IREMAINDER}}\index{IREMAINDER} {\tt IREMAINDER} returns the integer remainder from the Euclidean division of two integers.\\ {\tt IREMAINDER} is found in the {\tt MATH} key's Integer menu.\\ Typing: $${\tt IREMAINDER(148,5)}$$ gives: $${\tt 3}$$ {\tt IREMAINDER} works with integers or with Gaussian integers, which is what distinguishes it from {\tt MOD}.\\ {\sc Example} $${\tt IREMAINDER(2+3 \cdot i,1+i)}$$ returns {\tt i}\\ Try calculating: $${\tt IREMAINDER(148!,5!+2 )}$$ (The symbol $!$ is obtained by typing ${\tt SHIFT \ \times}$). In Step by step mode, the division is carried out as if in longhand, using the so-called "gallows" algorithm (see \ref{sec:iquot} for an example). \subsection{Test of pseudo-primality : {\tt ISPRIME?}}\index{ISPRIME?} {\tt ISPRIME?(N)} returns {\tt 1.} (TRUE) if {\tt N} is a pseudo-prime, and {\tt 0.} (FALSE) if {\tt N} is not prime.\\ {\sc Definition}: For numbers less than $10^{14}$, pseudo-prime and prime mean the same thing. But for numbers greater than $10^{14}$, a pseudo-prime is a number with a large probability of being prime (cf. Rabin's Algorithm, section 7.6).\\ Typing: $${\tt ISPRIME?(13) }$$ gives: $${\tt 1.}$$ Typing: $${\tt ISPRIME?(14) }$$ gives: $${\tt 0.}$$ \subsection{The least common multiple : {\tt LCM}}\index{LCM} {\tt LCM} returns the least common multiple of two integers.\\ Typing: $${\tt LCM(18,15) }$$ gives: $${\tt 90}$$ \subsection{Remainder of the Euclidean division : {\tt MOD}}\index{MOD} {\tt MOD} is an infix function that has two integers as arguments.\\ {\tt MOD} returns the remainder of the Euclidean division of the arguments.\\ Typing: $${\tt 3\ MOD\ 2}$$ produces the result: $${\tt 1}$$ {\tt MOD} accepts real numbers ($7.5 \ \bmod \ 2.1 \ =\ 1.2$), but not Gaussian integers (cf section \ref{sec:mod}). \subsection{The smallest pseudo-prime greater than {\tt N} : {\tt NEXTPRIME}}\index{NEXTPRIME} {\tt NEXTPRIME(N)} returns the smallest pseudo-prime greater than {\tt N}.\\ Typing: $${\tt NEXTPRIME(75)}$$ gives: $${\tt 79}$$ %Nouveau \subsection{Resolution of $a^2+b^2=p$ with integers: {\tt PA2B2}}\index{PA2B2} {\tt PA2B2} decompose a prime integer $p$ congruent to 1 modulo 4, as follows: $p= a^2+b^2$. \\ The calculator gives the result as $a+b \cdot i$\\ Typing : $${\tt PA2B2(17)}$$ gives : $${\tt 4+i }$$ that is, $17=4^2+1^2$\\ Typing : $${\tt PA2B2(29)}$$ gives : $${\tt 5+2 \cdot i }$$ that is, $29=5^2+2^2$ \subsection{The greatest pseudo-prime less than {\tt N} : {\tt PREVPRIME}}\index{PREVPRIME} {\tt PREVPRIME(N)} returns the greatest pseudo-prime less than {\tt N}.\\ Typing: $${\tt PREVPRIME(75)}$$ gives: $${\tt 73}$$ \section{Modular Calculations} All the functions in this section are found in the {\tt MATH} key's {\tt Modular} menu except {\tt DIV2MOD}, which is in the {\tt SHIFT MATH (CMDS)} key's menu.\\ You can carry out calculations in modulo {\tt p} that is, in ${\tt Z/pZ}$  or in ${\tt Z/pZ[X]}$.\\ {\sc Note}: For some commands, {\tt p} must be prime.\\ {\sc All the examples in this section use a value for {\tt p} of {\tt 13}}.\\ It's assumed that you have already typed: $${\tt MODSTO(13)}$$ or that you have used the {\tt SHIFT SYMB (SETUP)} keystroke to switch {\tt MODULO} to {\tt 13}. The representation is the symetric representation (-5 instead of 8 modulo 13). \subsection{Addition in Z/pZ or in Z/pZ[X] : {\tt ADDTMOD}}\index{ADDTMOD} {\tt ADDTMOD} performs an addition in $Z/pZ$.\\ Typing: $${\tt ADDTMOD(2,18)}$$ gives: $${\tt -6}$$ {\tt ADDTMOD} performs an addition in  $Z/pZ[X]$.\\ Typing: $${\tt ADDTMOD(11X+5,8X+6)}$$ gives: $${\tt 6X-2}$$ %Nouveau \subsection{Quotient and remainder in Z/pZ[X]: {\tt DIV2MOD}}\index{DIV2MOD} The arguments of {\tt DIV2MOD} are two polynomials {\tt A[X]} and {\tt B[X]}. The result is a {\tt AND} containing both the quotient and the remainder of the euclidean division of {\tt A[X]} by {\tt B[X]} in $Z/pZ[X]$.\\ Typing: $${\tt DI2VMOD(X^3+X^2+1,2\cdot X^2+4)}$$ you obtain in $Z/13Z$ : $${\tt -(6X+6)\ AND\ -(2X+1)}$$ since $X^3+X^2+1=(2\cdot X^2+4) \cdot \frac{X+1}{2}+\frac{5\cdot X-4}{4}$\\ $2*(-6)=1 \bmod 13$ and \\ $4*(-3)=1 \bmod 13$ \subsection{Division in Z/pZ or in Z/pZ[X] : {\tt DIVMOD}}\index{DIVMOD} The arguments are two integers $A$ and $B$. When $B$ has an inverse in $Z/pZ$, the result is $\frac{A}{B}$ simplified as $Z/pZ$, else there is an error message.\\ Typing: $${\tt DIVMOD(5,3)}$$ gives: $${\tt 6}$$ The arguments are two polynomials $A[X]$ and $B[X]$. The result is a rationale fraction $\frac{A[X]}{B[X]}$ simplified as $Z/pZ[X]$.\\ Typing: $${\tt DIVMOD(2X^2+5,5X^2+2X-3)}$$ gives: $${\tt -\frac{4X+5}{3X+3}} $$ \subsection{To expand and simplify expressions in  Z/pZ or in Z/pZ[X] : {\tt EXPANDMOD}}\index{EXPANDMOD} {\tt EXPANDMOD} has as an argument a integer expression.\\ {\tt EXPANDMOD} expands this expression in $Z/pZ$.\\ Typing: $${\tt EXPANDMOD(2*3+5*4)}$$ gives: $${\tt 0}$$ {\tt EXPANDMOD} has as an argument a polynomial expression.\\ {\tt EXPANDMOD} expands this expression in $Z/pZ[X]$.\\ Typing: $${\tt EXPANDMOD(( 2X^2+12).( 5X-4))}$$ gives: $${\tt -(3X^3-5X^2+5X-4)}$$ \subsection{To factor polynomials in Z/pZ[X] : {\tt FACTORMOD}}\index{FACTORMOD} {\tt FACTORMOD} has as an argument a polynomial.\\ {\tt FACTORMOD} factors this polynomial in $Z/pZ[X]$, providing that ${\tt p \leq 97}$ and $p$ is prime.\\ Typing: $${\tt FACTORMOD(-(3X^3-5X^2+5X-4))}$$ gives: $${\tt -((3X-5)(X^2+6))}$$ \subsection{GCD of the two polynomials in Z/pZ[X] : {\tt GCDMOD}}\index{GCDMOD} {\tt GCDMOD} has two polynomials as arguments.\\ {\tt GCDMOD} calculates the GCD of the two polynomials in $Z/pZ[X]$.\\ Typing: $${\tt GCDMOD(2X^2+5,5X^2+2X-3)}$$ gives: $${\tt -(6X-1)}$$ \subsection{Inverse in Z/pZ : {\tt INVMOD}}\index{INVMOD} {\tt INVMOD} has as an argument an integer.\\ {\tt INVMOD} calculates the inverse of the integer in $Z/pZ$.\\ Typing: $${\tt INVMOD(5)}$$ gives (since $5\times-5=-25=1\ (\bmod\ 13)$) : $${\tt -5}$$ \subsection{To put a value in the {\tt MODULO} variable : {\tt MODSTO}}\index{MODSTO} You use the {\tt MODSTO} command to set the value of the {\tt MODULO} variable {\tt p}.\\ {\sc The examples in this section all use a value for {\tt p} of {\tt 13}} that is, they assume that you have already typed: $${\tt MODSTO(13)}$$ \subsection{Multiplication in Z/pZ or Z/pZ[X] : {\tt MULTMOD}}\index{MULTMOD} {\tt MULTMOD} performs a multiplication in $Z/pZ$.\\ Typing: $${\tt MULTMOD(11,8)}$$ gives: $${\tt -3}$$ {\tt MULTMOD} performs a multiplication in $Z/pZ[X]$.\\ Typing: $${\tt MULTMOD(11X+5,8X+6)}$$ gives: $${\tt -(3X^2-2X-4)}$$ \subsection{Power in Z/pZ or in Z/pZ[X] : {\tt POWMOD}}\index{POWMOD} {\tt POWMOD(A,N)} calculates {\tt A} to the power of {\tt N} in $Z/pZ[X]$, and {\tt POWMOD(A(X),N)} calculates {\tt A(X)} to the power of {\tt N} in $Z/pZ[X]$.\\ The {\tt MODULO} variable {\tt p} must be a prime number less than 100.\\ Typing: $${\tt POWMOD(11,195)}$$ gives if ${\tt p=13}$: $${\tt 5}$$ In effect: $11^{12}=1 \ \bmod \ 13$ so $11^{195}=11^3=5 \ \bmod \ 13$\\ Typing: $${\tt POWMOD(2X+1,5)}$$ gives: $${\tt 6 \cdot X^5+2 \cdot X^4+2 \cdot X^3+X^2-3 \cdot X+1}$$ since:\\ $10=-3 \ (\bmod\ 13) \ \ 40=1\ (\bmod\ 13)\ \ 80=2 \ (\bmod\ 13)$\\ $ 32=6\ (\bmod\ 13)$. \subsection{Subtraction in Z/pZ or in Z/pZ[X] : {\tt SUBTMOD}}\index{SUBTMOD} {\tt SUBTMOD} performs a subtraction in $Z/pZ$. \\ Typing: $${\tt SUBTMOD(29,8)}$$ gives: $${\tt -5}$$ {\tt SUBTMOD} performs a subtraction in $Z/pZ[X]$. \\ Typing: $${\tt SUBTMOD(11X+5,8X+6)}$$ gives: $${\tt 3X-1}$$ \section{Rational Numbers} Calculate: $$\frac{123}{12}+\frac{57}{21}$$ After you press {\tt ENTER}, you get the result: $$\frac{363}{28}$$ If you now invoke the {\tt XNUM}\index{XNUM} function on the {\tt REWRITE} menu, or if you press the {\tt NUM} key, you get the following result: $$12.9642857143$$ If you mix the two representations, for example: $$\frac{1}{2}+0\ .\ 5$$ the calculator asks to go into {\tt approx} mode to do the calculation. After responding {\tt yes}, you get: $$1.$$ You'll then need to go back into exact mode ({\tt CFG}, etc... or by typing {\tt SHIFT NUM}). %Nouveau \subsection{The $n$-th Bernoulli's number : {\tt IBERNOULLI}}\index{IBERNOULLI} {\tt IBERNOULLI} has as an integer $n$ as argument.\\ {\tt IBERNOULLI} returns the $n^{th}$ Bernoulli's number $B(n)$.\\ We have : $$\frac{t}{e^t-1}=\sum_{n=0}^{+\infty} \frac{B(n)}{n!}t^n$$ Remember that the Bernoulli's polynomials $B_k$ are defined as : $$B_0=1$$ $$B_k{'}(x)=kB_{k-1}(x)$$ $$\int_0^1B_k(x)dx=0$$ Then, the Bernoulli's numbers are defined as:\\ $B(n)=B_n(0)$\\ Typing : $${\tt IBERNOULLI(6)}$$ gives : $${\tt \frac{1}{42}}$$ \subsection{Whole-number and fractionnal part : {\tt PROPFRAC}}\index{PROPFRAC} {\tt PROPFRAC} is found in the {\tt MATH} key's {\tt POLYNOMIAL} menu.\\ ${\tt PROPFRAC(\frac{A}{ B})}$ writes the fraction $\frac{A}{B}$ in the form: $$Q+\frac{R}{B}\ \ avec \ \ 0\leq R 0$, {\tt TCHEBYCHEFF} returns the polynomial $T_n$ such that: $$T_n[x]= \cos(n \cdot \arccos(x))$$ For $n \geq 0$, we have: $$T_n(x)=\sum_{k=0}^{[n/2]}C_n^{2k}(x^2-1)^kx^{n-2k}$$ For $n \geq 0$ we also have: $$(1-x^2)T_n^{''}(x)-xT_n^{'}(x)+n^2T_n(x)=0$$ For $n \geq 1$, we have: $$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$$ If n < 0, {\tt TCHEBYCHEFF} returns the 2nd-species Tchebycheff polynomial: $$T_n[x]=\frac{\sin(n \cdot \arccos(x))}{\sin(\arccos(x))}$$ Typing: $${\tt TCHEBYCHEFF(4)}$$ gives: $${\tt 8 \cdot X^4-8 \cdot X^2+1}$$ In effect:\\ $\cos( 4 \cdot x)=Re((\cos(x)+i \cdot \sin(x))^4)$\\ $\cos( 4 \cdot x)=\cos(x)^4-6 \cdot \cos(x)^2 \cdot (1-\cos(x)^2)+((1-\cos(x)^2)^2$\\ $\cos(4 \cdot x)=T_4(\cos(x))$ Typing: $${\tt TCHEBYCHEFF(-4)}$$ gives: $${\tt 8 \cdot X^3-4 \cdot X}$$ In effect:\\ $\sin(4 \cdot x)=\sin(x) \cdot (8 \cdot \cos(x)^3-4 \cdot \cos(x))$. \section{Functions} All the functions in this section can be found in the {\tt DIFF} menu on the menu bar, except {\tt DEF}, which is in the {\tt ALGB} menu, and {\tt IFTE}, which is in the {\tt MATH} key's Tests menu. \subsection{Define a function : {\tt DEF}}\index{DEF} {\tt DEF} has as argument an equality between (1) the name of a function (with parentheses containing the variable), and (2) an expression defining the function.\\ {\tt DEF} defines this function and returns the equality.\\ Typing: $${\tt DEF(U(N)=2^N+1)}$$ produces the result: $${\tt U(N)=2^N+1}$$ Typing: $${\tt U(3)}$$ then returns: $${\tt 9}$$ \subsubsection{Another {\tt DEF} example}\index{DEF} Calculate the first six Fermat numbers $F_1..F_6$ and say whether they're prime.