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\title{\verb?ALGB 2.3?.}
\author{
 Bernard Parisse\\ Institut Fourier (CNRS UMR 5582)\\ 
 Universit\'e de Grenoble I\\ F-38402 St Martin d'H\`eres C\'edex\\ 
 T\'el. (33 | 0) 4 76 51 43 14 \\ 
 \protect\verb|Bernard.Parisse@ujf-grenoble.fr| \\ 
}







\newtheorem {th}{Th\'eor\`eme}

\newtheorem {cor}[th]{Corollaire}

\newtheorem {prop}[th]{Proposition}

\newtheorem {lemme}[th]{Lemme}

\newtheorem {rem}[th]{Remarque}

\newcommand {\etoile}{($^{\ast}$)}

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\protect\noindent {\protect\verb|#1|} \hfill {\protect\verb|#2|} \\ 
\protect\verb|BYTES: #3| \hfill \protect\verb|CRC: #4h| \\ 

{\protect\em Explication:\/}\\  #6

{\protect\em Exemple d'utilisation:\/}\\ 
 }

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{\newpage
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\samepage \begin{tabular}{|l|l|l|l|l|l|}\hline
True data & Example & User & Example & Internal & Example \\  \hline
Integer & 5 & real & 5 & real & 5 \\ 
Float & 5.02 & real & 5.02 & real & 5 \\ 
Complex & $1+2i$  & symbolic & \verb|'1+2*i'|& complex & \verb|(1,2)| \\  
Complex & (1.1,2.3) & complex & \verb|(1.1,2.3)| & complex & \verb|(1.1,2.3)| \\ 
Fractions & $\frac{2}{3}$   & symbolic & \verb|'2/3'| & symbolic & \verb|'2/3'| \\ 
Irr. quadr. & $1+2\sqrt{5}$ & symbolic & \verb|'1+2*V3'| & program & \verb|<< 1 2 5 >>|\\ 
Unknowns & $a$, $x$ ... & variables & \verb|A| \verb|X| &list & {\em variable}\/ \\ 
Symbolics & $a+x^2$   & symbolic & \verb|'A+X^2'| & list & {\em variable}\/ \\ 
Lists & \verb|{| 1 $i$ \verb|}| & list & \verb|{ 1 'i' }| & list & \verb|{ 1 (0,1) }|\\ 
Array & \verb|[ [ 1 2 ] [ 3 4 ] ]| & array &  \verb|[ [ 1 2 ] [ 3 4 ] ]| &
array &  \verb|[ [ 1 2 ] [ 3 4 ] ]| \\ 
Symb. array & \verb|{ { 1 2 } { 3 4 }| & list & \verb|{ { 1 2 } { 3 4 }| &
list & \verb|{ { 1 2 } { 3 4 }| \\  \hline
\end{tabular}
}

{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 69
\label{rem:e_flag13}

\begin{itemize}
\item 
If flag 13 is cleared (real mode), all global names and all non
   rational functions are considered as real w.r.t. the instructions
   \verb|RE|, \verb|IM|, \verb|CONJ|. This could lead to false 
simplifications if a global
   name stays for a complex, or if a non-rational inverse function is
   called with a usually forbidden real argument, like
   \verb|LN(-1)| or \verb|ASIN(2)|
   Solution: in the first case, replace your global name, say 'Z',
   by 'X+iY'.
   In the second case, set flag 13.

\item  In internal mode, symbolics returned by \verb|ALGB| may contain lists. 
This is 
   not allowed by the HP compiler/decompiler, hence the stack display may 
   show something like:
   \verb|'UNKNOWN/2'| or \verb|'UNKNOWN/UNKNOWN'| or \verb|'UNKNOWN+X'|
   (e.g. by typing \verb|{ 1 2 } { 3 4 } NDXF|)
   Caution, you can {\em not}\/  edit such objects.
   Solution: use \verb|FXND| to know the numerator and denominator of the
   above fractions, or use another stack display like \verb|EQSTK| (by Mika
   Heiskanen) or \verb|JAVA| (by Richard Steventon and Andre Schoorl).
\end{itemize} 
\end{rem}
}

