config root man

% $Id: usersch2.tex 7841 2006-04-10 07:25:25Z kb $
% Copyright (c) 2000  The PARI Group
%
% This file is part of the PARI/GP documentation
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\chapter{Specific Use of the gp Calculator}

\section{Introduction}

Originally, \tet{gp} was designed as a debugging device for the PARI system
library, and not much thought had been given to making it user-friendly. The
situation has changed, and \kbd{gp} is very useful as a stand-alone tool. The
operations and functions available in PARI and \kbd{gp} are described in
the next chapter. In the present one, we describe the specific use of the
\kbd{gp} programmable calculator.

\emacs If you have GNU Emacs, you can work in a special Emacs shell,
described in \secref{se:emacs}. Specific features of this Emacs shell are
indicated by an EMACS sign in the left margin.

\subsec{Startup}

To start the calculator, the general command line syntax is:

\kbd{gp [-s \var{stacksize}] [-p \var{primelimit}] [\var{files}]}

\noindent
where items within brackets are optional. The [\var{files}] argument is a
list of files written in the GP scripting language, which will be loaded on
startup. The ones starting with a minus sign are \emph{flags}, setting some
internal parameters of \kbd{gp}, or \var{defaults}. See \secref{se:defaults}
below for a list and explanation of all defaults, there are many more than
just those two. These defaults can be changed by adding parameters to the
input line as above, or interactively during a \kbd{gp} session or in a
preferences file (also known as \tet{gprc}).

If a \idx{preferences file} (or \kbd{gprc}, to be discussed in
\secref{se:gprc}) can be found, \kbd{gp} then read its and execute the
commands it contains. This provides an easy way to customize \kbd{gp}.
The \var{files} argument is processed right after the \kbd{gprc}.

A copyright message then appears which includes the version number, and a lot
of useful technical information. After the copyright, the computer writes the
top-level help information, some initial defaults, and then waits after
printing its prompt, which is '\kbd{?~}' by default . Whether extended
on-line help and line editing are available or not is indicated in this
\kbd{gp} banner, between the version number and the copyright message.
Consider investigating the matter with the person who installed \kbd{gp} if
they are not. Do this as well if there is no mention of the GMP kernel.

\subsec{Getting help}

To get help, type a \kbd{?} and hit return. A menu appears, describing the
eleven main categories of available functions and how to get more detailed
help. If you now type \kbd{?$n$} with $1\le n\le11$, you get the list of
commands corresponding to category $n$ and simultaneously to Section $3.n$ of
this manual. If you type \kbd{?}\var{functionname} where \var{functionname}
is the name of a PARI function, you will get a short explanation of this
function.

If extended help (see \secref{se:exthelp}) is available on your system,
you can double or triple the \kbd{?} sign to get much more: respectively the
complete description of the function (e.g.~\kbd{??sqrt}), or a list of
\kbd{gp} functions relevant to your query (e.g.~ \kbd{???"elliptic curve"}
or \kbd{???"quadratic field"}).

If \kbd{gp} was properly installed (see Appendix~A), a line editor is
available to correct the command line, get automatic completions, and so on.
See \secref{se:readline} or \kbd{??readline} for a short summary of the line
editor's commands.

If you type \kbd{?\bs} you will get a short description of the metacommands
(keyboard shortcuts).

Finally, typing \kbd{?.} will return the list of available (pre-defined)
member functions. These are functions attached to specific kind of objects,
used to retrieve easily some information from complicated structures (you can
define your own but they won't be shown here). We will soon describe these
commands in more detail.

As a general rule, under \kbd{gp}, commands starting with \b\ or with some
other symbols like \kbd{?} or \kbd{\#}, are not computing commands, but are
metacommands which allow you to exchange information with \kbd{gp}. The
available metacommands can be divided into default setting commands
(explained below) and simple commands (or keyboard shortcuts, to be dealt
with in \secref{se:meta}).

\subsec{Input}

Just type in an instruction, e.g. \kbd{1 + 1}, or \kbd{Pi}. No action is
undertaken until you hit the \kbd{<Return>} key. Then computation starts, and
a result is eventually printed. To suppress printing of the result, end the
expression with a \kbd{;} sign. Note that many systems use \kbd{;} to
indicate end of input. Not so in \kbd{gp}: this will hide the result from you!
(Which is certainly useful if it occupies several screens.)

\subsec{Interrupt, Quit}

Typing \kbd{quit} at the prompt ends the session and exits \kbd{gp}. At any
point you can type \kbd{Ctrl-C} (that is press simultaneously the
\kbd{Control} and \kbd{C} keys): the current computation is interrupted and
control given back to you at the \kbd{gp} prompt, together with a message
like
\bprog
  *** gcd: user interrupt after 840 ms.
@eprog\noindent
telling you how much time ellapsed since the last command was typed in and
in which GP function the computation was aborted. It does not mean that that
much time was spent in the function, only that the evaluator was busy
processing that specific function when you stopped it.

\section{The general gp input line}

The \kbd{gp} calculator uses a purely interpreted language GP. The structure
of this language is reminiscent of LISP with a functional notation,
\kbd{f(x,y)} rather than \kbd{(f x y)}: all programming constructs,
such as \kbd{if}, \kbd{while,} etc\dots are functions\footnote{*}{Not exactly,
since not all their arguments need be evaluated. For instance it would be
stupid to evaluate both branches of an \kbd{if} statement: since only one
will apply, only this one is evaluated.}, and the main loop does not really
execute, but rather evaluates (sequences of) expressions. Of course, it is by
no means a true LISP.

\subsec{Introduction}. User interaction with a \kbd{gp} session proceeds as
follows. First, one types a sequence of characters at the \kbd{gp} prompt;
see \secref{se:readline} for a description of the line editor. When you hit
the \kbd{<Return>} key, \kbd{gp} gets your input, evaluates it, then prints
the result and assigns it to an ``history'' array if it is not void (see
next section).

More precisely, you input either a metacommand or a sequence of
expressions. Metacommands, described in \secref{se:meta}, are not part of the
GP language and are simple shortcuts designed to alter gp's internal state
(such as the working precision or general verbosity level), or speed up
input/output.

An \idx{expression}\sidx{expression sequence} is formed by combining
constants, variables, operator symbols, functions (including user-defined
functions) and control statements. It always has a value, which can be any
PARI object. There is a distinction between lowercase and uppercase. Also,
outside of character strings, blanks are completely ignored in the input to
\kbd{gp}. An expression is evaluated using the conventions about operator
priorities and left to right associativity.

Several expressions are combined on a single line by separating them with
semicolons ('\kbd{;}'). Such an expression sequence will be called simply a
\var{seq}. A \var{seq} also has a value, which is the value of the last
expression in the sequence. Under \kbd{gp}, the value of the \var{seq}, and
only this last value, becomes an history entry. The values of the other
expressions in the \var{seq} are discarded after the execution of the
\var{seq} is complete, except of course if they were assigned into variables.
In addition, the value of the \var{seq} is printed if the line does not end
with a semicolon \kbd{;}.

\subsec{The gp history}.

This is not to be confused with the history of your \emph{commands},
maintained by readline. It only contains their non-void results, in sequence.
Several inputs only act through side effects and produce a void result, for
instance a \kbd{print} statement, a \kbd{for} loop, or a function definition.