\\ So, you want to calculate $F(k)=2^{2^k}+1$ for $k = 1..6$.\\ Typing the formula : $$2^{2^2}+1$$ gives a result of 17. You can then invoke the {\tt ISPRIME?()} \index{ISPRIME?} command, which is found in the {\tt MATH} key's {\tt Integer} menu.\\ The response is {\tt 1.}, which means {\tt TRUE}. Using the history (which you access by pressing the {\tt SYMB} key), you put the expression $2^{2^2}+1$ into the Equation Editor with {\tt ECHO}, and change it to: $$2^{2^3}+1$$ Or better, define a function F(K) by selecting {\tt DEF} from the {\tt ALGB} menu (on the menu bar), and typing: $${\tt DEF(F(K)=2^{2^K}+1)}$$ The response is ${\tt 2^{2^K}+1}$, and {\tt F} is now listed amongst the variables (which you can verify using the VARS key.)\\ For $K = 5$, you then type: $${\tt F(5)}$$ which gives: $${\tt 4294967297}$$ You can factor $F(5)$ with {\tt FACTOR}, which you'll find in the {\tt ALGB} menu on the menu bar.\\ Typing: $${\tt FACTOR(F(5))}$$ gives $$641 \cdot 6700417$$ Typing: $${\tt F(6)}$$ gives: $$18446744073709551617$$ Using {\tt FACTOR} to factor it then yields: $$274177\cdot67280421310721$$ {\sc Note}: Pay careful attention to the position of the dot in : $$2\ .\ 5 \verb|the approximate representation of| \frac{5}{2}$$ versus $$2\cdot5=10$$ \subsubsection{Define a function with conditions : {\tt IFTE}}\index{IFTE} {\tt IFTE} has three arguments: a Boolean (note the use of {\tt = =} to test for equality), and two expressions $expr1,\ expr2$.\\ {\tt IFTE} evaluates the condition, then returns $expr1$ if the condition is true, or $expr2$ if the condition is false.\\ Typing: $${\tt STORE(2,N)}$$ $${\tt IFTE(N==0,1,\frac{N+1}{N})}$$ produces the result: $$ {\tt \frac{3}{2}}$$ It is easy to define functions using {\tt IFTE}. For example:\\ $${\tt DEF(F(X)=IFTE(X==0,1,\frac{SIN(X)}{X}))}$$ defines the function $f$ such that:\\ $$f(x)=\left\{ \begin{array}{rl} 1 &\mbox{si } x=0\\ \displaystyle \frac{\sin(x)}{x}&\mbox{si } x \neq 0 \end{array}\right.$$ \subsection{Derivative and partial derivative : {\tt DERIV}}\index{DERIV} {\tt DERIV} has two arguments: an expression (or a function) and a variable.\\ {\tt DERIV} returns the derivative of the expression (or the function) with respect to the variable given as the second parameter (used for calculating partial derivatives).\\ Example:\\ Calculate: $$\frac {\partial (x \cdot y^2 \cdot z^3+x \cdot y)}{\partial z}$$ Typing: $${\tt DERIV(X \cdot Y^2 \cdot Z^3+X \cdot Y\ ,\ Z)}$$ gives: $${\tt 3 \cdot X \cdot Y^2 \cdot Z^2}$$ \subsection{Derivative : {\tt DERVX}}\index{DERVX} {\tt DERVX} has one argument: an expression.\\ {\tt DERVX} calculates the derivative of the expression with respect to the variable stored in {\tt VX}.\\ For example, given: $$f(x)=\frac {x}{x^2-1}+\ln(\frac {x+1}{x-1})$$ Calculate the derivative of $f$.\\ Type: $${\tt DERVX(\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))} $$ or, if you have stored the definition of $f(x)$ in {\tt F} that is, if you have typed: $${\tt STORE(\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}),F)}$$ then type:\\ $${\tt DERVX(F) }$$ or, if you have defined $F(X)$ using {\tt DEF} that is, if you have typed: $${\tt DEF(F(X)=\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))}$$ then type: $${\tt DERVX(F(X)) }$$ The result is a complicated expression. Pressing {\tt ENTER} simplifies it, giving: $${\tt -\frac {3 \cdot X^2-1}{X^4-2 \cdot X^2+1}} $$ \subsection{Fourier coefficients : {\tt FOURIER}}\index{FOURIER}\label{sec:fourier} {\tt FOURIER} has two parameters: an expression $f(x)$ and a whole number $N$.\\ {\tt FOURIER} returns the Fourier coefficient $c_N$ of $f(x)$, considered to be a function defined over interval $[0,T]$ and with period $T$ ($T$ being equal to the contents of the variable {\tt PERIOD}).\\ If $f(x)$ is a discrete series, then: $$f(x)=\sum_{N=-\infty}^{+\infty} c_N e^\frac{2iNx\pi}{T}$$ Example: \\ Determine the Fourier coefficients of a periodic function $f$ with period $2\cdot \pi$ and defined over interval $[0\ 2 \cdot \pi[$ by $f(x) = x^2$.\\ Typing: $${\tt STORE(2 \cdot \pi\ , \ PERIOD)}$$ $${\tt FOURIER(X^2,N)}$$ The calculator do not know that {\tt N} is a whole number, so you have to replace ${\tt EXP(2*i*N*\pi)}$ with 1 and then simplify the expression. We get: $${\tt \frac{2 \cdot i \cdot N \cdot \pi+2}{N^2}}$$ \noindent So if $N \neq 0$, then: $$c_N=\frac{2 \cdot i \cdot N \cdot \pi+2}{N^2}$$ Typing: $${\tt FOURIER(X^2,0)}$$ gives: $${\tt \frac{4 \cdot {\pi}^2}{3} }$$ so if $N = 0$, then: $$c_0=\frac{4 \cdot {\pi}^2}{3}$$ \subsection{Partial integration : {\tt IBP}}\index{IBP} {\tt IBP} has two parameters: an expression of the form $u(x) \cdot v'(x)$ and $v(x)$.\\ {\tt IBP} returns the {\tt AND} of $u(x) \cdot v(x)$ and of $-v(x) \cdot u'(x)$ that is, the terms that one must calculate when one performs a partial integration.\\ It remains then to calculate the integral of the second term of the {\tt AND}, then add it to the first term of the {\tt AND} to obtain a primitive of $u(x) \cdot v'(x)$.\\ Typing: $${\tt IBP(LN(X),X) }$$ gives: $${\tt X \cdot LN(X)\ AND \ -1}$$ One completes the integration by calling {\tt INTVX}: $${\tt INTVX(X \cdot LN(X) AND -1)}$$ which produces the result: $${\tt X\cdot LN(X)-X}$$ {\sc Remark}: If the first {\tt IBP} (or {\tt INTVX}) parameter is an {\tt AND} of two elements, {\tt IBP} concerns itself only with the second element of the {\tt AND}, adding the integrated term to the first element of the {\tt AND} (so that you can perform multiple {\tt IBP} in succession). \subsection{Integrals : ${\tt \int}$}\index{$\int$} To calculate an integral, you can use the symbol $\int$ which is found on the keyboard and it can be produced by typing : {\tt SHIFT d/dX}.\\ You have just to enter the arguments (you can move the cursor with the arrow keys $\rhd$ and $\lhd$). For more details cf. \ref{sec:sum}.\\ You can put $+\infty$ or $-\infty$ as limit of the integral.\\ {\sc Example}\\ For the following expression, determine the limit when a approaches +(: $$ \int _2^a (\frac {x}{x^2-1}+\ln(\frac {x+1}{x-1}))\ dx$$ In the Equation Editor, type:\\ $${\tt \int _2^{+\infty} (\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))\ dX} $$ {\sc Note}: To obtain the symbol ${\tt {+\infty}}$, type: $${\tt {(-) \ (-) \ \infty}\ (SHIFT\ 0)\ }$$ This produces: $${\tt +\infty-\frac{7 \cdot LN(3)}{2} }$$ and, after simplification: $$+\infty$$ %Nouveau \subsection{Primitive and definied integral : {\tt INTVX}}\index{INTVX} {\tt INTVX} has one argument : une expression.\\ {\tt INTVX} calculates a primitive of its argument with respect to the variable stored in {\tt VX}.\\ Exercise 1\\ Calculate a primitive of $\sin(x) \times \cos(x)$.\\ Typing: $${\tt INTVX(SIN(X) \cdot COS(X))}$$ gives, in Step by step mode: $${\tt COS(X) \cdot SIN(X)}$$ $${\tt Int[u'*F(u)]\ with\ u=SIN(X)}$$ Pressing {\tt OK} then sends the result to the Equation Editor: $${\tt \frac{SIN(X)^2}{2}}$$ Exercise 2\\ Given: $$f(x)=\frac {x}{x^2-1}+\ln(\frac {x+1}{x-1})$$ Calculate a primitive of $f$.\\ Type: $${\tt INTVX(\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))} $$ Or, if you have stored the definition of $f (x)$ in {\tt F}, that is, if you have already typed: $${\tt DEF(F(X)=\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))}$$ then type: $${\tt INTVX(F)} $$ Or, if you have used {\tt DEF} to define {\tt F(X)}, that is, if you have already typed: $${\tt DEF(F(X)=\frac {X}{X^2-1}+LN(\frac {X+1}{X-1}))}$$ then type: $${\tt INTVX(F(X)) }$$ The result in all cases is equivalent to: $$ {\tt X \cdot LN(\frac {X+1}{X-1})+ \frac{3}{2} \cdot LN(|X-1|)+\frac{3}{2} \cdot LN(|X+1|)} $$ We obtain absolute values only in {\tt Rigourous} mode.\\ Exercise 3\\ Calculate: $$\int \frac {2}{x^6+2 \cdot x^4+x^2} \ dx $$ Typing: $${\tt INTVX(\frac {2}{X^6+2 \cdot X^4+X^2})}$$ gives a primitive: $${\tt -3 \cdot ATAN(X)-\frac {2}{X}-\frac {X}{X^2+1}} $$ {\sc Remark}:\\ You can also type: $${\tt \int_1^X \frac {2}{X^6+2 \cdot X^4+X^2} \ dX} $$ which gives the primitive which is zero for $X=1$: $${\tt -3\cdot ATAN(X)-\frac {2}{X}-\frac {X}{X^2+1}+\frac{3\cdot \pi+10}{4}}$$ Exercise 4\\ Calculate:\\ $$\int \frac {1}{\sin(x)+\sin(2 \cdot x )} \ dx $$ Typing: $${\tt INTVX(\frac {1}{SIN(X)+SIN(2 \cdot X )})}$$ gives the result: $${\tt \frac {1}{6}\cdot LN(|COS(X)-1|)+\frac {1}{2}\cdot LN(|COS(X)+1|)+}$$ $${\tt \frac {-2}{3} \cdot LN(|2\cdot COS(X)+1|)} $$ {\sc Remark}: If the argument to {\tt INTVX} is the {\tt AND} of two elements, {\tt INTVX} concerns itself only with the second element of the {\tt AND}, and adds the result to the first argument. \subsection{To calculate limits : {\tt LIMIT} or {\tt lim}}\index{LIMIT}\index{lim}\label{sec:limit} {\tt LIMIT} or {\tt lim} has two arguments: an expression dependent on a variable, and an equality (a variable = the value to which you want to calculate the limit).\\ You can omit the name of the variable and the sign {\tt =}, when this name is in {\tt VX}).\\ It is often preferable to use a quoted expression:\\ {\tt QUOTE(expression)}\index{QUOTE}, to avoid rewriting the expression in normal form (i.e., not to have a rational simplification of the arguments) during the execution of the {\tt LIMIT} command.\\ Typing, for example: $${\tt LIMIT(QUOTE((2X-1) \cdot EXP(\frac{1}{X-1})),X=+\infty)}$$ gives: $$+\infty$$ To find a right limit you type, for example: $${\tt LIMIT(\frac{1}{X-1},QUOTE(1+0))}$$ gives (if {\tt X} is the current variable) : $$+\infty$$ To find a leftt limit you type, for example: $${\tt LIMIT(\frac{1}{X-1},QUOTE(1-0))}$$ gives (if {\tt X} is the current variable) : $$-\infty$$ Ii is not necessary to quote the second argument, when it is written with {\tt =} for example :\\ $${\tt LIMIT(\frac{1}{X-1},X=1+0)}$$ gives : $$+\infty$$ {\sc Exercises}\\ For $n > 2$ in the following expression, find the limit as $x$ approaches 0: $$ \frac{n\cdot \tan(x)-\tan(n\cdot x)}{\sin(n\cdot x)-n\cdot \sin(x)}$$ You can use the {\tt LIMIT} command to do this.\\ Typing: $${\tt LIMIT \left( \frac{N \cdot TAN(X)-TAN(N \cdot X)}{SIN(N \cdot X)-N \cdot SIN(X)}, 0 \right)}$$ gives: $${\tt 2 }$$ {\sc Note} To find the limit as $x$ approaches $a^+$ (resp $a^-$), the second argument is written :\\ ${\tt X=A+0}$ (resp ${\tt X=A-0}$), see also \ref{sec:limit}.