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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rightarrow \frac{1}{4}\ln(x-2)
-\frac{1}{4}\ln(x+2)-\left(\frac{1}{4}\ln(\alpha -2)-\frac{1}{4}
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\rightarrow \frac{1}{4}\ln(x-2)
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{\newpage
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\samepage \setbox\sizebox=\hbox{$x\ln(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline967: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$1/(\sin(x)+2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline973: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\ln \ln (x), \quad \frac{1}{e^{\ln(x)^2+1}-1}, \quad
x^3e^{\left(\frac{x+1}{x+2}\right)} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sqrt{x^2-1}  , \quad e^{\ln(x)^2+1} \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-\sin(x)/x^2+\cos(x)/x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline975: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\int _1^2 \frac{1}{x^3+1}  = 
\frac{\ln(3)-2\ln(2)}{6} +\frac{\pi }{18}\sqrt{3} \end{displaymath}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\left\{\begin{array}{lllllll}
y_1 ' (x) &= &  y_1(x) &- &  y_2(x) &+& 1 \\ 
y_2 ' (x) &=& 2 y_1(x)& +& 4 y_2(x)& +& e^x
\end{array} \right. \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}a y'' + b y' + c y = f(x)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(a p^2 + b p + c) L(y) = L(f) (p) + a (p f(0) + f'(0)) + b f(0)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}( L(f) (p) + a (p f(0) + f'(0)) + b f(0) / ( a p^2 + b p + c) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y' = A y + b \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n \times n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1017: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}(p Id - A ) L(y) = L(b) + y(0) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L(y) = ( p Id - A )^-1 ( L(b) + y(0) )\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}( p Id - A )^-1 ( L(b) + y(0) ) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left\{ \begin{array}{rlc}
y_1 &=& -1/2*e^x-2/3+2*e^{2x}-5/6*e^{3x} \\ 
y_2 &=& 1/3 - 2* e^{2x} + 5/3*e^{3x}
\end{array} \right. \end{displaymath}
}

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{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 158
\label{rem:e_cosn}
$\cosh(nx)$  and $\sinh(nx)$ as polynomials of $\cosh(x)$ and $\sinh(x)$ 
have the same expression.
\end{rem}
}

{\newpage
\clearpage
\samepage \begin{displaymath}3 \rightarrow (-1+4c^2)s \quad  -3c+4c^3 \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sin(3x)=(-1+4\cos(x)^2)\sin(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1119: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cos(3x)=-3*\cos(x)+4*\cos(x)^3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1121: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\det(A)^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1161: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}(x I_n -A)P(A)=M(x) I_n = M(x) I_n - M(A) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}A=\left(\begin{array}{lcr}
 1&-1 & 0\\ 
0 & 1 & -1 \\ 
-1 & 0 & 1
 \end{array}\right) \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{3\pm\sqrt{3}i }{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1175: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\{ (1,1,1), (1,\frac{-1\mp i\sqrt{3} }{2}, \frac{-1\pm i\sqrt{3} }{2})
\} \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0, (3+\sqrt{3}i)/2, (3-\sqrt{3}i)/2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1177: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}\left(\begin{array}{ccc}
1 &2 &1 \\ 
2 & 0 &0 \\ 
1 & 0 &3
 \end{array}\right)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}A=\left(\begin{array}{ccc}
 3& -1& 1\\ 
 2 & 0 & 1\\ 
 1 & -1 & 2
 \end{array}\right),\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 202
\label{rem:e_diag}
If flag 12 is set (normal state), then eigenvectors and characteristic
vectors are divided by their common greatest divisor. In case of Jordan
chains, this means that the image of the $i$-th vector of the chain by 
$A-\lambda I_n$  is not necesseraly equal (but always colinear) to the 
$i+1$-th vector of the chain.
\end{rem}
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\left\{ \begin{array}{ccccc}
mx &+&y &=&-2 \\ 
mx &+ &(m-1)y&=&2
 \end{array}\right.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(2m-m^2)x=2m, \quad \quad (2-m)y=-4.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}0x+0y=4 \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 214
\label{rem:e_syst}
\verb|SYST| and \verb|SOLG| order the unknown so that principal
unknown are followed by auxiliary unknown. For the $m=0$ case, the
list of unknown returned is \verb|{ Y -1 X }|, this means that \verb|Y|
is a principal unknown and \verb|X| an auxiliary one.
\end{rem}
}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}A=\left(\begin{array}{cccc}
1 & 2 \\ 
3 & 4
 \end{array}\right) \,\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}l=\left(\begin{array}{rlc}
 1 &0 \\ 
 -3 & 1
\end{array}\right)
\quad \quad
U= \left(\begin{array}{rlc}
1 & 2\\ 
0 & -2
\end{array}\right) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left(\begin{array}{cccc}
1 & 2& 4 & 6\\ 
-1 & 3 & 5 & 7 \\ 
2 & 1 & 0 & 1 \\ 
2 & 6 & 9 & 14
 \end{array}\right)\ ,\end{displaymath}
}