The successive elements of the history array are called \kbd{\%1}, \kbd{\%2},
\dots As a shortcut, the latest computed expression can also be
called \kbd{\%}, the previous one \kbd{\%`},
the one before that \kbd{\%``} and so on. The total number of history entries
is \kbd{\%\#}.

When you suppress the printing of the result with a semicolon, its history
number will not appear either, so it is often a better idea to assign it to a
variable for later use than to mentally recompute what its number is. Of
course, on the next line, just use \kbd{\%} as usual.

This history ``array'' is in fact better thought of as a queue: its size is
limited to 5000 entries by default, after which \kbd{gp} starts forgetting
the initial entries. So \kbd{\%1} becomes unavailable as \kbd{gp} prints
\kbd{\%5001}. You can modify the history size using \tet{histsize}.

\subsec{Special editing characters}.\sidx{editing characters} A GP program
can of course have more than one line. Since \kbd{gp} executes your commands as
soon as you have finished typing them, there must be a way to tell it to
wait for the next line or lines of input before doing anything. There are
three ways of doing this.

The first one is simply to use the \idx{backslash character} \kbd{\bs} at the
end of the line that you are typing, just before hitting \kbd{<Return>}. This
tells \kbd{gp} that what you will write on the next line is the physical
continuation of what you have just written. In other words, it makes \kbd{gp}
forget your newline character. You can type a \kbd{\bs} anywhere. It is
interpreted as above only if (apart from ignored whitespace characters) it is
immediately followed by a newline. For example, you can type
\bprog
? 3 + \
4
@eprog
\noindent instead of typing \kbd{3 + 4}.

The second one is a slight variation on the first, and is mostly useful when
defining a user function (see \secref{se:user_defined}): since an equal sign
can never end a valid expression, \kbd{gp} disregards a newline immediately
following an \kbd{=}.
\bprog
? a =
123
%1 = 123
@eprog

The third one cannot be used everywhere, but is in general much more useful.
It is the use of braces \kbd{\obr} and  \kbd{\cbr}.\sidx{brace characters}
An opening brace (\kbd{\obr}) \emph{at the beginning of a line}
(modulo spaces as usual) signals that you are typing a multi-line
command, and newlines are ignored until you type a closing brace
\kbd{\cbr}. There is an important, but easily obeyed, restriction:
inside an open brace-close brace pair, all your input lines are
concatenated, suppressing any newlines. Thus, all newlines should occur after
a semicolon (\kbd{;}), a comma (\kbd{,}) or an operator (for clarity's sake,
we don't recommend splitting an identifier over two lines in this way). For
instance, the following program
\bprog
{
  a = b
  b = c
}
@eprog

\noindent would silently produce garbage, since this is interpreted as
\kbd{a=bb=c} which assigns the value of \kbd{c} to both \kbd{bb} and
\kbd{a}.

\section{The PARI types}

\noindent
We see here how to input values of the different data types known to PARI.
Recall that blanks are ignored in any expression which is not a string (see
below).

\misctitle{A note on efficiency.}
The following types are provided for convenience, not for speed:
\typ{INTMOD}, \typ{FRAC}, \typ{PADIC}, \typ{QUAD}, \typ{POLMOD},
\typ{RFRAC}. Indeed, they always perform a reduction of some kind after
each basic operation, even though it is usually more efficient to perform
a single reduction at the end of some complex computation. For instance,
in a convolution product $\sum_{i+j = n} x_i y_j$ in $\Z/N\Z$ (common
when multiplying polynomials!), it is wasteful to perform $n$ reductions
modulo $N$. In short, basic individual operations on these types are fast,
but recursive objects with such components could be handled more efficiently:
programming with libpari will save large constant factors here, compared to
GP.

\subsec{Integers} \sidx{integer}
(type \tet{t_INT}): type the integer (with an initial
\kbd{+} or \kbd{-}, if desired) with no decimal point.

\subsec{Real numbers} \sidx{real number}
(type \tet{t_REAL}): type the number with a decimal
point. The internal precision of the real number is the supremum of the
input precision and the default precision. For example, if the default
precision is 28 digits, typing \kbd{2.} gives a number with internal
precision 28, but typing a 45 significant digit real number gives a
number with internal precision at least 45, although less may be printed.

You can also use scientific notation with the letter \kbd{E} or
\kbd{e}, in which case the (non leading) decimal point may be omitted (like
\kbd{6.02 E 23} or \kbd{1e-5}, but \emph{not} \kbd{e10}). By definition,
\kbd{0.E $N$} (or \kbd{0 E $N$}) returns a real $0$ of (decimal) exponent
$N$, whereas \kbd{0.} returns a real 0 ``of default precision'' (of exponent
$-\tet{realprecision}$), see \secref{se:whatzero}.

\subsec{Intmods}\sidx{intmod}
(type \tet{t_INTMOD}): to enter $n \mod m$, type
\kbd{Mod(n,m)}, \emph{not} \kbd{n\%m}. Internally, all operations are done
on integer representatives belonging to $[0,m-1]$.

Note that this type is available for convenience, not for speed: each
elementary operation involves a reduction modulo $m$.

\subsec{Rational numbers}\sidx{rational number}
(types \tet{t_FRAC}): all fractions are automatically reduced to lowest
terms, so it is impossible to work with reducible fractions. To enter $n/m$
just type it as written. As explained in \secref{se:gdiv}, division is
\emph{not} performed, only reduction to lowest terms.\label{se:FRAC}

Note that this type is available for convenience, not for speed: each
elementary operation involves computing a gcd.

\subsec{Complex numbers}\sidx{complex number} (type \tet{t_COMPLEX}): to
enter $x+iy$, type \kbd{x + I*y} (\emph{not} \kbd{x+i*y}). The letter \tet{I}
stands for $\sqrt{-1}$. Recall from Chapter 1 that $x$ and $y$ can be of type
\typ{INT}, \typ{REAL}, \typ{INTMOD}, \typ{FRAC}, or \typ{PADIC}.

\subsec{$p$-adic numbers}\sidx{p-adic number}\label{se:padic} (type
\tet{t_PADIC}): to enter a $p$-adic number, simply write a rational or
integer expression and add to it \kbd{O($p$\pow $k$)}, where $p$ and $k$
are integers. This last expression indicates three things to \kbd{gp}:
first that it is dealing with a \typ{PADIC} type (the fact that $p$ is an
integer, and not a polynomial, which would be used to enter a series, see
\secref{se:series}), secondly the prime $p$, and finally the number of
significant $p$-adic digits $k$.

Note that it is not checked whether $p$ is indeed prime but results are
undefined if this is not the case: you can work on 10-adics if you want,
but disasters will happen as soon as you do something non-trivial like
taking a square root. Note that \kbd{O(25)} is not the same as
\kbd{O(5\pow 2)}; you want the latter!

For example, you can type in the $7$-adic number

\kbd{2*7\pow(-1) + 3 + 4*7 + 2*7\pow 2 + O(7\pow3)}

\noindent exactly as shown, or equivalently as \kbd{905/7 + O(7\pow3)}.