\\ For the following expression, find the limit as $x$ approaches $+\infty$ : $$\sqrt{x+\sqrt{x+\sqrt x}}-\sqrt x$$ Typing: $${\tt LIMIT (\sqrt{X+\sqrt{X+\sqrt{X}}}-\sqrt{X}, +\infty)} $$ produces (after a short wait): $$\frac{1}{2}$$ {\sc Note}: \\ The symbol $\infty$ is obtained by typing {\tt SHIFT 0}.\\ To obtain ${\tt {-\infty}}$ type: $${\tt (-)\ \infty}$$ To obtain ${\tt {+\infty}}$ type:\\ $${\tt (-) \ (-)\ \infty}$$ You can also find the symbol ${\tt \infty}$ in the {\tt MATH} key's {\tt Constant} menu. \subsection{Plot of an expression : {\tt PLOT}}\index{PLOT} {\tt PLOT} has an expression as argument.\\ {\tt PLOT} asks you to select an {\tt Aplet} and then stores this expression in an {\tt Aplet} variable.\\ For example, type: $${\tt PLOT(X^2+X)}$$ You select the {\tt Aplet} {\tt Function}, and you choose to save the expression ${\tt X^2+X}$ in {\tt F2}. In the history the answer is : $${\tt X^2+X}$$ To obtain the plot of the expression stored in {\tt F2}, you have to quit the {\tt CAS} (press {\tt HOME}) and then open the {\tt Aplet} {\tt Function}, select the function {\tt F2}, then press on {\tt PLOT} %Nouveau \subsection{Add the plot of an expression :{\tt PLOTADD}}\index{PLOTADD} {\tt PLOTADD} has an expression as argument.\\ {\tt PLOTADD} asks you to select an {\tt Aplet} and then stores this expression in an {\tt Aplet}-variable.\\a Typing: $${\tt PLOTADD(X^2-X)}$$ You select the {\tt Aplet} {\tt Function}, and you choose to save the expression ${\tt X^2-X}$ in {\tt F3}. In the history the answer is : $${\tt X^2-X}$$ To obtain the plot of the expressions stored in {\tt F2} and {\tt F3}, you have to quit the {\tt CAS} (press {\tt HOME}), to open the {\tt Aplet} {\tt Function}, then select the functions {\tt F2} and {\tt F3}, and press on {\tt PLOT}. \subsection{Evaluate a primitive : {\tt PREVAL}}\index{PREVAL} {\tt PREVAL} has three parameters: an expression {\tt F(VX)} dependent on the variable contained in {\tt VX}, and two expressions {\tt A} and {\tt B}.\\ For example if {\tt VX} contains {\tt X}, and if {\tt F} is a function, {\tt PREVAL (F(X),A,B)} returns {\tt F(B)-F(A)}.\\ {\tt PREVAL} is used for calculating an integral defined from a primitive: it evaluates this primitive between the two limits of the integral.\\ Typing: $${\tt PREVAL(X^2+X,2,3)} $$ gives: $${\tt 6}$$ \subsection{Primitive and definied intégral : {\tt RISCH}}\index{RISCH} {\tt RISCH} has two parameters: an expression and the name of a variable.\\ {\tt RISCH} returns a primitive of the first parameter with respect to the variable specified in the second parameter.\\ Typing: $${\tt RISCH((2 \cdot X^2+1) \cdot EXP(X^2+1),X) }$$ gives: $${\tt X \cdot EXP(X^2+1)}$$ {\sc Remark}: If the {\tt RISCH} parameter is the {\tt AND} of two elements, {\tt RISCH} concerns itself only with the second element of the {\tt AND}, and adds the result to the first argument.. %Nouveau \subsection{Discrete antiderivative : {\tt SIGMA}}\index{SIGMA} {\tt SIGMA} has two arguments: the first argument is a function $f(x)$ of a variable $x$ given as the second argument. {\tt SIGMA} returns the discrete antiderivative of the input function, that is, the function $G$ that satisfies the relation: $G(x+1)-G(x)=f(x)$.\\ For example, typing : $${\tt SIGMA(X \cdot X!,X)}$$ gives : $${\tt X!}$$ because $(X+1)!-X!=X\cdot X!$ %Nouveau \subsection{Discrete antiderivative : {\tt SIGMAVX}}\index{SIGMAVX} {\tt SIGMAVX} has as argument a function $f$ of the current variable {\tt VX}.\\ {\tt SIGMAVX} returns the discrete antiderivative of the input function, that is a function $G$ that satisfies the relation: $G(x+1)-G(x)=f(x)$.\\ For example, type: $${\tt SIGMAVX(X^2)}$$ You obtain: $${\tt \frac{2\cdot X^3-3\cdot X^2+X}{6}}$$ because $2(X+1)^3-3(X+1)^2+X+1-2X^3+3X^2-X=6X^2$ \subsection{Variation table : {\tt TABVAR}}\index{TABVAR} {\tt TABVAR} has as a parameter an expression with a rational derivative.\\ {\tt TABVAR} returns the variation table for the expression in terms of the current variable.\\ Typing: $${\tt TABVAR(LN(X)+X)} $$ In Step by step mode, this gives:\\ ${\tt F=:(LN(X)+X)}$\\ ${\tt F'=:(\frac{1}{X}+1}$\\ ${\tt \rightarrow :\frac{X+1}{X}} $\\ {\tt Variation table}: $$\left[\begin{array}{cccccc} -\infty & ? & 0 & + & +\infty & X\\ ? & ? & -\infty & \uparrow & +\infty & F \end{array}\right]$$ \section{Taylor and asymptotic Expansions} All functions in this section can be found in the {\tt DIFF} menu on the menu bar.\\ It is customary to write the expansions in ascending order by exponent of the variable; you set this up by choosing ${\tt 1+x+x^2...}$ in {\tt CFG}. \subsection{Division in increasing order by exponent : {\tt DIVPC}}\index{DIVPC} {\tt DIVPC} has three arguments: two polynomials {\tt A(X)} and {\tt B(X)} (where {\tt B(0)}$\neq 0$), and a whole number {\tt n}.\\ {\tt DIVPC} returns the quotient {\tt Q(X)} of the division of {\tt A(X)} by {\tt B(X)}, in increasing order by exponent, and with deg{\tt (Q)$\leq $ n} or {\tt Q = 0}.\\ {\tt Q[X]}( is then the limited $n$th-order expansion of ${\tt \frac{A[X]}{B[X]}}$ in the vicinity of {\tt X = 0}.\\ Typing: $${\tt DIVPC(1+X^2+X^3,1+X^2,5)}$$ gives: $${\tt 1+X^3-X^5}$$ {\sc Note}: When the calculator asks to go into "increasing powers" mode, respond {\tt yes}. \subsection{Limited $n$th-order expansion : {\tt SERIES}}\index{SERIES} {\tt SERIES} has three arguments : an expression dependent on a variable, an equality (the variable $x$ = the value $a$ to which you want to calculate the expansion) and a whole number (the order $n$ of the limited expansion).\\ You can omit the name of the variable and the sign {\tt =}, when this name is in {\tt VX})\\ {\tt SERIES} returns the limited $n$th-order expansion of the expression in the vicinity of $x=a$. \begin{itemize} \item Expansion in the vicinity of {\tt x = a}\\ Example:\\ Give a limited 4th-order expansion of $\cos(2\times x)^2$ in the vicinity of $x=\frac{\pi}{6}$.\\ For this you use the {\tt SERIES} command.\\ Typing: $${\tt SERIES( COS(2 \cdot X)^2 , X=\frac{\pi}{6} , 4 )}$$ gives: $${\tt \ (\frac{1}{4}-\sqrt 3 h+2h^2+\frac {8 \sqrt {3}}{3}h^3- \frac {8}{3}h^4 +O(\frac{h^5}{4})|_{\ \displaystyle h=X- \frac{ \pi}{6}}}$$ \item Expansion in the vicinity of {\tt x=+$\infty$} or {\tt x=-$\infty$}\\ Example 1:\\ Give a 5th-order expansion of $\arctan(x)$ in the vicinity of {\tt x=+$\infty$}, taking as infinitely small $h=\frac{1}{x}$.\\ Typing: $${\tt SERIES(ATAN(X),X=+\infty,5)}$$ gives: $${\tt (\frac{\pi}{2}-h+\frac{h^3}{3}-\frac{h^5}{5}+O(\frac{\pi \cdot h^6}{2})|_{\ \displaystyle h=\frac{1}{X}}}$$ Example 2:\\ Give a 2nd-order expansion of $(2x-1)e^\frac{1}{x-1}$ in the vicinity of {\tt x=+$\infty$}, taking as infinitely small $h=\frac{1}{x}$.\\ Typing: $${\tt SERIES((2X-1) \cdot EXP(\frac{1}{X-1}),X=+\infty,3)}$$ gives: $${\tt (\frac{12+6h+12h^2+17h^3}{6 \cdot h}+O(2 \cdot h^3)|_{\ \displaystyle h=\frac{1}{X}}}$$ Example 3:\\ Give a 2nd-order expansion of $(2x-1)e^\frac{1}{x-1})$ in the vicinity of {\tt x=-$\infty$}, taking as infinitely small $h=-\frac{1}{x}$.\\ Typing: $${\tt SERIES((2X-1) \cdot EXP(\frac{1}{X-1}),X=-\infty,3)}$$ gives: $${\tt (\frac{-12+6h-12h^2+17h^3}{6h}+O(-(2h^3))|_{\displaystyle h=-\frac{1}{X}}}$$ \item Unidirectional expansion\\ You must be in {\tt Rigourous} mode (Press {\tt DIFF}, then {\tt CFG}, then choose {\tt Rigourous} and then {\tt Quit config.}).\\ To perform an expansion in the vicinity of $x=a$ where $ \ x>a$, use a positive real (such as 4. or 4.0) for the order; to perform an expansion in the vicinity of $x=a$ where $ \ x\frac{1}{2}z2-z$. When $m$ moves on the circle $C$, $M$ will move on a curve $\Gamma$. The object of this exercice is to study and plot $\Gamma$. \begin{enumerate} \item Let $t \in [-\pi , \pi]$ and $m$ the point on $C$ of affix $z=e^{i \cdot t}$. Find the coordinates of $M$ in terms of $t$. \item Compare $x(-t)$ with $x(t)$ and $y(-t)$ with $y(t)$. Show that $\Gamma$ has a symmetry center. \item Compute $x'(t)$ and find the variations of $x$ over $[0,\pi]$. \item Same question for $y$. \item Show the variations of $x$ and $y$ in the same table. \item Put the points of $\Gamma$ corresponding to $t=0, \pi/3, 2\pi/3, \pi$ and draw the tangent to $\Gamma$ at these points. Draw the part of $\Gamma$ corresponding to $t\in [0,\pi]$. Draw $\Gamma$. \end{enumerate} \subsubsection{Correction} The object of this exercise is to trace the curve $\Gamma$ described by $M$ and given by $\frac{1}{2}\cdot z^2-z $, when $m$, given by $z$ describes a circle \textit{C} with center $O$ and having a radius of 1. Let $t$ be a real number in $[-\pi , \pi]$, and let $m$ be the point of \textit{C} given by $z=e^{i \cdot t}$. \begin{enumerate} \item Calculating the coordinates of $M$:\\ First, make {\tt t} the {\tt VX} variable (SHIFT SYMB (SETUP) keys and then {\tt Indep var : t})\\ Then, enter the expression $\frac{1}{2}\cdot z^2-z$ into the Equation Editor.\\ In the Equation Editor, type: $${\tt \ ALPHA \ Z \ x^y \ 2 \ \rhd\ \div\ 2\ \rhd\ -\ ALPHA \ Z \ \rhd\ \rhd\ }$$ The expression ${\tt \displaystyle \frac{Z^2}{2}-Z}$ is selected.\\ Since $z=e^{i \cdot t}$, invoke {\tt SUBST} and complete the second argument by typing: $${\tt SUBST(\frac{Z^2}{2}-Z,Z=EXP(i\times t))}$$ which gives: $${\tt \frac{EXP(i\cdot t)^2}{2} - EXP(i\cdot t)} $$ Then, you linearise the expression with the command:\\ {\tt LIN}\\ The result (after accepting the switch to Complex mode) is: $${\tt \frac{1}{2} \cdot EXP(2\cdot i\cdot t)+-1\cdot EXP(i\cdot t) }$$ \begin{itemize} \item Now store the result with the {\tt STORE} command by typing: $${\tt STORE(\frac{1}{2} \cdot EXP(2\cdot i\cdot t)+-1\cdot EXP(i\cdot t),M)}$$ then pressing {\tt ENTER}.\\ To calculate the real part of this expression, enter the command: $${\tt RE} $$ which returns: $${\tt \frac {COS ( t\cdot 2)-2\cdot COS(t)}{2}}$$ We suppose that {\tt VX} contains {\tt t} ({\tt SHIFT HOME (MODES)} then {\tt Indep var : t}).\\ Then, define the function $x(t)$ by invoking {\tt DEF}:\\ {\sc Note}: You will need to type {\tt = X(t)}, then exchange {\tt X(t)} with the expression ${\tt \frac {COS ( t\cdot 2)-2\cdot COS(t)}{2}}$. To do this, highlight {\tt X(t)} with $\rhd$ and type ${\tt SHIFT \lhd}$. Highlight the entire expression and select the {\tt DEF} command: $${\tt DEF\ (X(t) = \frac {COS ( t\cdot 2)-2\cdot COS(t)}{2})}$$ Then press {\tt ENTER}.