{\newpage
\clearpage
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\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{tabular}{|l|l|l|l|} \hline
Program & Option &   Input         &           Output    \\ \hline
RED & RANG  &  1: matrix       &   2: pivots, 1: redcued matrix   \\ \hline
RED &RED   &  1: matrix        &  2: pivots, 1: rref-ed matrix  \\ \hline
RED &DET  &  1: matrix  & 3: pivots, 2: reduced matrix , 1:determinant  \\ \hline
RED &INVL  &  1: matrix        & 2: pivots, 1: inverse         \\ \hline
RED &SYST  &  2: matrix, 1: list of unknowns &  4: list unknowns,\\ 
    &      &   &  3: list of principals and auxiliary unknowns \\ 
    &      &   &  2: list of pivots \\ 
    &      &   &  1: list of tagged algebraics \\ \hline
RED &SOLGEN & 2: matrix, 1: list of unknowns & 5: pivots, 4: list of unknowns, \\ 
  & &  &  3: list of principal and auxiliary unknowns \\ 
 & &  &  2: list of tagged algebraics, 1: result \\ \hline
\end{tabular}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}A=\left(\begin{array}{lcr}
 4 & 1\\ 
 1 & -3
 \end{array} \right) ,\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}A=
\left(\begin{array}{lcr}
 4 & 0\\ 
 1 & 1
 \end{array} \right) \times
\left(\begin{array}{lcr}
 '1/4' & 0\\ 
 0 & '-13/4'
 \end{array} \right) \times
\left(\begin{array}{lcr}
 4 & 1\\ 
 0 & 1
 \end{array} \right). \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}q(x,y)=4x^{2}+2xy-3y^{2} = \frac{1}{4} (4x+y)^{2}-\frac{13}{4} y^{2}. \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 284
\label{rem:e_quadr}
\begin{itemize}
\item If you want to use \verb|AXQ| with large array (more than 4x4),
   add global identifiers to the VXYZ list.
\item  
If you want to save time, use numeric mode (hit <->NUM> in user mode)!
\end{itemize}
\end{rem}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}1.20710678118 \rightarrow  \frac{1}{2} +\frac{1}{2} \sqrt{2} \quad
      1.5  \rightarrow  \frac{3}{2} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(X+1) = -(X^2+2X+1)*(-1) + (X^2+3X+2)*1 \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}.5 \rightarrow  \frac{1}{2}, \quad 
  \pi \rightarrow \mbox{\verb|NUM|} \rightarrow  \{ 3 7 15 1 292 \} \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\pi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1345: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\stepcounter{section}
\stepcounter{subsection}
{\newpage
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\samepage \setbox\sizebox=\hbox{$\leftrightarrow$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1351: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\stepcounter{subsection}
{\newpage
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{\newpage
\clearpage
\samepage \begin{rem}% latex2html id marker 322
\label{rem:e_factor}
Factorization of polynomial is very important in several mathematical
functions, like symbolic integration or matrix diagonalization. It is
important to understand the mecahnism used by \verb|ALGB| to perform
this tasks.