Note that this type is available for convenience, not for speed:
internally, \typ{PADIC}s are stored as $p$-adic units modulo some $p^k$.
Each elementary operation involves updating $p^k$ (multiplying or
dividing by powers of $p$) and a reduction mod $p^k$. In particular
additions are slow.
\bprog
    ? n = 1+O(2^20);   for (i=1,10^5, n++)
    time = 86 ms.
    ? n = Mod(1,2^20); for (i=1,10^5, n++)
    time = 48 ms.
    ? n = 1;           for (i=1,10^5, n++)
    time = 38 ms.
@eprog

\subsec{Quadratic numbers}\sidx{quadratic number} (type \tet{t_QUAD}): first,
you must define the default quadratic order or field in which you want to
work. This is done using the \tet{quadgen} function, in the following way.
Write something like
\bprog
    w = quadgen(d)
@eprog\noindent
where \kbd{d} is the \emph{discriminant} of the quadratic order in
which you want to work (hence $d$ is congruent to $0$ or $1$ modulo $4$). The
name \kbd{w} is of course just a suggestion, but corresponds to traditional
usage. You can use any variable name that you like. However,
quadratic numbers are always printed with a \kbd{w}, regardless of the
discriminant. So beware, two numbers can be printed in the same way and not
be equal. However \kbd{gp} will refuse to add or multiply them for example.

Now $(1,w)$ is the ``canonical'' integral basis of the quadratic order
(i.e.~$w=\sqrt{d}/2$ if $d\equiv 0 \mod 4$, and $w=(1+\sqrt{d})/2$ if
$d\equiv 1 \mod 4$, where $d$ is the discriminant), and to enter $x+yw$ you
just type \kbd{x + y*w}.

\subsec{Polmods}\sidx{polmod} (type \tet{t_POLMOD}): exactly as
for intmods, to enter $x \mod y$ (where $x$ and $y$ are polynomials),
type \kbd{Mod(x,y)}, not \kbd{x\%y}. Note that when $y$ is an irreducible
polynomial in one variable, polmods whose modulus is $y$ are simply
algebraic numbers in the finite extension defined by the polynomial $y$.
This allows us to work easily in \idx{number field}s, finite extensions of
the $p$-adic field $\Q_p$, or \idx{finite field}s.

Note that this type is available for convenience, not for speed: each
elementary operation involves a reduction modulo $y$.

\label{se:rempolmod}
\misctitle{Important remark.}\sidx{variable (priority)} 
Mathematically, the variables\sidx{variable} occurring in a polmod are not
free variables. But internally, a congruence class in $R[t]/(y)$ is
represented by its representative  of lowest degree, which is a \typ{POL} in
$R[t]$, and computations occur with polynomials in the variable $t$. PARI
will not recognize that \kbd{Mod(y, y\pow2 + 1)} is ``the same'' as
\kbd{Mod(x, x\pow2 + 1)}, since \kbd{x} and \kbd{y} are different variables.

To avoid inconsistencies, polmods must use the same variable in internal
operations (i.e.~between polmods) and variables of lower priority for
external operations, typically between a polynomial and a polmod. See
\secref{se:priority} for a definition of ``priority'' and a discussion of
(PARI's idea of) multivariate polynomial arithmetic.
For instance:
\bprog
    ? Mod(x, x^2+ 1) + Mod(x, x^2 + 1)
    %1 = Mod(2*x, x^2 + 1)    \\@com $2i$ (or $-2i$), with $i^2=-1$
    ? x + Mod(y, y^2 + 1)
    %2 = x + Mod(y, y^2 + 1)  \\@com in $\Q(i)[x]$
    ? y + Mod(x, x^2 + 1)
    %3 = Mod(x + y, x^2 + 1)  \\@com in $\Q(y)[i]$
@eprog\noindent
The first two are straightforward, but the last one may not be what you
want: \kbd{y} is treated here as a numerical parameter, not as a polynomial
variable.

If the main variables are the same, it is allowed to mix \typ{POL} and
\typ{POLMOD}s. The result is the expected \typ{POLMOD}. For instance 
\bprog
    ? x + Mod(x, x^2 + 1)
    %1 = Mod(2*x, x^2 + 1)
@eprog

\subsec{Polynomials}\sidx{polynomial}\label{se:pol}
(type \tet{t_POL}): type the polynomial in a natural way, not
forgetting to put a ``$*$'' between a coefficient and a formal variable
(this $*$ does not appear in beautified output). Any \idx{variable} name
can be used except for the reserved names \kbd{I} (used exclusively for the
square root of $-1$), \kbd{Pi} ($3.14\dots$), \kbd{Euler} (Euler's
constant), and all the function names: predefined functions, as described
in Chapter~3 (use \b{c} to get the complete list of them) and user-defined
functions, which you ought to know about (use \b{u} if you are subject to
memory lapses). The total number of different variable names is limited to
$16384$ and $65536$ on 32-bit and 64-bit machines respectively, which
should be enough. If you ever need hundreds of variables, you should
probably be using vectors instead. See \secref{se:priority} for a discussion
of multivariate polynomial rings.

\subsec{Power series}\sidx{power series}\label{se:series}
(type \tet{t_SER}): type a rational function or polynomial expression and add
to it \hbox{\kbd{O(\var{expr}\pow $k$)}}, where \var{expr} is an expression
which has non-zero valuation (it can be a polynomial, power series, or a
rational function; the most common case being simply a variable name). This
indicates to \kbd{gp} that it is dealing with a power series, and the desired
precision is $k$ times the valuation of \var{expr} with respect to the main
variable of \var{expr}. (To check the ordering of the variables, or to modify
it, use the function \kbd{reorder}; see~\secref{se:reorder}.)

\misctitle{Caveat.} Power series with inexact coefficients sometimes have a
non-intuitive behaviour: if $k$ significant terms are requested, an inexact
zero is counted as significant, even if it is the coefficient of lowest
degree. This means that useful higher order terms may be disregarded.
If the series precision is insufficient, errors may occur (mostly division by
$0$), which could have been avoided by a better global understanding of the
computation:
\bprog
    ? A = 1/(y + 0.); B = 1. + O(y);
    ? B * denominator(A)
    %2 = 0.E-28 + O(y)
    ? A/B
    ***   division by zero
    ? A*B
    ***   Warning: normalizing a series with 0 leading term.
    ***   division by zero
    ? A*(1/B)
    ***   Warning: normalizing a series with 0 leading term.
    %3 = 1.000000000000000000000000000*y^-1 + O(1)
@eprog\noindent
If a series with a zero leading coefficient must be inverted, then as a
desperation measure that coefficient is discarded, and a warning is issued:
\bprog
    ? C = 0. + y + O(y^2);
    ? 1/C
    ***   Warning: normalizing a series with 0 leading term.
    %2 = y^-1 + O(1)
@eprog\noindent
The last result could be construed as a bug since it is a priori impossible
to deduce such a result from the input ($0.$ may represent any sufficiently
small real number). But it was thought more useful to try and go on with an
approximate computation than to raise an early exception.

In the first example above, to compute \kbd{A*(1/B)}, the denominator of
\kbd{A} was converted to a power series, then inverted.

\subsec{Rational functions}\sidx{rational function}
(types \tet{t_RFRAC}): as for fractions, all rational
functions are automatically reduced to lowest terms. All that was
said about fractions in \secref{se:FRAC} remains valid here.

\subsec{Binary quadratic forms of positive or negative discriminant}%
\sidx{binary quadratic form}
(type \tet{t_QFR} and \tet{t_QFI}):
these are input using the function \kbd{Qfb} (see Chapter~3). For example
\kbd{Qfb(1,2,3)} creates the binary form $x^2+2xy+3y^2$. It is imaginary (of
internal type \typ{QFI}) since $2^2 - 4*3 = -8$ is negative.