\\ To calculate the imaginary part of this expression, type the command:\\ {\tt IM(M)}\\ which returns: $${\tt \frac {SIN ( t\cdot 2)-2\cdot SIN(t)}{2}}$$ Then, define the function $y(t)$ (in the same way as you defined $x(t)$): $${\tt DEF( Y(t) =\frac {SIN ( t\cdot 2)-2\cdot SIN(t)}{2}}$$ Then press {\tt ENTER}. \end{itemize} \item To find an axis of symmetry for $\Gamma$, calculate $x(-t)$ and $y(-t)$ by typing: $${\tt X(-t)\ ENTER}$$ which produces: $${\tt \frac {COS ( t\cdot 2)-2\cdot COS(t)}{2}}$$ In other words: $x(-t) = x(t)$\\ Then type: $${\tt Y(-t)\ ENTER}$$ which produces: $${\tt \frac {-SIN ( t\cdot 2)+2\cdot SIN(t)}{2}}$$ In other words: $y(-t) = -y(t)$\\ If $M_1 (x(t), y(t))$ is part of $\Gamma$, then $M_2 (x(-t),y(-t))$ is also part of $\Gamma$.\\ Since $M_1$ and $M_2$ are symmetrical with respect to the $x$-axis , we can deduce that the $x$-axis is an axis of symmetry for $\Gamma$.\\ \item Calculate x´(t):\\ Typing: $${\tt DERVX (X(t))}$$ returns: $${\tt \frac {2 \cdot (-2 \cdot SIN ( t\cdot 2)-2\cdot(- SIN(t)))}{4}}$$ or, after simplification (ENTER), the reponse is : $${\tt -SIN ( t)\cdot (2 \cdot COS(t)- 1)}$$ You can now define the function $x'(t)$ by invoking {\tt DEF}:\\ Note: You will need to type {\tt = X1(t)}, then exchange {\tt X1(t)} with the expression ${\tt -SIN ( t)\cdot (2 \cdot COS(t)- 1)}$. To do this, highlight {\tt X1(t)} ($\rhd$) and type ${\tt SHIFT \ \lhd}$. This returns: $${\tt DEF(\ X1(t)= -SIN ( t)\cdot (2 \cdot COS(t)- 1)) }$$ Then press {\tt ENTER}.\\ \item Calculate y'(t):\\ Typing: $${\tt DERVX (Y(t))}$$ returns: $${\tt \frac {2 \cdot (2 \cdot COS( t\cdot 2)-2\cdot COS(t))}{4}}$$ and, after simplification ({\tt ENTER}), the response is : $${\tt 2 \cdot COS(t)^2- COS(t)-1}$$ Invoke FACTOR to factor the expression: $${\tt FACTOR(2 \cdot COS(t)^2- COS(t)-1)} $$ then press ENTER, the response is: $${\tt (COS ( t)-1)\cdot (2 \cdot COS(t)+1)}$$ You can now define the function $y'(t)$ (in the same way as you defined $x'(t)$): $${\tt DEF(\ Y1(t)=(COS ( t)-1)\cdot (2 \cdot COS(t)+1))}$$ \item Variations of $x(t)$ and $y(t)$\\ For this, you trace $x(t)$ and $y(t)$ on the same graph.\\ We suppose that {\tt t} is contained in{\tt VX} ({\tt SHIFT SYMB (SETUP)}) Type {\tt X(t)} in the Equation Editor and press {\tt ENTER}.\\ Then press the {\tt PLO}T key.\\ Select {\tt Function} using the dialog box, and select {\tt F1} as the destination.\\ Then, do the same thing with {\tt Y(t)}, making {\tt F2} the destination.\\ To graph the functions: quit {\tt CAS} (using the {\tt ON (CANCEL)} button), choose the Function aplet, and check {\tt F1} and {\tt F2}. You will have to set the window's parameters ({\tt SHIFT PLOT}), then press {\tt PLOT} to see the graphs.\\ Back in the Equation Editor (press the {\tt HOME} key, then {\tt CAS} on the menu bar), we can get exact outputs from the curve $\Gamma$ :\\ \begin{itemize} \item Values of $x(t)$ and $y(t)$\\ Find the values of $x(t)$ and $y(t)$ for $t=0,\frac{\pi}{3},\frac{2 \cdot \pi}{3},\pi$ by typing in succession ({\tt ENTER} is pressed twice in most cases for further simplification):\\ $${\tt X(0) \ ENTER}$$ Response: $\frac{-1}{2}$ $${\tt X(\frac{\pi}{3}) \ ENTER\ ENTER}$$ Response: $\frac{-3}{4}$ $${\tt X(2 \times \frac{\pi}{3}) \ ENTER\ ENTER}$$ Response: $\frac{1}{4}$ $${\tt X(\pi) \ ENTER\ ENTER}$$ Response: $\frac{3}{2}$ $${\tt Y(0) \ ENTER}$$ Response: $0$\\ $${\tt Y(\frac{\pi}{3}) \ ENTER\ ENTER}$$ Response: $\frac{-\sqrt3}{4}$ $${\tt Y(2 \times \frac{\pi}{3}) \ ENTER\ ENTER}$$ Response: $\frac{-3 \cdot \sqrt 3}{4}$ $${\tt Y(\pi) \ ENTER\ ENTER}$$ Response: $0$ \item Slope of the tangents ($m=\frac{y'(t)}{x'(t)}$)\\ Find the values of $\frac{y'(t)}{x'(t)}$ for $t=0,\frac{\pi}{3},\frac{2 \cdot \pi}{3},\pi$ by typing in succession: $${\tt LIMIT(\frac{Y1(t)}{X1(t)},t=0) \ ENTER}$$ Response: $0$ $${\tt LIMIT(\frac{Y1(t)}{X1(t)},t=\pi \div 3) \ ENTER}$$ Response (answer {\tt YES} when asked {\tt UNSIGNED INF. SOLVE?}): $\infty$ $${\tt LIMIT(\frac{Y1(t)}{X1(t)},t=2 \times \pi \div 3) \ ENTER}$$ Response: $0$ $${\tt LIMIT(\frac{Y1(t)}{X1(t)},t=\pi) \ ENTER}$$ Response (answer {\tt YES} when asked {\tt UNSIGNED INF. SOLVE?}): (\\ Here, then, are the variations of $x(t)$ and $y(t)$: $$\begin{array}{|c|ccccccc|}\hline t & 0 & & \frac{\pi}{3} & & \frac{2\pi}{3} & &\pi \\\hline x'(t)&0&-&0&+&\sqrt3&+&0\\\hline x(t)&\frac{-1}{2} & \downarrow & \frac{-3}{4} & \uparrow &\frac{1}{4} & \uparrow &\frac{3}{2}\\\hline y(t)&0&\downarrow & -\frac{\sqrt3}{4} &\downarrow & \frac{-3\sqrt3}{4} & \uparrow & 0\\\hline y'(t)&0&-&-1&-&0&+&2\\\hline m &0&&\infty&&0&&\infty\\\hline \end{array}$$ \item The curve $\Gamma $ :\\ Now graph the parametric curve.\\ In the Equation Editor, type ${\tt X(t)+i \times Y(t)}$, then press {\tt ENTER}.\\ Then press:\\ {\tt PLOT}, and select {\tt Parametric} using the dialog box, specifying {\tt X1,Y1} as the destination. \\ To make the graph of the curve $\Gamma $ : quit {\tt CAS} (using the {\tt HOME} button), then choose the {\tt Parametric} aplet. Check {\tt X1(T), Y1(T)}, and choose default values in {\tt PLOT SETUP}. \end{itemize} \end{enumerate} \subsection{Exercise 2 (specialty)} \subsubsection{Terms of exercise 2} For $n$ a strictly positive integer, we define \[ a_n=4\times 10^n-1, \quad b_n=2\times 10^n-1, \quad c_n=2\times 10^n+1\] \begin{enumerate} \item a/ Compute $a_1,b_1,c_1,a_2,b_2,c_2,a_3,b_3$ and $c_3$.\\ b/ Find how many digits the decimal representation of $a_n$ and $c_n$ have. Show that $a_n$ and $c_n$ are divisible by 3.\\ c/ Using a list of prime smaller than 100, show that $b_3$ is prime.\\ d/ Show that for every integer $n>0$, $b_n\times c_n=a_{2n}$. Deduce the prime factor decomposition of $a_6$.\\ e/ Show that GCD($b_n,c_n$)=GCD($c_n,2$). Deduce that $b_n$ and $c_n$ are prime together. \item Consider the equation: $$(1)\ \ b_3x+c_3y=1$$ where the integers $x$ and $y$ are the unknown.\\ a/ Show that (1) has at least one solution\\ b/ Apply Euclide's algorithm to $c_3$ and $b_3$ and find a solution to (1).\\ c/ Find all solutions of (1) \end{enumerate} \subsubsection{Correction} Define the following for a natural whole number: $$a_n=4 \times 10^n-1,\ b_n=2 \times 10^n-1 \mbox{ and } c_n=2 \times 10^n+1$$ Begin by typing: $${\tt DEF(A(N)=4 \cdot 10^N-1)} $$ $${\tt DEF(B(N)=2 \cdot 10^N-1)} $$ $${\tt DEF(C(N)=2 \cdot 10^N+1)} $$ \begin{enumerate} \item Now do the following:\\ Calculate \begin{itemize} \item a) Simply type (in succession): $${\tt A(1)}$$ Result: 39 $${\tt B(1)}$$ Result: 19 $${\tt C(1)}$$ Result: 21 $${\tt A(2)}$$ Result: 399 $${\tt B(2)}$$ Result: 199 $${\tt C(2)}$$ Result: 201 $${\tt A(3)}$$ Result: 3999 $${\tt B(3)}$$ Result: 1999\\C(3)\\ $${\tt C(3)}$$ Result: 2001 \item b) Number of digits, and divisibility\\ In this, the calculator is used only to try out different values of $n$ ... \\ Show that the whole numbers $k$ such that: $$10^n \leq k < 10^{n+1} $$ have $(n+1)$ digits in decimal notation.\\ We have: $$10^n\ <\ 3 \cdot 10^n\ <\ a_n\ < \ 4 \cdot 10^n\ <\ 10^{n+1}$$ $$ 10^n\ <\ b_n\ < \ 2 \cdot 10^n\ < \ 10^{n+1}$$ $$10^n\ <\ 2 \cdot 10^n\ <\ c_n\ < \ 3 \cdot 10^n\ <\ 10^{n+1}$$ so $a_n,\ b_n,\ c_n$ have $(n+1)$ digits in decimal notation. \\ Moreover, $d_n=10^n-1$ is divisible by 9, since its decimal notation can only end in 9.\\ We also have: $$a_n=3 \cdot 10^n\ +\ d_n$$ and $$c_n=3 \cdot 10^n\ -\ d_n$$ so $a_n$ and $c_n$ are both divisible by 3.\\ \item c) $b_3$ is a prime number\\ Typing: $${\tt ISPRIME? (B(3))}$$ gives: $${\tt 1.}$$ which means true.\\ To prove that $b_3=1999$ is a prime number, it is necessary to show that 1999 is not divisible by any of the prime numbers less than or equal to $\sqrt{1999}$.\\ As $1999<2025=45^2$, that means testing the divisibility of 1999 by $n = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41$.\\ 1999 is not divisible by any of these numbers, so we can conclude that 1999 is prime. \item d) $a_{2n}=b_n \times c_n$\\ Typing: $${\tt B(N)\cdot C(N)}$$ produces: $${\tt 4 \cdot {(10^N)}^2-1}$$ after applying the EXP2POW command. This is the value $a_{2n}$.\\ Decomposition of $a_6$ into its prime factors:\\ Typing: $${\tt FACTOR(A(6))}$$ yields: $${\tt 3 \cdot 23 \cdot 29 \cdot 1999}$$ \item e) $b_n$ and $c_n$ are relatively prime.\\ In this part, the calculator is useful only for trying out different values of $n$...\\ To show that $c_n$ and $b_n$ are relatively prime, it is enough to remark that\\ $$c_n=b_n+2$$ That means that the common divisors of $c_n$ and $b_n$ are the common divisors of $b_n$ and 2, as well as the common divisors of $c_n$ and 2. $b_n$ and 2 are relatively prime because $b_n$ is a prime number different from 2. So: $$PGCD(c_n,\ b_n)=PGCD(c_n,\ 2)=PGCD(b_n,\ 2)=1$$ \end{itemize} \item Given the equation: $$b_3 \cdot x + c_3 \cdot y =1$$ \begin{itemize} \item a) It has at least one solution, as it is actually a form of Bézout's Identity.\\ In effect, Bézout's Theorem says:\\ If $a$ and $b$ are relatively prime, there exist $x$ and $y$ such that: $$a \cdot x + b \cdot y =1$$ Therefore, the equation: $$b_3 \cdot x + c_3 \cdot y =1$$ has at least one solution.\\ \item b) Typing: $${\tt IEGCD(B(3),C(3))}$$ returns: $${\tt 1000\ AND\ -999 =1}$$ In other words: $$b_3 \times 1000 +c_3 \times (- 999)=1$$ so we have a particular solution:\\ $x=1000, \ y=-999$.\\ The rest can be done on paper:\\ $ c_3\ =\ b_3\ +\ 2$ and $b_3\ =\ 999\ \times \ 2 \ +\ 1$\\ so, $b_3\ =\ 999\ \times \ (c_3 \ -\ b_3) \ +\ 1$, or: $$b_3 \times 1000 +c_3 \times (- 999)\ =\ 1$$ \item c) In this part, the calculator is not used for finding the general solution.\\ We have: $$b_3 \cdot x + c_3 \cdot y\ =\ 1$$ and $$b_3 \times 1000 +c_3 \times (- 999)\ =\ 1$$ so, by subtraction: $$b_3 \cdot (x-1000 )+ c_3 \cdot (y+999)\ =\ 0$$ or: $$b_3 \cdot (x-1000 )\ =\ - c_3 \cdot (y+999)\ $$ According to Gauss's Theorem, $c_3$ is prime with $b_3$, so $c_3$ is a divisor of $(x-1000)$.\\ So, there exists $ k\ \in Z$ such that: $$(x-1000)=k \times c_3 $$ and $$-(y+999)=k \times b_3$$ Solving for x and y, we get: $$x\ =\ 1000\ +\ k \times\ c_3$$ and $$y\ =\ -999\ -\ k \times\ b_3$$ for $\ k\ \in Z$\\ This gives us: $$b_3 \cdot x+ c_3 \cdot y =b_3 \times 1000 +c_3 \times (- 999)\ =\ 1$$ The general solution for all $k\ \in Z$ is therefore: $$x\ =\ 1000\ +\ k \times\ c_3$$ $$y\ =\ -999\ -\ k \times\ b_3$$ \end{itemize} \end{enumerate} \subsection{Exercise 3 (non-specialty)} \subsubsection{Terms of exercise 2} For $n$ an integer, define: $$u_n=\int _0 ^2\ \frac{2x+3}{x+2}\ e^\frac{x}{n}\ dx$$ \begin{enumerate} \item a/ Define $g$ over $[0,2]$ by $\displaystyle g(x)=\frac{2x+3}{x+2}$.\\ Find the variations of $g$ over $[0,2]$. Show that for every real $x$ in $[0,2]$, $\displaystyle\frac{3}{2} \leq g(x) \leq \frac{7}{4}$.