  Let's begin by recalling some mathematicals facts:
\begin{itemize}
\item Theorem (d'Alembert):\\ 
A polynomial of degre $n$  has exactly $n$ complex roots (counted with multiplicity).
\item   Theorem (Galois):\\ 
There is no formula for solving a generic polynomial of degre $\leq $  5
(by algebraic operations and extraction of $n$-th roots)
\end{itemize}
  This means that you can not root a multivariate polynomial of order
greater than 5 (for such polynomials, systems like Maple, Reduce, Axiom
Mathematica or Mupad use algebraic extension), and that you can only root 
numerically a univariate
polynomial of order $\geq $ 5. Note that the generic solution
of a polynomial of order 3 is still complicated and of order 4 very
complicated. I think that it is not possible to handle the generic
solution of polynomials of order 3 or 4 on the HP48 in a raisonable
amount of time.
Summary: only polynomial of order 2 are generically solved by \verb|ALGB|.

However, in some situations, you can root exactly 
polynomials of order $\leq $ 3,
by searching multiple roots and by finding obvious roots (or obvious factor). 
The rooting algorithm of \verb|ALGB| search first multiple roots by computing
the \verb|GCD| of the polynomial and his first derivative (this is
the \verb|SQFFext| algorithm in the source of \verb|ALGB|). Of course
flag 12 must be set for this step to be done.
  If a univariate
polynomial has only integer (or rational) coefficients, you can find all
rational solutions of this polynomial by testing a finite set of rationals
(of the form numerator/denominator where numerator is a divisor of
the constant coefficient and denominator a divisor of the leading
coefficient). This is implemented in \verb|ALGB| by the nullnamed 
\verb|XLIB EVIDENText| which is called if flag 14 is set.

If exact solving fails, \verb|ALGB| calls the numeric solver (which is the
\verb|PROOT| function in \verb|ALGBG.LIB| or the Bairstow or Laguerre 
algorithm in
\verb|ALGB.LIB|) and tries to find second order polynomial with integer
coefficients by coupling 2 approximate solutions (this was an idea of
Mika Heiskanen implemented in \verb|POLYLIB|).

Summary: \verb|ALGB| should find all rationals and quadratic irrationals roots
of a univariate polynomial (unless the polynomial is bad conditionned).

  For multivariate polynomials, the two first steps are achieved
(\verb|EVIDENText| and \verb|SQFFext|). \verb|ALGB| should find 
all rational multivariate
roots of a polynomial (1st order factors). 
Unfortunately, \verb|ALGB| does not implement the exhaustiv search of
all 2nd order (or greater) multivariate rationals factors. This can be
performed using the \verb|FCTR| function of the \verb|ALG48| library.

  Summary: \verb|ALGB| should find 1st order factors of multivariate 
polynomials.

  Abstract of the Sysrpl \verb|XLIB|s (include \verb|alextdec.h| and 
\verb|alhash.h| in your source code to use them) to factorize:
\begin{itemize}
\item \verb|EVIDENText|: finds rational roots
\verb|SQFFext|: finds square-free factorization of a polynomial
\verb|SOLVext|: roots a univariate polynomial numerically and tries to rebuild
quadratic irrationals roots
\end{itemize}
  Abstract of the users commands to factorize:
\begin{itemize}
\item \verb|COLC|:\\ 
 for symbolic input calls \verb|SQFFext|, then \verb|EVIDENText|, does {\bf not}  call  \verb|SOLVext|,\\ 
for list input calls only \verb|SQFFext|
\item \verb|FROOT|: calls \verb|SQFFext| then \verb|EVIDENText| then 
\verb|SOLVext|,
\item \verb|FACTO|: calls \verb|SQFFext| then \verb|EVIDENText|, does not 
call \verb|SOLVext|
\end{itemize}
Flags 12 and 14 may be cleared to skip respectively \verb|SQFFext| and 
\verb|EVIDENText|.
\end{rem}
}