Although imaginary forms could be positive or negative definite, only
positive definite forms are implemented.

In the case of forms with positive discriminant (type \typ{QFR}), you may add
an optional fourth component (related to the regulator, more precisely to
Shanks and Lenstra's ``distance''), which must be a real number. See also the
function \kbd{qfbprimeform} which directly creates a prime form of given
discriminant (see Chapter~3).

\subsec{Row and column vectors}\sidx{row vector}\sidx{column vector} (types
\tet{t_VEC} and \tet{t_COL}): to enter a row vector, type the components
separated by commas ``\kbd{,}'', and enclosed between brackets
``\kbd{[}$\,$'' and ``$\,$\kbd{]}'', e.g.~\kbd{[1,2,3]}. To enter a column
vector, type the vector horizontally, and add a tilde ``\til'' to transpose.
\kbd{[ ]} yields the empty (row) vector. The function \tet{Vec} can be used
to transform any object into a vector (see Chapter~3).

\subsec{Matrices} (type \tet{t_MAT}):\sidx{matrix} to enter a matrix, type
the components line by line, the components being separated by commas
``\kbd{,}'', the lines by semicolons ``\kbd{;}'', and everything enclosed in
brackets ``\kbd{[}$\,$'' and ``$\,$\kbd{]}'', e.g. \kbd{[x,y; z,t; u,v]}.
\kbd{[ ; ]} yields the empty (0x0) matrix. The function \tet{Mat} can be used
to transform any object into a matrix (see Chapter 3).

Note that although the internal representation is essentially the same (only
the type number is different), a row vector of column vectors is \emph{not} a
matrix; for example, multiplication will not work in the same way.

Note also that it is possible to create matrices (by conversion of empty
column vectors and concatenation, or using the \kbd{matrix} function) with a
given positive number of columns, each of which has zero rows. It is not
possible to create or represent matrices with zero columns and a nonzero
number of rows.

\subsec{Lists} (type \tet{t_LIST}):\sidx{list} lists cannot be input
directly; you have to use the function \kbd{listcreate} first, then
\kbd{listput} each time you want to append a new element (but you can access
the elements directly as with the vector types described above). The function
\kbd{List} can be used to transform (row or column) vectors into lists (see
Chapter~3).

\subsec{Strings} (type \tet{t_STR}):\sidx{string}\sidx{character string} to
enter a string, just enclose it between double quotes \kbd{"}, like this:
\kbd{"this is a string"}. The function \kbd{Str} can be used to transform any
object into a string (see Chapter~3).

\subsec{Small vectors} (type \tet{t_VECSMALL}): this is an internal type,
used to code in an efficient way vectors containing only small integers, such
as permutations. Most \kbd{gp} functions will refuse to operate on these
objects.

\subsec{Note on output formats}. A zero real number is printed in \kbd{e}
format as $0.Exx$ where $xx$ is the (usually negative) \emph{decimal}
exponent of the number (cf.~\secref{se:whatzero}). This allows the user to
check the accuracy of that particular zero.

When the integer part of a real number $x$ is not known exactly because the
exponent of $x$ is greater than the internal precision, the real number is
printed in \kbd{e} format.

Note also that in beautified format, a number of type integer or real is
written without enclosing parentheses, while most other types have them.
Hence, if you see the expression \kbd{(3.14)}, it is not of type real, but
probably of type complex with zero imaginary part, or polynomial of degree $0$
(to be sure, use \b{x} or the function \kbd{type}).

\section{GP operators}\label{se:operators}

\noindent Loosely speaking, an \idx{operator} is a function (usually
associated to basic arithmetic operations) whose name contains only
non-alphanumeric characters. In practice, most of these are simple functions,
which take arguments, and return a value; assignment operators also have side
effects. Each of these has some fixed and unchangeable priority, which means
that, in a given expression, the operations with the highest priority is
performed first. Operations at the same priority level are performed in the
order they were written, i.e.~from left to right. Anything enclosed between
parenthesis is considered a complete subexpression, and is resolved
independently of the surrounding context. For instance, assuming that
\var{op}$_1$, \var{op}$_2$, \var{op}$_3$ are standard binary operators with
increasing priorities (think of \kbd{+}, \kbd{*}, \kbd{\pow} for instance),
$$ x~\var{op}_1~y~\var{op}_2~z~\var{op}_2~x~\var{op}_3~y $$ is
equivalent to $$ x~\var{op}_1~((y~\var{op}_2~z)~\var{op}_2~
(x~\var{op}_3~y)).$$

GP contains quite a lot of different operators, some of them unary (having only
one argument), some binary, plus special selection operators. Unary operators
are defined for either prefix (preceding their single argument: \var{op}~$x$)
or postfix (following the argument: $x$~\var{op}) position, never both (some
are syntactically correct in both positions, but with different meanings).
Binary operators all use the syntax $x$~\var{op}~$y$. Most of them are well
known, some are borrowed from C~syntax, and a few are specific to GP. Beware
that some GP operators may differ slightly from their C counterparts. For
instance, GP's postfix \kbd{++} returns the \emph{new} value, like the prefix
\kbd{++} of~C, and the binary shifts \kbd{<<}, \kbd{>>} have a priority which
is different from (higher than) that of their C counterparts. When in doubt,
just surround everything by parentheses. (Besides, your code will be
more legible.)

\noindent Here is the complete list in order of decreasing \idx{priority},
binary unless mentioned otherwise:

\def\point#1{\noindent $\bullet$ #1\hfill\break\indent\strut} \point{Priority
10}
%
\kbd{++} and \kbd{--} (unary, postfix): \kbd{$x$++} assigns the value $x+1$
to $x$, then returns the new value of $x$. This corresponds to the C
statement \kbd{++$x$} (there is no prefix \kbd{++} operator in GP).
\kbd{$x$--} does the same with $x-1$.

\point{Priority 9}
%
\kbd{\var{op}=}, where \var{op} is any simple binary operator (i.e.~a binary
operator with no side effects, i.e.~one of those defined below) which is not
a boolean operator (comparison or logical). \kbd{x~\var{op}=~$y$} assigns
$(\kbd{x}~\var{op}~y)$ to~\kbd{x}, and returns the new value of~\kbd{x}.
This is \emph{not} a reference to the \idx{variable}~\kbd{x}, i.e.~an
\kbd{lvalue}; thus
\bprog
  (x += 2) = 3
@eprog\noindent
is invalid.

\point{Priority 8}
%
\kbd{=} is the assignment operator. The result of \kbd{x~=~$y$} is the value
of the expression~$y$, which is also assigned to the variable~\kbd{x}. This
is \emph{not} the equality test operator; a statement like \kbd{x~=~1} is
always true (i.e.~non-zero), and sets \kbd{x} to~1. The right hand side of
the assignment operator is evaluated before the left hand side. If the left
hand side cannot be modified, raise an error.