\\ b) Show that for every real $x$ in $[0,2]$: $\displaystyle\frac{3}{2} e^\frac{x}{n}\leq g(x)e^\frac{x}{n} \leq \frac{7}{4}e^\frac{x}{n}$.\\ c) After integration, show that : $$\frac{3}{2}n (e^\frac{2}{n}-1)\leq u_n \leq \frac{7}{4} n (e^\frac{2}{n}-1)$$ \\ d) Using $\displaystyle \lim_{h \rightarrow 0}\frac{e^h-1}{h}=1$.\\ show that if $u_n$ has a limit $L$ as $n \rightarrow \infty$, then $\displaystyle 3 \leq L \leq \frac{7}{2}$.\\ \item a) Show that for every $x$ in $[0,2]$: $\displaystyle \frac{2x+3}{x+2}=2-\frac{1}{x+2}$.\\ Find the value of $I=\int _0 ^2\ \frac{2x+3}{x+2}\ dx$.\\ b) Show that for every $x$ in $[0,2]$: $\displaystyle 1 \leq e^\frac{x}{n}\leq e^\frac{2}{n}$.\\ Deduce that $\displaystyle 1 \leq u_n \leq e^\frac{2}{n}I$.\\ c) Show that ($u_n$) is convergent and find it's limit $L$. \end{enumerate} \subsubsection{Correction} Before you begin, check that the calculator is in exact real mode with {\tt X} as the current variable; if not, select {\tt Default cfg} in {\tt CFG}.\\ Given: $$u_n=\int _0 ^2\ \frac{2x+3}{x+2}\ e^\frac{x}{n}\ dx$$ \begin{enumerate} \item Do the following: \item \begin{itemize} \item Variation of $g(x)=\frac{2x+3}{x+2} \mbox{ for } x\in [0,2]$\\ Typing: $${\tt DEF(G(X)=\frac{2X+3}{X+2})}$$ then: $${\tt TABVAR(G(X))}$$ yields: $$\begin{array}{cccccc} -\infty & + & -2 & + & +\infty & X\\ 2 & \uparrow & \infty & \uparrow & 2 & F \end{array}$$ The first line gives the sign of {\tt g'(x)} according to {\tt x}, and the second line the variations of {\tt g (x)}. Note that for {\tt TABVAR} the function is always called {\tt F}.\\ We can deduce, then, that $g$(x) increases over $[0, 2]$.\\ If you were in Step by step mode (for this, choose {\tt Step/step} and then {\tt OK} on the {\tt CFG} menu bar), you would obtain (although the function is labelled {\tt F}): $${\tt F=: \frac{2 \cdot X+3}{X+2}}$$ Press {\tt ENTER} : $${\tt F':=\frac{2 \cdot (X+2)-(2 \cdot X+3)}{SQ(X+2)}}$$ Using the down-arrow $\triangledown$, scroll down the screen: $${\tt \rightarrow \frac{1}{(X+2)^2}}$$ Then press {\tt ENTER} to obtain the table of variations.\\ If you are not in Step by step mode, you can also request the calculation of the derivative by typing: $${\tt DERVX(G(X))}$$ which produces the preceding calculation.\\ To calculate $g(0)$ and $g(2)$, type: $${\tt G(0)}$$ Response: $\frac{3}{2}$ $${\tt G(2)}$$ Response: $\frac{7}{4}$\\ whence the inequality: $$\frac{3}{2} \leq g(x) \leq \frac{7}{4} \ \mbox{ for } x\in [0,2]$$. \item b) The calculator is not needed here... simply stating that : $$e^\frac{x}{n} \geq 0 \ \mbox{ pour } x\in [0,2] $$ is sufficient to show that, for $x\in [0,2]$, we have: $$\frac{3}{2}e^\frac{x}{n}\leq g(x)e^\frac{x}{n}\leq \frac{7}{4}e^\frac{x}{n}$$ \item c) To integrate the preceding inequality, type: $${\tt \int _0^2\ e^\frac{X}{N}\ dX} $$ which produces: $${\tt N \cdot e^\frac{2}{N} -N } $$ We can then deduce that: $$\frac{3}{2}(n e^\frac{2}{n}-n) \leq u_n \leq \frac{7}{4}(ne^\frac{2}{n}-n)$$ To justify the preceding calculation, it is necessary to assume that $n \cdot e^\frac{x}{n}$ is a primitive of $ e^\frac{x}{n}$.\\ If you are not sure, you can type: $${\tt INTVX(EXP(\frac{X}{N}))}$$ The simplified result is: ${\tt N \cdot e^\frac{X}{N}}$ \item d) To find the limit of $(n e^\frac{2}{n}-n)$ when $n \rightarrow +\infty$: $${\tt LIMIT(N \cdot EXP(\frac{2}{N}) -N\ ,\ N=+\infty)}$$ The result is: $${\tt 2}$$ {\sc Note} :\\ The variable {\tt VX} is set equal to {\tt N}; use the {\tt SHIFT SYMB (SETUP)} keys to reset {\tt VX} to {\tt X}.\\ To check the result, we can say that: $$\lim_{x \rightarrow 0}\frac{e^x-1}{x}=1$$ and that therefore: $$\lim_{n \rightarrow +\infty}\frac{e^\frac{2}{n}-1}{\frac{2}{n}}=1$$ or, simplifying: $$\lim_{n \rightarrow +\infty}(e^\frac{2}{n}-1)\cdot n=2$$ If the limit $L$ of $u_n$ exists as $n$ tends to $+\infty$ in the inequalities in 1b), we get: $$\frac{3}{2} \cdot 2 \leq L \leq \frac{7}{4} \cdot 2$$ \end{itemize} \item \begin{itemize} \item a) Do the following:\\ $g(x)=2-\frac{1}{x+2}$ and the calculation of $I= \int _0 ^2 g(x)dx$\\ Typing: $${\tt PROPFRAC(G(X))}$$ returns: $${\tt 2-\frac{1}{X+2}}$$ To calculate the integral $I$, type: $${\tt \int _0 ^2 G(X)dX}$$ This produces: $${\tt -(LN(2)-4)}$$ Working by hand: $2x+3=2(x+2)-1$, so : $$g(x)=2-\frac{1}{x+2}$$ Then, integrating term by term between 0 and 2 produces: $$\int_0^2 g(x)dx=[2x-\ln(x+2)]_{x=0}^{x=2}$$ that is, since $\ln4=2\ln2$ : $$\int_0^2 g(x)dx=4- \ln2$$ \item b) The calculator is not needed here... simply stating that $e^\frac{x}{n}$ increases for $x \in [0,2]$ is sufficient to yield the inequality: $$1 \ \leq \ e^\frac{x}{n} \ \leq e^\frac{2}{n} $$ Then, through multiplication, $g(x)$ being positive over $[0, 2]$, we get: $$g(x)\ \leq \ g(x)e^\frac{x}{n}\ \leq g(x)e^\frac{2}{n} $$ and then, integrating: $$I \ \leq u_n \ \leq \ e^\frac{2}{n}I $$ \item c) Convergence of $u_n$\\ Find the limit of $e^\frac{2}{n}$ when $n \rightarrow +\infty$: $${\tt LIMIT(EXP(\frac{2}{N})\ , \ N=+\infty)}$$ This returns: $${\tt 1}$$ In effect, $\frac{2}{n}$ tends to 0 as $n$ tends to $+\infty$, so $e^\frac{2}{n}$ tends to $e^0=1$ as $n$ tends to $+\infty$.\\ As $n$ tends to $+\infty$, $u_n$ is the portion between $I$ and a quantity that tends to $I$ (cf. inequalities 2b)).\\ So then, $u_n$ converges, and its limit is $I$.\\ We have therefore shown that: $$L=I=4- \ln2$$ \end{itemize} \end{enumerate} \section{Conclusion} You can see that, by that making good use of the {\tt HP40GS} calculator , you can use it for a large proportion of the calculations...\\ It is necessary to note, however, that arithmetic does require more reasoning. Here, the calculator enables you to check your work... %%%%% HP40GS END CAS %%%%% \chapter{Programming} \section{Implementation}\label{sec:programme} \subsection{How to edit and save a program} >From the {\tt HOME} screen, to get access to the catalogue of programs, press the {\tt SHIFT 1 (PROGRAM)} keys.\\ A screen then appears containing the list of available programs and a menu bar ({\tt EDIT NEW SEND RECV RUN)}.\\ To create a new program, press {\tt F2 (NEW)}.\\ You are asked for the name of the program.\\ {\sc Note}: If you are not in {\tt Alpha} mode, press {\tt F4 (A..Z)} to go into it!\\ Type its name, then press {\tt F6 (OK)}.\\ You enter your program, and your work is automatically saved when you leave the editor by pressing {\tt HOME} or {\tt SHIFT 1 (PROGRAM)}.\\ \subsection{How to correct a program} If the syntax is invalid, the calculator displays:\\ {\tt Invalid Syntax Edit program?} Respond by pressing {\tt F6 (YES)}.\\ The calculator automatically positions the cursor where the compiler has detected the error. All you need do then is correct it! \subsection{How to run a program} To run a program, open the catalogue of programs by pressing {\tt SHIFT 1 (PROGRAM)}.\\ A screen is then displayed containing the list of available programs and the menu bar {\tt EDIT NEW SEND RECV RUN}.\\ Highlight the name of the program and press {\tt F6 (RUN)}. \subsection{How to modify a program} To modify a program (and overwrite the old one), open the catalogue of programs by pressing {\tt SHIFT 1 (PROGRAM)}. A screen is then displayed containing the list of available programs and the menu bar {\tt EDIT NEW SEND RECV RUN}.\\ Highlight the name of the program and press {\tt F1 (EDIT)}.\\ If you want to save and edit the old program under a new name, then:\\ - Open the catalogue of programs ({\tt SHIFT 1 (PROGRAM)})\\ - Press {\tt F2 (NEW)} and type the new name, then press {\tt F6 (OK)}. The Editor opens.\\ - Press {\tt VARS}, then the letter {\tt P} to highlight {\tt Program}.\\ - Using the arrows, highlight the name of the old program, then press {\tt F4 (VALUE)} (to select {\tt VALUE} on the menu bar), then {\tt F6 (OK)}.\\ This copies the text of the old program into the editor. \section{Comments} It is important to comment your programs.\\ In writing algorithms, a comment commonly starts with {\tt //} and ends at the end of the line.\\ In the {\tt HP40GS}, a comment starts with {\tt @} and ends at the end of the line or at the next {\tt @}, whichever comes first.\\ {\sc Note}:\\ Do not forget to put a space after the {\tt @}.\\ The character @ is obtained by typing {\tt SHIFT VAR (CHARS)}, then highlighting the character and choosing {\tt ECHO} on the menu bar (or {\tt ECHO1 ECHO1 OK} to have two characters {\tt @}). \subsection{{\tt SHIFT 1 (PROGRAM)} keys} This key combination, if pressed in {\tt HOME}, displays the screen : \\ {\tt PROGRAM CATALOG}\\ It shows:\\ - a list of the programs that you've written, and\\ - a menu bar containing the commands:\\ {\tt EDIT NEW RUN SEND RECV}.\\ {\tt EDIT} enables you to edit the highlighted program,\\ {\tt NEW} enables you to create a new program,\\ {\tt RUN} enables you to run the highlighted program (cf \ref{sec:programme}),\\ {\tt SEND} and {\tt RECV} are functions that enable your calculator to talk to your computer or another calculator.\\ For example:\\ If you press {\tt SEND} on the menu bar, it asks:\\ {\tt HP39/40 (Wire)} or {\tt Disk Drive}\\ You highlight {\tt HP39/40 (Wire)} to send a program to another {\tt HP40GS}, or you highlight {\tt Disk Drive} to send a program to a computer.\\ Then press {\tt OK} on the menu bar.\\ For example, here is how you connect a Linux computer to the {\tt HP40GS} so as to use the program C-Kermit version 7 (which you can find at the URL {\tt www.columbia.edu/kermit}, or which you can download via anonymous ftp from the site {\tt kermit.columbia.edu}):\\ Connect the calculator to a data transfer cable.\\ On the computer, type:\\ {\tt kermit}\\ {\tt set line /dev/ttyS0} (or {\tt S1...} depending on the number of your serial port)\\ {\tt set speed 9600}\\ {\tt set carrier-watch off}\\ {\tt serv}\\ On the {\tt HP40GS}:\\ Highlight the program called {\tt NAME}, then press {\tt SEND} on the menu bar and highlight {\tt Disk drive}. Then press {\tt OK} on the menu bar, and the program called {\tt NAME} on the {\tt HP40GS} is copied into your computer. \\ OR:\\ Press {\tt RECV} on the menu bar and highlight {\tt Disk drive}. Then press {\tt OK} on the menu bar, and the calculator displays a list of the {\tt HP40GS} programs on your computer. (Naturally, you have to have already created a directory on your computer where the {\tt HP40GS} programs are stored).\\ You then highlight {\tt GCD}, and the program called {\tt GCD} on your computer is copied into the {\tt HP40GS}.