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\appendix
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{\newpage
\clearpage
\samepage \begin{tabular}% latex2html id marker 356
{|l|c|c|c|}\hline
Name & Function        &  Arguments           &     Returns  \\ \hline
\verb|{ALGB}|  &(\verb|{VAR}|) & &  \\  \hline
ARIT & Directory       &nothing  &               nothing \\ 
CST  & Cst Menu        &nothing   &               $\{$ l $\}$   \\ 
fr & English doc & nothing & string \\ 
FUNC  &Directory       &nothing    &              nothing\\ 
GETALL&Get All Modules &nothing     &             nothing\\ 
INIT  &Initialization  &nothing      &            nothing\\ 
INTG  &Directory       &nothing  &                nothing\\ 
MATR  &Directory       &nothing   &               nothing\\ 
PURG  &Purge ALGB      &nothing    &              nothing\\ 
QUAD  &Directory       &nothing     &             nothing\\ 
risch & Symbolic integration & $s$ & $s$ \\ 
SCST  &CST-Menu string &nothing      &            string\\ 
SETFR &Set French Flags&nothing       &           nothing\\ 
TSIMP & Transcendental simplification & $s$ & $s$ \\ 
UKEYS &User keys string&nothing        &        string\\ 
us & English doc & nothing & string \\ 
\hline

\verb|{ALGB ARIT}|  &(\verb|{VAR}|) & &  \\  \hline
CAB  & Bezout $ax+by=c$ & 3,2,1: $c$, $a$, $b$ & 1:1 [3,2: $x$, $y$] ou 1:  0 \\ 
EULER & Euler indicatrix & $n$ & $n$ \\ 
FCLI  & $\rightarrow$ continued fraction & \% & $f$ or \% or \{ l \} \\ 
LIFC  & continued fraction $\rightarrow $ & \{ l \} & 2: $N$, 1: $D$ \\ 
ORND  & Round an object & 2: $o$, 1: $D$  & $o$ \\ 
XFRC & To quadratic irrational & $o$ & $o$ \\ 
\hline

\verb|{ALGB FUNC}| & (\verb|{VAR}|) & &  \\  \hline
COSN & $\cos, \sin(nx) \rightarrow P(\cos x, \sin x)$ &
$n > 0$ & 2: $s$, 1: $s$ \\ 
     &  & $n<0$ & 2: \{ $p$ \}, 1: \{ $p$ \} \\ 
DER & derivative & 2: $s$, 1: $v$ & 1: $s$ \\  
TAYLR & Taylor & 3: $s$, 2: $v$, 1: $n$ & 3: $n$, 2: \{ $p$ \}, 1: $s$ \\ 
\hline

\verb|{ALGB INTG}|  &(\verb|{VAR}|) & &  \\  \hline
INTG & Symbolci integration & $s$ & $s$ \\ 
LIN & Linearization & C\%, $s$ ou \{ $p$ \} & $s$ \\ 
PEVAL  & Evaluation & 3: primitive, 2,1:bornes & $s$ \\ 
PRIMIT & Last primitive & Rien & $s$  \\ 
VX & integration variable & Rien & $v$ \\ 
\hline

\verb|{ALGB MATR}|  &(\verb|{VAR}|) & &  \\  \hline
cross & Wedge product & 2: $x$, 1: $y$ & $x \wedge y $ \\ 
DET & Determinant & endomorphism & determinant \\ 
DIAG & Diagonalisation (Faddeev) & endomorphism & 7 \`a 1: {\em cf.\/} section \ref{sec:linear} \\ 
MATRIX & Last matrix & nothing & matrix \\ 
PCAR & Characteristic polynomial & endomorphism & $s$ \\ 
RED & Gau\ss reduction & matrix or system & {\em cf.\/} section \ref{sec:linear} \\ 
\hline