\point{Priority 7} \kbd{[ ]} is the selection operator. \kbd{$x$[$i$]}
returns the $i$-th component of vector $x$; \kbd{$x[$i$,$j$]$},
\kbd{$x[,$j$]$} and \kbd{$x$[$i$,]} respectively return the entry of
coordinates $(i,j)$, the $j$-th column, and the $i$-th row of matrix $x$. If
the assignment operator (\kbd{=}) immediately follows a sequence of
selections, it assigns its right hand side to the selected component. E.g
\kbd{x[1][1] = 0} is valid; but beware that \kbd{(x[1])[1] = 0} is not
(because the parentheses force the complete evaluation of \kbd{x[1]}, and the
result is not modifiable).

\point{Priority 6}
%
\kbd{'} (unary, prefix): quote its argument (a variable name) without
evaluating it.
\bprog
? a = x + 1; x = 1;
? subst(a,x,1)
  ***   variable name expected: subst(a,x,1)
                                        ^---
? subst(a,'x,1)
%1 = 2
@eprog
%
\kbd{\pow}: powering.

\kbd{'} (unary, postfix): derivative with respect to the main variable. If
$f$ is a (GP or user) function, $f'(x)$ is allowed. If $x$ is a scalar, the
operator performs \idx{numerical derivation}, defined as $(f(x+\varepsilon) -
f(x-\varepsilon)) / 2\varepsilon$ for a suitably small epsilon depending on
current precision. It behaves as $(f(x))'$ otherwise.

\strut\kbd{\til} (unary, postfix): vector/matrix transpose.

\kbd{!} (unary, postfix): factorial. $x\kbd{!}=x(x-1)\cdots 1$.

\kbd{.}\var{member} (unary, postfix): \kbd{$x$.\var{member}} extracts
\var{member} from structure $x$ (see~\secref{se:member}).

\point{Priority 5}
%
\kbd{!} (unary, prefix): logical \var{not}. \kbd{!$x$} return $1$ if $x$ is
equal to $0$ (specifically, if \kbd{gcmp0($x$)==1}), and $0$ otherwise.

\kbd{\#} (unary, prefix): cardinality; \kbd{\#$x$} returns \kbd{length($x$)}.

\point{Priority 4}
%
\kbd{+}, \kbd{-} (unary, prefix): \kbd{-} toggles the sign of its argument,
\kbd{+} has no effect whatsoever.

\point{Priority 3}
%
\kbd{*}: multiplication.

\kbd{/}: exact division (\kbd{3/2}=$3/2$, not $1.5$).

\kbd{\bs}, \kbd{\%}: Euclidean quotient and remainder, i.e.~if $x =
qy + r$, with $0\le r < y$ (if $x$ and $y$ are polynomials, assume instead
that $\deg r< \deg y$ and that the leading terms of $r$ and $x$ have the
same sign), then $\kbd{x \b{ } y} = q$, $\kbd{x\%y} = r$.

\kbd{\bs/}: rounded Euclidean quotient for integers (rounded towards
$+\infty$ when the exact quotient would be a half-integer).

\kbd{<<}, \kbd{>>}: left and right binary shift: \kbd{x<<n}$~=~x * 2^n$
if $n>0$, and $x \b{/} 2^{-n}$ otherwise. Right shift is defined by
\kbd{x>>n}$~=~$\kbd{x<<(-n)}.

\point{Priority 2}
%
\kbd{+}, \kbd{-}: addition/subtraction.

\point{Priority 1}
%
\kbd{<}, \kbd{>}, \kbd{<=}, \kbd{>=}: the usual comparison operators,
returning 1 for \kbd{true} and 0 for \kbd{false}. For instance,
\kbd{x<=1} returns $1$ if $x\le 1$ and $0$ otherwise.

\kbd{<>}, \kbd{!=}: test for (exact) inequality.

\kbd{==}: test for (exact) equality.

\point{Priority 0}
%
\kbd{\&}, \kbd{\&\&}: logical \var{and}.

\kbd{|}, \kbd{||}: logical (inclusive) \var{or}. Any sequence of logical
\var{or} and \var{and} operations is evaluated from left to right,
and aborted as soon as the final truth value is known. Thus, for instance,
\bprog
  x && test(1/x)
  type(p) == "t_INT" && isprime(p)
@eprog\noindent
will never produce an error since the second argument need not (and will not)
be processed when the first is already zero (false).

\misctitle{Remark:} For optimal efficiency, you should use the
\kbd{++}, \kbd{--} and \var{op}\kbd{=} operators whenever possible:
\bprog
? a = 200000;
? i = 0; while(i<a, i=i+1)
time = 4,919 ms.
? i = 0; while(i<a, i+=1)
time = 4,478 ms.
? i = 0; while(i<a, i++)
time = 3,639 ms.
@eprog
\noindent For the same reason, the shift operators should be preferred to
multiplication:
\bprog
? a = 1<<20000;
? i = 1; while(i<a, i=i*2);
time = 5,255 ms.
? i = 1; while(i<a, i<<=1);
time = 988 ms.
@eprog

\section{Variables and symbolic expressions}\sidx{variable}
\subsec{Variable names}. In GP you can use up to 16383 variable names (up to
65535 on 64-bit machines). A valid identifier name starts with a letter and
contain only valid keyword characters: \kbd{\_} or alphanumeric characters
([\kbd{\_A-Za-z0-9}]). You may not use built-in function names; see the
list with \b{c}, including the constants \kbd{Pi}, \kbd{Euler} and
$\kbd{I}=\sqrt{-1}$.

Note that GP names are case sensitive. This means for instance that the
symbol \kbd{i} is perfectly safe to use, and will not be mistaken for
$\sqrt{-1}$, and that \kbd{o} is not synonymous to \kbd{O}.

We will see in \secref{se:user_defined} that it is possible to restrict
the use of a given variable by declaring it to be \tet{global} or
\tet{local}. This can be useful to enforce clean programming style, but
is in no way mandatory.

\subsec{Vectors and matrices}. If the variable $x$ contains a vector or list,
$x[m]$ refers to its $m$-th entry. You can assign a result to $x[m]$
(i.e.~write something like $x[k]=\var{expr}$). If $x$ is a matrix, $x[m,n]$
referes to its $(m,n)$ entry; you can assign a result to $x[m,n]$, but
\emph{not} to $x[m]$. If you want to assign an expression to the $m$-th
column of a matrix $x$, use $x[,m]=\var{expr}$ instead. Similarly, use
$x[m,]=\var{expr}$ to assign an expression to the $m$-th row of $x$. This
process is recursive, so if $x$ is a matrix of matrices of \dots, an
expression such as \kbd{x[1,1][,3][4] = 1} is perfectly valid
(and actually identical to \kbd{x[1,1][4,3] = 1}), assuming
that all matrices along the way have compatible dimensions.

\subsec{Variables and polynomials}
  The main thing to understand is that PARI/GP is \emph{not} a symbolic
manipulation package. One of the main consequences of this fact is that all
expressions are evaluated as soon as they are written, they never stay in an
abstract form%
\footnote{*}{An obvious but important exception are character strings which
are evaluated essentially to themselves (type \typ{STR}). Not exactly
so though, since we do some work to treat the quoted characters correctly
(those preceded by a \b{)}.}.
%
As an important example, consider what happens when you use a variable name
\emph{before} assigning a value into it, \kbd{x} say. This is perfectly
acceptable, it is considered as a monomial of degree 1 in the variable
\kbd{x}.
\bprog
  ? p = x^2 + 1
  %1 = x^2 + 1
  ? x = 2;
  ? x^2 + 1
  %3 = 5
  ? p
  %4 = x^2 + 1
  ? eval(p)
  %5 = 5
@eprog\noindent
As is shown above, assigning a value to a variable, does not affect
polynomials that used it; to take into account the new variable's value, one
must use the function \kbd{eval} (see \secref{se:eval}). It is in general
preferable to use \kbd{subst}, rather than assigning values to polynomial
variables.