\\ For Windows users, the connectivity program is found on the URL :\\ {\tt http://www.hp.com/calculators/france}.\\ To find out more about the use of Kermit with HP calculators, visit the URL: {\tt http://www.columbia.edu/kermit/hp48.html} \section{Variables} \subsection{Their names} These are the containers in which you can store values, numbers, expressions, and other objects.\\ With the {\tt HP40GS}, only the 26 letters of the alphabet, and the Greek letter ${\tt \theta}$ are available for storing real numbers. \subsection{The concept of local variables}\index{PUSH}\index{POP} Local variables are used to protect user variables from modifications after a program call. This concept does not exist natively on the {\tt HP40GS} (only global variables are supported).\\ For advanced users, it is however possible to mimic the local variables mechanism using the {\tt CAS} history and {\tt PUSH} and {\tt POP}. Say for example that our program use the variables {\tt A, B, C}. At the beginning we can {\tt PUSH A} then {\tt PUSH B} then {\tt PUSH C}. This will save the values of {\tt A}, {\tt B} and {\tt C} in the {\tt CAS} history. At the end of the program we restore the value of {\tt A}, {\tt B} and {\tt C} using {\tt POP} and store the value in {\tt C}, {\tt B} and {\tt A}, it must be done {\bf in reverse order} since the {\tt CAS} history is a stack container (LIFO=Last In First Out). \subsection{The concept of parameters}\index{PUSH}\index{POP} It is not possible for programs written on the {\tt HP40GS} to natively pass parameters. Hence you cannot use the {\tt HP40GS} programming language to write functions having parameters.\\ For advanced users, it is possible to mimic parameters using the {\tt CAS} history. The caller {\tt PUSH} a list of parameters in the {\tt CAS} history, then runs the program, that will have as first instruction a {\tt POP} instruction. This mechanism does not work in conjonction with the local variables turnaround above. If you want to combine both concepts, you can choose a list variable for parameter transmission and return value, and use local variables as described above. \section{User entry} \subsection{Algorithmic syntax} So that the user is able to enter a value for the variable {\tt A} during the course of the program's execution, one writes in algorithms:\\ {\tt input A}\\ And for entering values into {\tt A} and {\tt B}, one writes:\\ {\tt input A,B} \subsection{{\tt HP40GS} syntax}\index{PROMPT}\index{PROMPTSTO}\label{sec:prompt} \noindent{\tt INPUT A;"TITRE";"A=";;0:}\\ If it annoys you to have to type all the punctuation in the {\tt INPUT} command, you might prefer to use the {\tt PROMPT} or {\tt PROMPTSTO} command instead (thanks to Jean Yves!).\\ {\tt PROMPTSTO(A)} or {\tt PROMPT A} opens a window that asks you to enter the value of A.\\ In the remainder of this guide, programs that were written before the existence of PROMPT use the subroutine {\tt IN}, which enables you to enter two values into {\tt A} and {\tt B}. \section{Output} \subsection{Algorithmic syntax} In algorithms one writes:\\ {\tt display "A=",A} \subsection{{\tt HP40GS} syntax} \noindent{\tt DISP 3;"A="A:} 3 represents the number of the line where{\tt A} will be displayed\\ Or:\\ {\tt MSGBOX "A="A:} \section{The sequence of statements, or "action"} An action is a sequence of one or more statements. \subsection{Algorithmic syntax} In algorithmic language, you use the space or the linebreak to terminate the statement. \subsection{{\tt HP40GS} syntax} \noindent {\tt:} indicates the end of the statement. \section{The assignment statement} A value or an expression is stored in a variable by means of an assignment. \subsection{Algorithmic syntax} In algorithms, one writes (for example):\\ {\tt 2*A->B}\\ to store {\tt 2*A} in {\tt B}. \subsection{{\tt HP40GS} syntax} The arrow is obtained by means of the ${\tt STO\triangleright}$ key on the menu bar.\\ One types (for example):\\ {\tt 2*A-> B} that mean :\\ press {\tt 2*A} then press ${\tt STO\triangleright}$ on the menu bar then press {\tt B}. \section{Conditional statements} \subsection{Algorithmic syntax} \noindent Conditional statements have two syntax :\\ {\tt If \emph{condition} then\\ \emph{action}\\ endif}\\ and also :\\ {\tt If \emph{condition} then\\ \emph{action1} else\\ \emph{action2}\\ endif}\\ Example:\\ {\tt If A = 10 ou A < B then \\ B-A->B else \\A-B->A \\endif} \subsection{{\tt HP40GS} syntax} \noindent{\tt IF \emph{condition} THEN\\ \emph{action} : \\ END:}\\ and also :\\ {\tt IF \emph{condition} THEN\\ \emph{action1} : ELSE\\ \emph{action2}:\\ END:}\\ {\sc Note} au {\tt ==} to denote the condition of equality.\\ Example :\\ {\tt IF A==10 OR AB : ELSE}\\ {\tt A-B-> A}\\ {\tt END:} \section{FOR statement} \subsection{Algorithmic syntax} \noindent{\tt For I = A to B do action endfor}\\ and also :\\ {\tt For I = A to B (step P) do action endfor} \subsection{{\tt HP40GS} syntax} \noindent{\tt FOR I = A TO B STEP 1 ; \emph{action} : END:}\\ and also :\\ {\tt FOR I = A TO B STEP P ; \emph{action} : END:} \section{WHILE statement} \subsection{Algorithmic syntax} \noindent{\tt While condition do action endwhile} \subsection{{\tt HP40GS} syntax} \noindent{\tt WHILE \emph{condition} REPEAT \emph{action} : END:} \section{Boolean Expressions} A condition is a function that has a Boolean as a value, that is, it is either {\tt TRUE} or {\tt FALSE}. \subsection{Algorithmic syntax} To express a condition, simply use the operators: {\tt = > > $\leq$ $\geq$ $\neq$} \subsection{{\tt HP40GS} syntax} {\sc Note}: for the {\tt HP40GS} calculator, equality is denoted by:\\ {\tt ==}\\ Otherwise, the operators are the same. \section{Logical Operators} \subsection{Algorithmic syntax} To express complex conditions, use the logical operators: {\tt or and not} \subsection{{\tt HP40GS} syntax} {\tt Or, and, not} are represented on the {\tt HP40GS} as {\tt OR, AND, NOT} \section{Lists} \subsection{Algorithmic syntax} In algorithms, you use {\tt \{ \}} to define a list.\\ For example, {\tt \{\}} designates a void list, and \{1, 2, 3\} is a list with three elements.\\ The {\tt +} is used to concatenate two lists, or a list and an element, or an element and a list:\\ {\tt \{1, 2, 3\} -> TAB}\\ {\tt TAB + 4 -> TAB} ({\tt TAB} now designates {\tt \{1, 2, 3, 4\}}\\ {\tt TAB[2]} indicates the second element of {\tt TAB}, here {\tt 2}. \subsection{{\tt HP40GS} syntax} The list variables have the following names: {\tt L0, L1, L2,... L9}.\\ You use {\tt \{ \}} to delimit a list.\\ For example, {\tt \{1, 2, 3\}} is a list with three elements.\\ But {\tt \{\}} does not designate a void list; to clear a list you must use the command {\tt CLRVAR}\index{CLRVAR}, e.g.:\\ {\tt CLRVAR L0} \\ will initialise the list {\tt L0} as void. You can also use the command {\tt SUB} which extract a sub-list (if {\tt L1=\{1,2,3,4\}} then {\tt SUB L2;L1;2;3} make the list {\tt L2=\{2,3\}}) and so, {\tt SUB L1;L1;2;1} to initialise the list {\tt L1} as void.\\ Following are some useful commands:\\ {\tt MAKELIST(I*I, I, 1, 10, 2)} designates a list of the squares of the first five odd whole numbers (2 indicates the step of {\tt I}).\\ {\tt L1(I)} designates the Ith element of the list.\\ {\tt CONCAT (L1, \{5\})} designates a list having the element {\tt 5} in addition to the elements of the list {\tt L1}.\\ You can also use:\\ {\tt AUGMENT(L1,5)}, which designates a list having the element {\tt 5} in addition to the elements of the list {\tt L1}.\\ {\tt SUB L2; L1; 2; 4} is a command that copies into {\tt L2} the elements of {\tt L1} having indices from 2 to 4.\\ {\sc Note} the difference between functions and commands:\\ Functions return a value, and they have parentheses enclosing their arguments (which are separated by commas), whereas\\ Commands do not return values, and their arguments are written directly after the name of the command (and are separated by semi-colons). \section{Example: The Sieve of Eratosthenes} \subsection{Description} To find the prime numbers less than or equal to $N$: \begin{enumerate} \item Write the numbers from 1 to $N$ in a list. \item Cross out 1, and let 2 be the first value of $P$.\\ If $P*P \leq N$, then taking the numbers from $P$ to $N$. \item Cross out all the multiples of $P$ from $P*P$ onward. \item Augment $P$ by 1, If $P*P$ is less than or equal to $N$, then taking the non-excluded numbers from $P$ to $N$. \item Change $P$ to the smallest non-excluded number between $P$ and $N$. \item Repeat steps 3, 4 and 5 until $P*P$ exceeds $N$. \end{enumerate} \subsection{Algorithmic syntax} \begin{verbatim} Function sieve(N) local TAB PRIMES I P //TAB and PRIMES are lists {} ->TAB {} ->PRIMES //do steps 1 and 2 for I = 2 to N do TAB+I -> TAB endfor 0 + TAB -> TAB 2-> P //TAB is the list 0 2 3 4 ..N //1 has been excluded by replacing it with 0 \end{verbatim}{\tt while P*P $\leq$ N do}\begin{verbatim} //exclude all multiples of P from P*P onward for I = P to FLOOR(N/P) do //FLOOR(N/P) designates the integer part of N/P 0 -> TAB[I*P] endfor P+1-> P //Find the smallest non-excluded number //between P and N \end{verbatim} {\tt {\ } while (P*P $\leq$ N) and (TAB[P]=0) do} \begin{verbatim} P+1 -> P endwhile endwhile //write the result to the list PRIMES for I = 2 to N do\end{verbatim} {\tt {\ } If TAB[I] $\neq$ 0 then} \begin{verbatim} PRIMES + I -> PRIMES endif endfor Result: PRIMES \end{verbatim} \subsection{{\tt HP40GS} syntax} Following is the program {\tt SIEVE}:\\ The user must input a value for N.\\ At the end, the list {\tt L2} contains the prime numbers less than or equal to {\tt N}. \begin{verbatim} INPUT N;"CRIBLE";"N=";;10: ERASE: MAKELIST(I,I,1,N,1) -> L1: 0 -> L1(1): 2->P:\end{verbatim} {\tt WHILE P*P $\leq$ N REPEAT}\begin{verbatim} FOR I = P TO INT(N/P) STEP 1; 0->L1(I*P): END: DISP 3;""L1: P+1->P:\end{verbatim} {\tt {\ } WHILE P*P $\leq $ N AND L1(P) == 0 REPEAT}\begin{verbatim} P+1->P: END: END: {2}->L2: @ on sait que 2 est premier FOR I=3 TO N STEP 1;\end{verbatim} {\tt {\ }IF L1(I) $\neq$ 0 THEN}\begin{verbatim} CONCAT(L2,{I}) ->L2: END: END: DISP 3 ;"PREM" L2: FREEZE: \end{verbatim} \chapter{Arithmetic Programs} \section{GCD and Euclid's Algorithm} Given the two whole positive numbers $A$ and $B$, find their $GCD$ (Greatest Common Divisor).\\ Euclid's Algorithm is based on the recursive definition of $GCD$:\\ \begin{eqnarray*} GCD(A,0) & = & A\\ GCD(A,B) & = & GCD(B, A \bmod B) if B \neq 0 \end{eqnarray*} where $A\ \bmod \ B$ designate the remainder of the Euclidean division of $A$ by $B$.