\verb|{ALGB QUAD}|  &(\verb|{VAR}|) & &  \\  \hline
AXQ & array to $s$ quadratic form  & \{ $m$ \} & $s$ \\ 
    &             & 2: \{ $m$ \}, 1: \{ $v$ \} & $s$ \\ 
GAUSS & Gau\ss\ quadratic form reduction & 1: $A$ & 2: $D$, 1: $P$ \\ 
      &                         & $s$  & 5: $D$,
                                         4: $P$, 3: $A$, 2: \{ $v$ \},1: $s$ \\ 
      &                         & 2: $s$, 1: \{ $v$ \} & 5: $D$,
                                         4: $P$, 3: $A$, 2: \{ $v$ \},1: $s$ \\ 
QXA & $s$ quadratic form to array & 2: $s$, 1: \{ $v$ \} & \{ $m$ \} \\ 
      &                             & $s$ & \{ $m$ \} \\ \hline
\end{tabular}
}

{\newpage
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\samepage \begin{tabular}{|l|c|c|c|}\hline
Name & Function        &  Arguments           &     Returns  \\ \hline
\verb|{ALGB.LIB}|  &(\verb|0:788|) & &  \\  \hline
AXL & array $\leftrightarrow $ list & [ ] ou \{ $m$ \} & \{ $m$ \} ou [ ] \\ 
add & Addition & 2: $o_2$, 1: $o_1$ & $o_2+o_1$\\ 
CHS & Change signe & $o$ & $-o$ \\ 
COLC & Factorization & $s$ & $s$ \\ 
conj & Conjugate & $o$ & $\overline{o} $ \\ 
DEGRE & Order& \{ $p$ \} & $n$ \\ 
DIV1 & Usual division & 2: $o_2$, 1: $o_1$  & $o_2/o_1$ \\ 
DIV2 & Euclidean division & 2: $o_2$, 1: $o_1$ & 2: $o_2$ div $o_1$,
1: $o_2$ mod $o_1$ \\ 
DIVIS & List of divisors & $o$ & \{ l \} \\ 
EXEC & Substitution or {\em doall}\/  & 2: \{ l \}, 1: program & 1: \{ l \} \\ 
     &                                & 2: $s$, 1: $o_1=o_2$ & $s$ \\ 
     &                & 3: $s$, 2: $\{$ l1 $\}$  1: $\{$ l2 $\}$   & $s$ \\  

EPSX0 & Strip expression & $o$ & $o$ \\   
EXPA & Simplification & $o$ & $o'$ \\ 
FACTO & Factorization & $o$ & 3: \{ $v$ \} 2: $f$ , 1: \{ $f_1 n_1 f_2 n_2$ ... \} \\ 
FCOEF & roots/poles $\rightarrow $ Fraction & \{ $r_1 n_1 r_2 n_2$ ... \} & \{ $f$ \} \\ 
FDER & Derivation& $f$  & $f$  \\ 
FINTG & Integration & $f$ & $s$\\ 
FROOTS & Factorisation & $o$ & 3: \{ $v$ \} 2: $f$ , 1: \{ $s_1 n_1 s_2 n_2$ ... \} \\ 
FXND & Split a fraction & $f=N/D$ & 2: $N$, 1: $D$ \\ 
GCD1  & Greatest common divisor & 2: $o_2$, 1: $o_1$ & GCD($o_2$,$o_1$)\\ 
GCD3  & GCD (solves $au+bv=d$) 	&	2,1: $a$ , $b$ & GCD$(a,b)=d$, $u$, $v$ \\ 
HORN & Horner scheme & 2:$p$ , 1: r & 3:  $p/(X-r)$ , 2: $r$, 1: $P(r)$ \\ 
idn &  identity     &     real integer or matrix   & identity matrix \\ 
im & imaginary part & $o$ & $\Im (o)$ \\ 
L2S & Evaluation & 2: \{ $p$ \}, 1:$v$  & $p(v)$ \\ 
    &              & 2: \{ $p$ \},1: \{ $v$ \} & $p(v)$\\ 
LCM  & Least common multiple &  2: $o_2$, 1: $o_1$ & LCM($o_2$,$o_1$)\\ 