\subsec{Variable priorities, multivariate objects}.\sidx{variable (priority)}\label{se:priority}
PARI has no ``sparse'' representation of polynomials. So a
multivariate polynomial in PARI is just a polynomial (in one variable), whose
coefficients are themselves polynomials, arbitrary but for the fact that they
do not involve the main variable. All computations are then just done
formally on the coefficients as if the polynomial was univariate.

This is not symmetrical. So if I enter \kbd{x + y} in a clean session,
what happens ? This is understood as
$$ x^1 + y*x^0 \in (\Z[y])[x] $$
but how do we know that $x$ is ``more important'' than $y$ ? Why not $y^1 +
x*y^0$, which is the same mathematical entity after all ?

The answer is that variables are ordered implicitly by the \kbd{gp} interpreter:
when a new identifier (e.g~$x$, or $y$ as above) is input, the corresponding
variable is registered as having a strictly lower priority than any variable in
use at this point\footnote{**}{This is not strictly true: if an
identifier is interpreted as a user function, no variable is registered. Also,
the variable $x$ is predefined and always has the highest possible priority.}
%
. To see the ordering used by \kbd{gp} at any given time, type
\kbd{reorder()}\kbdsidx{reorder}.

Given such an ordering, multivariate polynomials are stored so that the
variable with the highest priority is the main variable. And so on,
recursively, until all variables are exhausted. A different storage pattern
(which could only be obtained via library mode) would produce an invalid
object, and eventually a disaster.

In any case, if you are working with expressions involving several variables
and want to have them ordered in a specific manner in the internal
representation just described, the simplest is just to write down the
variables one after the other under \kbd{gp} before starting any real computations.
You could also define variables from your GPRC to have a consistent
ordering of common variable names in all your \kbd{gp} sessions, e.g read in a file
\kbd{variables.gp} containing
\bprog
x;y;z;t;a;b;c;d;
@eprog

If you already have started working and want to change the names of the
variables in an object, use the function \tet{changevar}. If you only want to
have them ordered when the result is printed, you can also use the function
\tet{reorder}, but this won't change anything to the internal representation,
and is not recommended.

\misctitle{Important note:} PARI allows Euclidean division of multivariate
polynomials, but assumes that the computation takes place in the fraction
field of the coefficient ring (if it is not an integral domain, the result
will a priori not make sense). This can be very tricky; for instance
assume $x$ has highest priority (which is always the case), then
$y$:
\bprog
? x % y
%1 = 0
? y % x
%2 = y             \\@com these two take place in $\Q(y)[x]$
? x * Mod(1,y)
%3 = Mod(1, y)*x   \\@com in $(\Q(y)/y\Q(y))[x] \sim \Q[x]$
? Mod(x,y)
%4 = 0
@eprog
\noindent In the last example, the division by $y$ takes place in
$\Q(y)[x]$,
hence the \kbd{Mod} object is a coset in $(\Q(y)[x]) / (y\Q(y)[x])$, which
is the null ring since $y$ is invertible! So be very wary of variable
ordering when your computations involve implicit divisions and many
variables. This also affects functions like \tet{numerator}/\tet{denominator}
or \tet{content}:
\bprog
? denominator(x / y)
%1 = 1
? denominator(y / x)
%2 = x
? content(x / y)
%3 = 1/y
? content(y / x)
%4 = y
? content(2 / x)
%5 = 2
@eprog
\noindent Can you see why ? Hint: $x/y = (1/y) * x$ is in $\Q(y)[x]$ and
denominator is taken with respect to $\Q(y)(x)$; $y/x = (y*x^0) / x$ is in
$\Q(y)(x)$ so $y$ is invertible in the coefficient ring. On the other hand,
$2/x$ involves a single variable and the coefficient ring is simply $\Z$.

These problems arise because the variable ordering defines an \emph{implicit}
variable with respect to which division takes place. This is
the price to pay to allow \kbd{\%} and \kbd{/} operators on polynomials
instead of requiring a more cumbersome \kbd{divrem($x$, $y$, \var{var})}
(which also exists). Unfortunately, in some functions like \tet{content} and
\tet{denominator}, there is no way to set explicitly a main variable like in
\tet{divrem} and remove the dependence on implicit orderings. This will
hopefully be corrected in future versions.

\subsec{Multivariate power series}
Just like multivariate polynomials, power series are fundamentally
single-variable objects. It is awkward to handle many variables at once,
since PARI's implementation cannot handle multivariate error terms like
$O(x^i y^j)$. (It can handle the polynomial $O(y^j) \times x^i$ which is
a very different thing, see below.)

The basic assumption in our model is that if variable $x$ has higher
priority than $y$, then $y$ does not depend on $x$: setting $y$ to a
function of $x$ after some computations with bivariate power series does
not make sense a priori. This is because implicit constants in
expressions like $O(x^i)$ depend on $y$ (whereas in $O(y^j)$ they can not
depend on $x$). For instance
\bprog
  ? O(x) * y
  %1 = O(x)
  ? O(y) * x
  %2 = O(y)*x
@eprog\noindent
Here is a more involved example:
\bprog
  ? A = 1/x^2 + 1 + O(x); B = 1/x + 1 + O(x^3);
  ? subst(z*A, z, B)
  %2 = x^-3 + x^-2 + x^-1 + 1 + O(x)
  ? B * A
  %3 = x^-3 + x^-2 + x^-1 + O(1)
  ? z * A
  %4 = z*x^-2 + z + O(x)
@eprog\noindent
The discrepancy between \kbd{\%2} and \kbd{\%3} is surprising. Why does
\kbd{\%2} contain a spurious constant term, which cannot be
deduced from the input ? Well, we ignored the rule that forbids to
substitute an expression involving high-priority variables
to a low-priority variable. The result \kbd{\%4} is correct according to
our rules since the implicit constant in $O(x)$ may depend on $z$. It is
obviously wrong if $z$ is allowed to have negative valuation in $x$. Of
course, the correct error term should be $O(xz)$, but this is not
possible in PARI.

\section{User defined functions}\sidx{user defined functions}
\label{se:user_defined}

\subsec{Definition}.
It is easy to define a new function in GP, which can then be used like any
other function. The syntax is as follows:

\kbd{name(}\var{list of formal variables}\kbd{) = %
local(}\var{list of local variables}\kbd{);} \var{seq}

\noindent which looks better written on consecutive lines:
\bprogpart
name($x_0$, $x_1$, @dots) =
{
  local($t_0$, $t_1$, @dots);
  local(@dots);

  @dots
}
@eprog \noindent
(the first newline is disregarded due to the preceding \kbd{=} sign, and the
others because of the enclosing braces). Both lists of variables are
comma-separated and allowed to be empty. The \tet{local} statements can be
omitted; as usual \var{seq} is any expression sequence.

\noindent \kbd{name} is the name given to the function and is subject to the
same restrictions as variable names. In addition, variable names are not
valid function names, you have to \kbd{kill} the variable first (the converse
is true: function names can't be used as variables, see \secref{se:kill}).
Previously used function names can be recycled: you are just redefining the
function. The previous definition is lost of course.