\\ Here is the description of this algorithm:\\ Perform the successive Euclidean divisions: \begin{eqnarray*} A=& B \times Q_1+R_1 & 0 \leq R_1 < B \\ B=& R_1 \times Q_2+R_2 & 0 \leq R_2 < R_1 \\ R_1=& R_2 \times Q_3+R_3 & 0 \leq R_3 < R_2 \\ ....... \end{eqnarray*} After a finite number of steps, there exists a whole number n such that: $ R_n = 0$.\\ One then has:\\ $GCD(A,B) = GCD(B, R_1) = ...=$\\ $GCD(R_{n-1}, R_n) = GCD(R_{n-1},0) = R_{n-1}$. \subsection{Algorithmic syntax} - Iterative version\\ If $B \neq 0$, calculate $R=A \bmod B$. Then, make $A$ equal to $B$, and $B$ equal to $R$, and repeat the calculation until $B=0$. The $GCD$ is then $A$. \begin{verbatim} Function GCD(A,B) Local R\end{verbatim} {\tt While B $\neq$ 0 do}\begin{verbatim} A mod B ->R B ->A R ->B endwhile Result: A endfunction \end{verbatim} - Recursive version\\ One simply writes the recursive definition given earlier. \begin{verbatim} Function GCD(A,B)\end{verbatim} {\tt If B $\neq$ 0 then}\begin{verbatim} Result: GCD(B,A mod B) Else Result: A endif endfunction \end{verbatim} \subsection{{\tt HP40GS} syntax}\label{sec:pgcdc} - Iterative version for two whole numbers\\ First, type the subroutine {\tt IN}, which enables the user to enter two numbers {\tt A} and {\tt B} : \begin{verbatim} INPUT A;"A";;;1: INPUT B;"B";;;1: ERASE:\end{verbatim} Then type the {\tt GCD} program: \begin{verbatim}RUN IN: DISP 3;"PGCD "{A,B}:\end{verbatim} {\tt WHILE B $\neq$ 0 REPEAT } \begin{verbatim}A MOD B ->R: B ->A: R ->B: END: DISP 4;"PGCD "A: FREEZE:\end{verbatim} - Recursive version for two whole numbers {\tt A} and {\tt B}\\ You cannot create recursive functions on the {\tt HP40GS}... but you can create the program {\tt GCDR} : \begin{verbatim}DISP 3;"PGCD "{A,B}: FREEZE:\end{verbatim} {\tt IF B $\neq$ 0 THEN} \begin{verbatim}A MOD B ->R: B ->A: R ->B: PGCDR: ELSE DISP 3;"PGCD "A: FREEZE: END:\end{verbatim} First, the values of {\tt A} and {\tt B} are stored.\\ The program {\tt GCDR} displays the GCD that it is in the process of calculating.\\ The recursive call {\tt GCDR} returns you to the {\tt GCDR} program, which you must execute by pressing {\tt RUN} on the menu bar.\\ The {\tt GCDR} program then displays the intermediate GCD calculations.\\ You can also replace {\tt GCDR} in the preceding program with {\tt RUN GCDR} to avoid having to press {\tt RUN} on the menu bar, and to suppress the display of the intermediate values, so that you can use this program in a larger program that caters for input and output:\\ >From the recursive program {\tt GCDR}, we can derive the recursive program {\tt PR} :\\ \noindent{\tt IF B $\neq$ 0 THEN} \begin{verbatim} A MOD B ->R: B ->A: R ->B: RUN PR: END:\end{verbatim} The program {\tt PR} can be inserted into a larger program catering for input and output: \begin{verbatim} PROMPT A: PROMPT B: RUN PR: ERASE: MSGBOX A: \end{verbatim} - Iterative version for two complex numbers.\\ If you use the symbolic calculation function {\tt IREMAINDER} in place of {\tt MOD} in the preceding programs, {\tt GCD} (or {\tt PR}) can then have Gaussian whole numbers as parameters on condition that you replace the names of the variables {\tt A}, {\tt B} and {\tt R} with {\tt Z1}, {\tt Z2} and {\tt Z3}, and that you change the test for completion. When creating the new version start by recalling the contents of the earlier version, as described in 6.1.4. Here is the iterative version: \begin{verbatim} PROMPT Z1: PROMPT Z2: ERASE: DISP 3;"PGCD "{Z1,Z2}:\end{verbatim} {\tt WHILE ABS(Z2) $\neq$ 0 REPEAT } \begin{verbatim}XNUM(IREMAINDER(XQ(Z1),XQ(Z2)) ->Z3: Z2 ->Z1: Z3 ->Z2: END: DISP 4;"PGCD "Z1: FREEZE: \end{verbatim} - Iterative version for two polynomials.\\ The variables {\tt E1, E2,...} enable you to store expressions, which is what you must do if you deal with polynomials! If you use the symbolic calculation function {\tt REMAINDER} in place of {\tt MOD} in the preceding programs, {\tt GCD} (or {\tt PR}) can then have polynomials as parameters on condition that you replace the names of the variables {\tt A}, {\tt B} and {\tt R} with {\tt E1, E2, E3}, and that you change the test for completion. The {\tt CAS} must be in {\tt Exact} and {\tt Direct} modes for this program. \begin{verbatim} PROMPT E1: PROMPT E2: ERASE:\end{verbatim} {\tt WHILE DEGREE(E2) $\neq$ -1 REPEAT } \begin{verbatim}REMAINDER(E1,E2) ->E3: E2 ->E1: E3 ->E2: END: DISP 4;"PGCD "E1: FREEZE: \end{verbatim} You enter (for example):\\ ${\tt E1=S1^2-1}$ and ${\tt E2=S1^2-2*S1+1}$ to find the GCD equal to {\tt 2*S1-2}. \section{Bézout's Identity} Bézout's function (A,B) returns the list $\{U, V, PGCD(A,B)\}$ (where $U$ and $V$ are such that:\\ $ A \times U + B \times V= PGCD(A,B)$. \subsection{Iterative version without lists} Euclid's Algorithm enables us to find a pair $U$ and $V$ such that:\\ $ A \times U + B \times V= PGCD(A,B) $\\ In effect, if we call $A_0$ and $B_0$ the values the $A$ and $B$ at the start, then we have:\\ \begin{eqnarray*} A & =A_0 \times U+B_0 \times V \ \mbox{ avec }\ U=1 \ \mbox{ and }\ V=0 \\ B & =A_0 \times W+B_0 \times X \ \mbox{ avec }\ W=0 \ \mbox{ and } \ X=1 \end{eqnarray*} You then derive $A,\ B,\ U,\ V,\ W,\ X$ in such a way that the two relations above are always satisfied.\\ If:\\ $ A=B \times Q+R \ \ \ 0 \leq R < B \ \ ( R = A \ mod \ B \ and \ Q = E(A / B )) $\\ then:\\ \begin{eqnarray*} R=A-B \times Q=A_0 \times (U-W \times Q)+B_0 \times (V-X \times Q)=\\ A_0 \times S+B_0 \times T \ \mbox{ avec } \ S=U-W \times Q \ \mbox{ and } \ T=V-X \times Q \end{eqnarray*} It remains then to repeat the process with:\\ $B$ in the role of $A$ ({\tt B->A\ W->U \ X->V}), and\\ $R$ in the role of $B$ ({\tt R->B \ S->W \ T->X})\\ whence the algorithm: \begin{verbatim} Function Bezout (A,B) local U,V,W, X, S, T, Q, R 1->U 0->V 0->W 1->X\end{verbatim} {\tt While B $\neq$ 0 do}\begin{verbatim} A mod B->R FLOOR(A/B)->Q //R=A-B*Q U-W*Q ->S V-X*Q ->T B->A W->U X->V R->B S->W T->X endwhile Result: {U, V, A} endfunction \end{verbatim} \subsection{Iterative version with lists} You can simplify the algorithmic syntax below by using fewer variables: for this, use the lists {\tt LA, LB, LR} to store the triplets {\tt \{U, V, A\}}, {\tt \{W, X, B\}} and {\tt \{S, T, R\}}. This is quite easy, as the calculator knows how to add lists of the same length (by adding elements with the same index), as well as how to multiply a list by a number (by multiplying each of the list's elements by the number). \begin{verbatim} Function Bezout (A,B) local LA LB LR {1, 0, A}->LA {0, 1, B}->LB\end{verbatim} {\tt While LB[3] $\neq$ 0 do}\begin{verbatim} LA-LB*E(LA[3]/LB[3])->LR LB->LA LR->LB endwhile Result : LA endfunction \end{verbatim} \subsection{Recursive version with lists} Bézout's Function can be recursively defined by: $ Bezout(A,0)=\{1,\ 0,\ A\} $ If $ B \neq 0 $, is it necessary to define $Bezout(A, B)$ as a function of $Bezout(B, R)$ when $ R=A-B \times Q \ \mbox{ and } \ Q=FLOOR((A/B)$. We have: \begin{eqnarray*} Bezout(B,R)=LT=\{W, X, pgcd(B,R)\} \\ \mbox{ avec }\ W \times B+X \times R=pgcd(B,R) \end{eqnarray*} with \begin{eqnarray*} W \times B+X \times (A-B \times Q) & = & gcd(B,R) \ \mbox{ or} \\ X \times A+(W-X \times Q) \times B & = & gcd(A,B). \end{eqnarray*} whence, if $B \neq 0$ and if $ Bezout(B,R)=LT$, we have:\\ $ Bezout(A,B)=\{LT[2],\ LT[1]-LT[2] \times Q,\ LT[3]\}.$ \begin{verbatim} function Bezout (A,B) local LT Q R\end{verbatim} {\tt If B $\neq$ 0 then}\begin{verbatim} FLOOR(A/B) ->Q A-B*Q->R Bezout(B,R)->LT Result: {LT[2], LT[1]-LT[2]*Q, LT[3]} else Result: {1, 0, A} endif endfunction \end{verbatim} \subsection{Recursive version without lists}\label{sec:recsl} If you use global variables for {\tt A, B, D, U, V, T}, you can view the function {\tt Bezour} as using {\tt A, B} to calculate the values for {\tt U, V, D (AU+BV=D)} by means of a local variable {\tt Q}.\\ One can then write: \begin{verbatim} program Bezour local Q\end{verbatim} {\tt If B $\neq$ 0 then}\begin{verbatim} FLOOR(A/B) ->Q A-B*Q->T B->A T->B Bezour U-V*Q->T V->U T->V else 1->U 0->V A->D endif\end{verbatim} \subsection{{\tt HP40GS} syntax} \noindent- Iterative version with lists\\ Here, you also use the program {\tt IN}, which enables the user to enter two numbers {\tt A} and {\tt B}: \begin{verbatim}INPUT A;"A";;;1: INPUT B;"B";;;1: ERASE:\end{verbatim} Now type the {\tt BEZOUT} program: \begin{verbatim} RUN IN: DISP 3;"BEZOUT "{A,B}: {1,0,A} ->L1: {0,1,B} ->L2:\end{verbatim} {\tt WHILE L2(3) $\neq$ 0 REPEAT} \begin{verbatim}L1-L2*FLOOR(L1(3)/L2(3)) ->L3: L2 ->L1: L3 ->L2: END: DISP 4;"U V PGCD "L1: FREEZE: \end{verbatim} - Recursive version without lists\\ Type the program {\tt BEZOUR}, using the commands (thanks to Bernard!!!):\\ {\tt PUSH}\index{PUSH} (the command {\tt PUSH(A)} stores the contents of {\tt A} on the {\tt CAS} stack), and\\ {\tt POP}\index{POP} (the {\tt POP} command retrieves values stored on the {\tt CAS} stack)\\ {\tt PUSH} and {\tt POP} can simulate the local variable {\tt Q} We have replaced in the translation (cf \ref{sec:recsl}) the variables {\tt R} and {\tt W} by the temporary variable {\tt T}.\\ \noindent{\tt PROGRAM BEZOUR}\\ {\tt IF B $\neq$ 0 THEN} \begin{verbatim}PUSH (FLOOR(A/B)): A MOD B->T: B->A: T->B: RUN BEZOUR: U-V*POP->T: V->U: T->V: ELSE 1->U: 0->V: A->D: END: \end{verbatim} {\tt PUSH (FLOOR(A/B))} has the effect of putting the different values of {\tt FLOOR(A/B)} onto a stack, and {\tt POP} recovers them.\\ {\tt T} is an auxiliary variable.\\ {\tt BEZOUR} takes, as user input, the values of the global variables {\tt A} and {\tt B}, and assigns values to the global variables {\tt U} and {\tt V} such that:\\ ${\tt A\cdot U+B\cdot V=GCD(A,B)}$. We can then write the final program {\tt BEZOURT}, which caters for the input of A and B and the output of {\tt \{U, V, D\}}:\\ \noindent{\tt PROGRAM BEZOURT} \begin{verbatim} PROMPT A: PROMPT B: RUN BEZOUR: ERASE: MSGBOX {U,V,D}: \end{verbatim} \noindent {\sc Remarks} (cf. iterative version for two complex \ref{sec:pgcdc}): \\ If you use the symbolic calculation functions {\tt IREMAINDER} and {\tt IQUOT(XQ(Z1),XQ(Z2))} in place of {\tt MOD} and {\tt FLOOR(A/B)} in the preceding programs, then {\tt BEZOUT} and {\tt BEZOUR} can take Gaussian integers as parameters, on condition that you replace the names of the variables {\tt A, B, R...} with {\tt Z1, Z2, Z3...}.