LGCD & GCD of a list & \{ l \} & $o$=GCD \\ 
LU2  & $LU$  decomposition  & $M$                    & $L^-1$, $U$  \\ 
LVAR & list of variables & $o$ & \{ $v$ \} \\ 
MAD & inverse, char. polyn., etc. & $o$ & 4: det, 3: $1/o$, 2: \{ $p$ \},
1: \{ $p$ \} \\ 
MMULT & special product & 3: $o_2$, $o_1$, $n$  & ``$o_2 \times o_1$''\\ 
MULT & product & 2: $o_2$, $o_1$ & $o_2 \times o_1$\\ 
NDXF & create a fraction & 2: $N$, 1: $D$ & $f=N/D$ \\ 
PF &   Partial fraction & $f$   & $\sum_i f_i $ \\ 
POWER & integral power & 2: $o$, 1: $n$  & $o^{n}$ \\ 
PTAYL & Taylor for polynomials & 2: $P(X)$, 1: $o$ &
$P(X-o )$ \\ 
re & real part & $o$ & $\Re(o)$ \\ 
rref &  Row reduction   &  $M$        &  \{ $s$ \}, reduced matrix\\ 
S2L & Symbolic to list & 2: $o$, 1: \{ $v$ \} & 2: \{ $v$ \},
1:\{ $p$ \} \\ 
                &                       & 2: $o$, 1: $v$  & \{ $p$ \} \\ 
SIMP2 & Simplification & 2: $o_2$, 1: $o_1$ & 2: $o_2'$, 1: $o_1'$ \\ 
SQRT & Square root & $n$ or C\% or $s$ & $n$ or C\% or $s$ \\ 
SUBT & Substraction& 2: $o_2$, 1: $o_1$ & $o_2-o_1$ \\ 
SXL & Conversion & Internal [user] & User [internal]\\ 
tEVAL & Execution time & ..., 1: $o$ & EVAL($o$), 1: time \\ 
TR & trace & [ ] or \{ $m$ \}$=(a_{ij})_{1\leq i,j\leq n} $ & $\sum _{i=1}^{n} a_{ii}$ \\ 
TRAN & transposed & [ ] or \{ $m$ \} & [ ] ou \{ $m$ \} \\ 
VER & Version & rien & \% 2.28 \\ 
XY & produit scalaire de 2 vecteurs & 2: $x$ 1: $y$ & $x.y$ \\ 
XNUM &$\rightarrow$ Numeric & $o$ & $o$ \\ 
XQ &$\rightarrow$ Rational & $o$ & $o$ \\  \hline

\end{tabular}
}

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\samepage \begin{tabular}{|l|l|l|}\hline
Princ. Key  & Second. Key &      Function \\ \hline
EVAL    &                      & L2S\\ 
SIN     &     Derivative       & FDER\\ 
COS     &     Integration      & FINTG\\ 
$\sqrt{\,}$      &     $\sqrt{\,}$ &               SQRT\\ 
$y^x$      &     $y^x$             &   POWER\\ 
$1/x$       &     $1/x$            &    MAD\\ 
$\pm$      &     $pm$         &       CHS\\ 
7       &     SOLVE           &  FROOTS\\ 
8       &     PLOT            &  TSIMP\\ 
9       &     ALGEBRA         &  EXPA\\ 
Division &    Division        &  DIV1\\ 
4        &    TIME            &  tEVAL\\ 
5	 &                    &  EXEC\\ 
6        &                    &  COLC\\ 
*        &                    &  MULT\\ 
1        &                    &  clears flag 13\\ 
2        &                    &  sets flag 13\\ 
3         &   UNITS           &  AXL\\ 
-        &    -               &  SUBT\\ 
SPC      &                    &  rref\\ 
+        &    +                & add \\  \hline
\end{tabular}
}

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