\noindent \var{list of formal variables} is the list of variables
corresponding to those which you will actually use when calling your
function. The number of actual parameters supplied when calling the function
has to be less than the number of formal variables. Arguments are passed by
value, not as variables: modifying a function's argument in the function
body is allowed, but does not modify its value in the calling frame. In fact,
a \emph{copy} of the actual parameter is assigned to the formal parameter
when the function is called.

Uninitialized formal variables are given a default value. An equal (\kbd{=})
sign following a variable name in the function definition, followed by any
expression, gives the variable a default value. The said expression gets
evaluated the moment the function is called, hence may involve the preceding
function parameters (a default value for $x_i$ may involve $x_j$ for $j <
i$). A variable for which you supply no default value is initialized to 
(the integer) zero. For instance
\bprog
  foo(x, y=2, z=3) = print(x ":" y ":" z)
@eprog
\noindent defines a function which prints its arguments (at most three of
them), separated by colons. This then follows the rules of default
arguments generation as explained at the beginning of
\secref{se:functions}.
\bprog
? foo(6,7)
6:7:3
? foo(,5)
0:5:3
? foo()
0:2:3
@eprog

\noindent \var{list of local variables} is the list of additional temporary
variables used in the function body. Note that if you omit some or all of
these local variable declarations, the non-declared variables will become
global, hence known outside of the function, and this may have undesirable
side-effects. On the other hand, in some cases it may also be what you want.
See \secref{se:varscope} for details. Local variables can be given a default
value as the formal variables.

\misctitle{Restrictions on variable use:} it is not allowed to use the same
variable name for different parameters of your function. Or to use a given
variable both as a formal parameter and a local variable in a given function.
Hence
\bprog
? f(x,x) = 1
  ***   user function f: variable x declared twice.
@eprog

\misctitle{Note:} The above syntax (using the \tet{local} keyword) was
introduced in version 2.0.13. The old syntax

\kbd{name(}\var{list of true formal variables, list of local variables}%
\kbd{) = }{\var{seq}}

\noindent is still recognized but is deprecated since genuine arguments and
local variables become undistinguishable.

\subsec{Use}. Once the function is defined using the above syntax, you can
use it like any other function, see the example with \kbd{fun} above. In
addition, you can also recall its definition exactly as you do for predefined
functions, that is by writing \kbd{?\var{name}}. This will print the list of
arguments, as well as their default values, the text of \var{seq}, and a
short help text if one was provided using the \kbd{addhelp} function (see
\secref{se:addhelp}). One small difference to predefined functions is that
you can never redefine the built-in functions, while you can redefine a
user-defined function as many times as you want.

Typing \b{u} will output the list of user-defined functions.

An amusing example of a user-defined function is the following. It is
intended to illustrate both the use of user-defined functions and the power
of the \kbd{sumalt} function. Although the \idx{Riemann zeta-function} is
included in the standard functions, let us assume that this is not the case
(or that we want another implementation). One way to define it, which is
probably the simplest, but certainly not the most efficient, is as
follows:\sidx{zeta function}
\bprog
zet(s) =
{ local(n); /* not needed, and possibly confusing (see below) */

  sumalt(n=1, (-1)^(n-1)*n^(-s)) / (1 - 2^(1-s))
}
@eprog
\noindent This gives reasonably good accuracy and speed as long as you are
not too far from the domain of convergence. Try it for $s$ integral between
$-5$ and $5$, say, or for $s=0.5+i*t$ where $t=14.134\dots$

\subsec{Recursive functions}. Recursive functions\sidx{recursion} can easily
be written as long as one pays proper attention to variable scope. Here is an
example, used to retrieve the coefficient array of a multivariate polynomial
(a non-trivial task due to PARI's unsophisticated representation for those
objects): \sidx{multivariate polynomial}
\bprog
coeffs(P, nbvar) =
{
  if (type(P) != "t_POL",
    for (i=1, nbvar, P = [P]);
    return (P)
  );
  vector(poldegree(P)+1, i, coeffs(polcoeff(P, i-1), nbvar-1))
}
@eprog

\noindent If $P$ is a polynomial in $k$ variables, show that after the
assignment {\tt v = coeffs(P,k)}, the coefficient of $x_1^{n_1}\dots
x_k^{n_k}$ in P is given by {\tt v[$n_1$+1][\dots][$n_k$+1]}.

The operating system automatically limits the \idx{recursion depth}:
\bprog
? dive(n) = if (n, dive(n-1))
? dive(5000);
  ***   deep recursion: if(n,dive(n-1))
                        ^---------------
@eprog
There is no way to increase the recursion limit (which may be different on
your machine) from within \kbd{gp}. To increase it before launching \kbd{gp},
you can use \tet{ulimit} or \tet{limit}, depending on your shell, and raise
the process available stack space (increase \tet{stacksize}).

\subsec{Function which take functions as parameters ?}
Use the following trick (neat example due to Bill Daly):

\bprog
calc(f, x) = eval( Str(f, "(x)") )
@eprog

\noindent If you call this with \kbd{calc("sin", 1)}, it will
return $\sin(1)$ (evaluated!).

\subsec{Defining functions within a function ?}
The first idea
\bprog
  init(x) = add(y) = x+y; mul(y) = x*y;
@eprog
\noindent does not work since in the construction \kbd{f() = }\var{seq}, the
function body contains everything until the end of the expression. Hence
executing \kbd{init} defines the wrong function \kbd{add}. The way out is to
use parentheses for grouping, to that enclosed subexpressions be
evaluated independently:
\bprog
  init(x) = ( add(y) = x+y ); ( mul(y) = x*y );
@eprog

\subsec{Variable scope}.\label{se:varscope}

Local variables should more appropriately be called \emph{temporary values}
since they are in fact local to the function declaring them \emph{and} any
subroutine called from within. In the following example
\bprog
  f() = local(y); ... ; g()
  g() = y + 1
@eprog
\noindent \kbd{g()} ``sees'' the $y$ introduced in \kbd{f()}. True lexical
scoping does not exist in GP. (See e.g.~the difference between \kbd{local}
and \kbd{my} in Perl.)

In an iterative constructs which use a variable name (\kbd{for$xxx$},
\kbd{prod$xxx$}, \kbd{sum$xxx$}, \kbd{vector}, \kbd{matrix}, \kbd{plot},
etc.) the given variable is also local to the construct. A value is pushed on
entry and poped on exit. So, it is not necessary for a function using such an
iterator to declare the variable as \kbd{local}. On the other hand, if you
exit the loop prematurely, e.g.~using the \kbd{break} statement, you must
save the loop index in another variable since its value prior the loop will
be restored upon exit: for instance
\bprog
  for(i = 1, n,
    if (ok(i), break);
  );
  if (i > n, return(failure));
@eprog
\noindent is incorrect, since the value of $i$ tested by the $(i > n)$ is quite
unrelated to the loop index.

Finally, the statement \sidx{global}\kbd{global(x, y, z, t)} (see
\secref{se:global}) declares the corresponding variables to be global. It is
then forbidden to use them as formal parameters or loop indexes as above,
since the parameter would ``shadow'' the variable. If speed is of the essence
and an object is large (e.g.~a \var{bnf}, a huge matrix), it should be
declared \kbd{global}, not passed as a parameter, since this saves an
expensive copy. It is possible to declare it local and use it as a global
variables from relevant subroutines, but \tet{global} is safer.