\\ If you use the symbolic calculation function {\tt REMAINDER} in place of {\tt MOD} in the preceding programs, then {\tt BEZOUT} and {\tt BEZOUR} can take polynomials as parameters, on condition that you replace the names of the variables {\tt A, B, R...} with {\tt E1, E2, E3...} and that you change the test for completion. \section{Decomposition into prime factors} \subsection{Algorithmic syntax} - Initial algorithm\\ Let $N$ be a whole number.\\ For all numbers $D$ from 2 to $N$, test the divisibility of $N$ by $D$.\\ If $D$ is a divisor of $N$, then find the divisors of $N/D$, and so on... with $N/D$ taking the role of $N$. The process stops when $N = 1$.\\ As divisors are found, they are stored in the list {\tt FACT}. \begin{verbatim} function facprimes(N) local D FACT 2->D {} -> FACT\end{verbatim} {\tt While N $\neq$ 1 do}\begin{verbatim} If N mod D = 0 then FACT + D -> FACT N/D -> N Else D+1 -> D endif endwhile Result: FACT endfunction \end{verbatim} - First improvement \\ One only tests the divisors D that are between 2 and $FLOOR(\sqrt N)$.\\ In effect, if $N = D1* D2$, then we can say:\\ Let $D1 \leq FLOOR(\sqrt N)$, and let $D2 \leq FLOOR(\sqrt N)$; otherwise, we would have:\\ $D1 * D2 \geq (FLOOR(\sqrt N) + 1)^2 > N$. \begin{verbatim} function facprimes(N) local D FACT 2-> D {} -> FACT\end{verbatim} {\tt While D*D $\leq$ N do} \begin{verbatim} If N mod D = 0 then FACT + D -> FACT N/D -> N Else D+1-> D endif endwhile FACT + N -> FACT Result: FACT endfunction \end{verbatim} - Second improvement \\ One looks to see whether 2 is a divisor of $N$, then one tests only the odd divisors $D$ that are between 3 and $FLOOR(\sqrt N)$.\\ In the list {\tt FACT}, each divisor is written followed by its exponent:\\ decomp(12)=\{2,2,3,1\}. \begin{verbatim} function facprimes(N) local K D FACT {}->FACT 0 -> K While N mod 2 = 0 do K+1 -> K N/2 -> N endwhile\end{verbatim} {\tt If K $\neq$ 0 then} \begin{verbatim} FACT + {2 K} -> FACT endif 3-> D\end{verbatim} {\tt While D*D $\leq$ N do} \begin{verbatim} 0 -> K While N mod D = 0 do K+1 -> K N/D -> N endwhile\end{verbatim} {\tt \ \ If K $\neq$ 0 then} \begin{verbatim} FACT + (D K( -> FACT endif D+2 -> D endwhile {\tt If N $\neq 1$ 1 then} \begin{verbatim} FACT + {N 1} -> FACT endif Result: FACT endfunction\end{verbatim} \subsection{{\tt HP40GS} syntax} This is a translation of the last algorithm.\\ The {\tt HP40GS} calculator does not understand \{\}, so to initialise {\tt L1} as an empty list you must type:\\ {\tt CLRVAR L1}\\ or {\tt SUB L1;L1;2;1}.\\ Here is the program {\tt FACTPRIMES}: \begin{verbatim} INPUT N;"N";;;1: ERASE: 0 ->K: CLRVAR L1: WHILE N MOD 2 == 0 REPEAT 1+K -> K: N/2 -> N: END:\end{verbatim} {\tt IF K $\neq$ 0 THEN}\begin{verbatim} {2,K} ->L1: END: 3 ->D:\end{verbatim} {\tt WHILE D*D $\leq$ N REPEAT}\begin{verbatim} 0 -> K: WHILE N MOD D == 0 REPEAT K+1 -> K: N/D -> N: END:\end{verbatim} {\tt IF K $\neq$ 0 THEN}\begin{verbatim} CONCAT (L1,{D,K}) -> L1: END: 2+D -> D: END:\end{verbatim} {\tt IF N $\neq$ 1 THEN}\begin{verbatim} CONCAT (L1, {N,1}) -> L1: END: DISP 3; "FACT" L1: FREEZE: \end{verbatim} \section{Calculating $ A^P \ mod\ N$}\label{sec:puimod} \subsection{Algorithmic syntax} - First algorithm\\ This algorithm uses two local variables, {\tt POWER} and {\tt I}.\\ Write an iterative program such that at each stage, {\tt POWER} represents ${\tt A^I \ (\bmod \ N)}$. \begin{verbatim} function powermod (A, P, N) local POWER, I 1 -> POWER for I = 1 to P do A*POWER mod N -> POWER endfor Result: POWER endfunction\ \end{verbatim} - Second algorithm\\ This algorithm uses a single local variable {\tt POW}, but it varies {\tt P} such that at each stage of the iteration we have:\\ ${\tt result = POW * A^P \ (bmod N)}$ \begin{verbatim} function powermod (A, P, N) local POW 1 -> POW While P>0 do A*POW mod N -> POW P-1 ->P endwhile Result: POW endfunction \end{verbatim} - Third algorithm\\ One can easily modify this program by taking into account that:\\ ${\tt A^{2*P} = (A*A)^P}$.\\ Therefore when P is even, we have the relation:\\ ${\tt PUI*A^P = PUI*(A*A)^{P/2} \ (\bmod \ N)}$\\ and when P is odd, we have the relation:\\ ${\tt PUI*A^P = PUI*A*A^{P-1}\ (\bmod \ N)}$.\\ We are left with a very fast algorithm for ${\tt A^P \ (\bmod \ N)}$. \begin{verbatim} function powermod (A, P, N) local POW 1->POW While P<0 do If P mod 2=0 then P/2->P A*A mod N->A Else A*POW mod N ->POW P-1->P endif endwhile Result: POW endfunction \end{verbatim} It goes without saying that if {\tt P} is odd, then {\tt P-1} is even.\\ One can then write: \begin{verbatim} function powermod (A, P, N) local POW 1->POW While P<0 do If P mod 2=1 then A*POW mod N->POW P-1->P endif P/2-(>P A*A mod N->A endwhile Result: POW endfunction \end{verbatim} \subsection{{\tt HP40GS} syntax} The calculation of ${\tt A^p \bmod N}$ is treated in the program on the probability method of Mr Rabin. Please refer to the {\tt HP40GS} version of that program (cf \ref{sec:rabin}). \section{The function "isprime"} \subsection{Algorithmic syntax} - Initial algorithm Write a Boolean function of {\tt N} that is equal to {\tt TRUE} when {\tt N} is prime and {\tt FALSE} when it is non-prime.\\ For this, find whether {\tt N} has a divisor $\neq$ 1 and $\leq$ ${\tt FLOOR(\sqrt N)}$ (the whole part of the square root of {\tt N}).\\ The case where {\tt N=1} is treated separately!\\ Here, the Boolean variable {\tt PRIME} is used, which is {\tt TRUE} by default, and which is set to {\tt FALSE} if a divisor of {\tt N} is found. \begin{verbatim} Function isprime(N) local PRIME, I, J\end{verbatim} ${\tt FLOOR((\sqrt N)->J}$ \begin{verbatim}If N = 1 then FALSE->PRIME else TRUE->PRIME endif 2->I\end{verbatim} {\tt While PRIME and I $\leq$ J do} \begin{verbatim} If N mod I=0 then FALSE->PRIME else I+1->I endif endwhile Result: PRIME endfunction \end{verbatim} - First improvement\\ Of course, we can test to see if {\tt N} is even, then look only for odd divisors of {\tt N}. \begin{verbatim} function isprime(N) Local PRIME, I, J\end{verbatim} ${\tt FLOOR(\sqrt N)->J}$\\ {\tt If (N = 1) or (N mod 2 = 0) and (N$\neq$2) then}\begin{verbatim} FALSE->PRIME else TRUE->PRIME endif 3->I\end{verbatim} {\tt While PRIME and I $\leq$(J do} \begin{verbatim} If N mod I = 0 then False -> PRIME} else I+2->I endif endwhile Result: PRIME endfunction \end{verbatim} - Second improvement\\ We can test to see if {\tt N} is divisible by 2 or 3, then look only for divisors of {\tt N} that are of the form $6 \times k-1 $ oo $6 \times k+1 $. \begin{verbatim} function isprime(N) local PRIME, I, J\end{verbatim} ${\tt FLOOR(\sqrt N)->J}$ \begin{verbatim}If (N = 1) or (N mod 2 = 0) or (N mod 3 = 0) then FALSE-(PRIME else TRUE->PRIME endif If N=2 or N=3 then TRUE->PRIME endif 5->I\end{verbatim} {\tt While PRIME and I $\leq$ J do} \begin{verbatim} If (N mod I = 0) or (N mod (I + 2) = 0) then FALSE->PRIME else I + 6 -> I endif endwhile Result: PRIME endfunction end{verbatim} \subsection{{\tt HP40GS} syntax} \begin{verbatim} INPUT N;"N";;;1: IF N MOD 2== 0 OR N MOD 3==0 OR N==1 THEN 0 ->P: ELSE 1->P: END: IF N==2 OR N==3 THEN 1->P: END: 5->I:\end{verbatim} ${\tt FLOOR(\sqrt {N})->J:}$\\ {\tt WHILE I $\leq$ J AND P REPEAT}\begin{verbatim} IF N MOD I==0 OR N MOD (I+2)==0 THEN 0 ->P: ELSE I+6 ->I: END: END: ERASE: DISP 5;P: FREEZE: \end{verbatim} \section{Mr Rabin's probability method}\label{sec:rabin} If $N$ is prime, then all numbers $K$ less than $N$ are prime with $N$, so according to Fermat's Little Theorem we have:\\ $ K^{N-1} = 1 \ (\bmod\ N)$\\ If $N$ is not prime, the integers $K$ such that:\\ $ K^{N-1} = 1 \ (\bmod\ N)$ \\ are very few indeed.\\ A number $N$ such that $ K^{N-1} = 1 \ (\bmod\ N)$ for 20 random tries of $K$ is called a pseudo-prime number. The probability method of Rabin consists of choosing a random number $K$ ($1< K < N $) and calculating:\\ $ K^{N-1} \ (\bmod\ N)$ \\ If $ K^{N-1} = 1 \ (\bmod\ N)$), another random number is tried, whereas if $ K^{N-1} \neq 1 \ (\bmod\ N)$, it is certain that $N$ is not prime.\\ If $ K^{N-1} = 1 \ (\bmod\ N)$ is obtained for 20 tries of $K$, one can conclude that $N$ is prime with a very small probability of error. Naturally, this method is used to test whether large numbers are pseudo-primes, but the command {\tt ISPRIME?} use the Miller-Rabin method. This method use also Fermat's Little Theorem and is also a probabilist method. It conclude that $N$ is prime with a very small probability of error (less than $(0.25)^{20}$ if you make 20 random tries of $K$ or on the order of $10^{-12}$). \subsection{Algorithmic syntax} Let us suppose that:\\ {\tt Random(N)} returns a random whole number between 0 and $N-1$.\\ Calculation of:\\ $ K^{N-1} \bmod\ N$\\ is carried out using the very fast algorithm developed earlier (cf page \pageref{sec:puimod}.\\ {\sc Note} the statement:\\ {\tt powermod(K, P, N)}, the function that calculates ${\tt K^P \bmod\ N}$. \begin{verbatim} Function isprime(N) local K, I, P 1->I 1->>P While P = 1 and I < 20 do Random(N-2)+2->K powermod(K, N-1, N)->P I+1->I endwhile If P =1 then Result: TRUE else Result: FALSE endif endfunction \end{verbatim} \subsection{{\tt HP40GS} syntax} \begin{verbatim} PROMPT N: RANDSEED TIME: 1->I: 1->P: WHILE I < 20 AND P==1 REPEAT FLOOR( RANDOM * (N-2))+2->K: N-1->M: @ Calcul de K puissance M mod N dans P. 1->P: WHILE 0 < M REPEAT IF M MOD 2 == 0 THEN M / 2 -> M : (K * K) MOD N ->K : ELSE K*P MOD N -> P: M - 1 -> M: END: END: @ P contient K puissance M mod N et M=N-1. I+1 ->I: END: ERASE: IF P==1 THEN DISP 3;"PREMIER " N: ELSE DISP 3;"NON PREMIER " N: END: FREEZE: \end{verbatim} {\sc Remark}: You can also use the computer algebra function {\tt POWMOD}, substituting:\\ {\tt MODSTO(N):}\\ {\tt POWMOD(K,N-1) -> P:}\\ for the statements between the comments ( @ ). This results in : \begin{verbatim} PROMPT N: RANDSEED TIME: 1->I: 1->P: WHILE I < 20 AND P==1 REPEAT FLOOR( RANDOM * (N-2))+2->K: MODSTO(N): POWMOD(K,N-1)-> P: I+1 ->I: END: ERASE: IF P==1 THEN DISP 3;"PREMIER " N: ELSE DISP 3;"NON PREMIER " N: END: FREEZE: \end{verbatim} \chapter{GNU Free Documentation License} \label{sec:fdl} {\tiny Version 1.1, March 2000 Copyright (C) 2000 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. 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If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. } \appendix \chapter{Index} Skip 2 pages. \twocolumn %\newpage {\tt \printindex} \chapter{Table of contents} Skip 2 pages %\newpage\twocolumn \tableofcontents \end{document}