It is strongly recommended to explicitly declare all \tet{global} variables
at the beginning of your program and all \tet{local} variable used inside a
given function, with the possible exception of loop indexes which are local
to their loop. If a function accesses a variable which is not one of its
formal parameters, the value used will be the one which happens to be on top
of the stack at the time of the call. This could be a ``global'' value, or a
local value belonging to any function higher in the call chain, and is in
general \emph{not} what you want to do. So, be warned.


Coming back to our previous example \kbd{zet}, since loop variables are not
visible outside their loops, the variable \kbd{n} need not be declared in the
function protoype.
\bprog
zet(s) = sumalt(n=1, (-1)^(n-1)*n^(-s)) / (1 - 2^(1-s))
@eprog

\noindent would be a better definition. One last example: what is
wrong with the following definition?
\bprog
FirstPrimeDiv(x) =
{ local(p);

  forprime(p=2, x, if (x%p == 0, break));
  p
}
? FirstPrimeDiv(10)
%1 = 0
@eprog
\noindent Well, the index $p$ in the \kbd{forprime} loop is local to
the loop and is not visible to the outside world. Hence, it does not survive
the \kbd{break} statement. More precisely, at this point the loop index is
restored to its preceding value, which is 0 (local variables are
initialized to 0 by default). To sum up, the routine returns the $p$
declared local to it, not the one which was local to \kbd{forprime} and ran
through consecutive prime numbers. Here is a corrected version:
\bprog
? FirstPrimeDiv(x) = forprime(p=2, x, if (x%p == 0, return(p)))
@eprog

\misctitle{Implementation note:} For the curious reader, here is how values
of variables are handled: a \idx{hashing function} is computed from the
variable name, and used as an index in \tet{hashtable}, a table of linked
list of structures (type \tet{entree}). The linked list is searched linearly
for the identifier (each list typically has less than 10 components). When
the correct \kbd{entree} is found, it points to the top of the stack of
values for that identifier if it is a variable, to the function itself if it
is a predefined function, and to a copy of the text of the function if it is
a user-defined function. When an error occurs, all of this maze (rather a
tree, in fact) is searched and restored to the state preceding the last call
of the main evaluator.

\section{Member functions}\sidx{member functions} \label{se:member}

Member functions use the `dot' notation to retrieve information from
complicated structures, by default: \tev{bid}, \tev{ell}, \tev{galois},
\tev{nf}, \tev{bnf}, \tev{bnr} and prime ideals. The syntax
\kbd{structure.member} is taken to mean: retrieve \kbd{member} from
\kbd{structure}, e.g.~\kbd{ell.j} returns the $j$-invariant of the elliptic
curve \kbd{ell}, or outputs an error message if \kbd{ell} doesn't have the
correct type.

To define your own member functions, use the syntax \var{structure.member =
function text}, where \var{function text} is written as the \var{seq} in a
standard user function (without local variables), whose only argument would
be \kbd{structure}. For instance, the current implementation of the \kbd{ell}
type is simply an horizontal vector, the $j$-invariant being the thirteenth
component. It could be implemented as

\bprog
x.j =
{
  if (type(x) != "t_VEC" || length(x) < 14,
    error("this is not a proper elliptic curve: " x)
  );
  x[13]
}
@eprog\noindent
Typing \b{um} will output the list of user-defined member functions.

You can redefine one of your own member functions simply by typing a new
definition for it. On the other hand, as a safety measure, you can't redefine
the built-in member functions, so typing the above text would in fact produce
an error (you'd have to call it e.g.~\kbd{x.myj} in order for \kbd{gp} to
accept it).

\misctitle{Warning:} contrary to user functions arguments, the structure
accessed by a member function is \emph{not} copied before being used.
Any modification to the structure's components will be permanent.

\misctitle{Warning:} it is advised not to apply a member whose name
starts with \kbd{e} or \kbd{E} to an integer constant, where it would be
confused with the usual floating point exponent. E.g compare
\bprog
  ? x.e2 = x+1
  ? 1.e2
  %1 = 100.000000000 \\@com taken to mean $1.0E2$.
  ? (1).e2
  %2 = 2
  ? 1.0.e2
  %3 = 2.00000000000
@eprog

\misctitle{Note:} Member functions were not meant to be too complicated or to
depend on any data that wouldn't be global. Hence they do no have parameters
(besides the implicit \kbd{structure}) or local variables. Of course, if you
need some preprocessing work in there, there's nothing to prevent you from
calling your own functions (using freely their local variables) from a member
function. For instance, one could implement (a dreadful idea as far as
efficiency goes):

\bprog
correct_ell_if_needed(x) =
{ local(tmp);

  if (type(x) != "t_VEC", tmp = ellinit(x))
    \\ @com some further checks
  tmp
}
x.j = correct_ell_if_needed(x)[13];
@eprog

\section{Strings and Keywords}\sidx{string}\sidx{keyword}
\label{se:strings}

\subsec{Strings}. GP variables can hold values of type character string
(internal type \typ{STR}). This section describes how they are actually used,
as well as some convenient tricks (automatic concatenation and expansion,
keywords) valid in string context.

As explained above, the general way to input a string is to enclose
characters between quotes~\kbd{"}. This is the only input construct where
whitespace characters are significant: the string will contain the exact
number of spaces you typed in. Besides, you can ``escape'' characters by
putting a \kbd{\bs} just before them; the translation is as follows
\bprog
   \e: <Escape>
   \n: <Newline>
   \t: <Tab>
@eprog
For any other character $x$, \b{$x$} is expanded to $x$. In particular, the
only way to put a \kbd{"} into a string is to escape it. Thus, for
instance, \kbd{"\bs"a\bs""} would produce the string whose content is
``a''. This is definitely \emph{not} the same thing as typing \kbd{"a"},
whose content is merely the one-letter string a.

You can concatenate two strings using the \tet{concat} function. If either
argument is a string, the other is automatically converted to a string if
necessary (it will be evaluated first).

\bprog
? concat("ex", 1+1)
%1 = "ex2"
? a = 2; b = "ex"; concat(b, a)
%2 = "ex2"
? concat(a, b)
%3 = "2ex"
@eprog

Some functions expect strings for some of their arguments: \tet{print} would
be an obvious example, \tet{Str} is a less obvious but useful one (see the
end of this section for a complete list). While typing in such an argument,
you will be said to be in \tev{string context}. The rest of this section is
devoted to special syntactical tricks which can be used with such arguments
(and only here; you will get an error message if you try these outside of
string context):

$\bullet$ Writing two strings alongside one another will just concatenate
them, producing a longer string. Thus it is equivalent to type in
\kbd{"a " "b"} or \kbd{"a b"}. A little tricky point in the first expression:
the first whitespace is enclosed between quotes, and so is part of a string;
while the second (before the \kbd{"b"}) is completely optional and \kbd{gp}
actually suppresses it, as it would with any number of whitespace characters
at this point (i.e.~outside of any string).

$\bullet$ If you insert any expression when a string is expected, it gets
``expanded'': it is evaluated as a standard GP expression, and the final
result (as would have been printed if you had typed it by itself) is then
converted to a string, as if you had typed it directly. For instance \kbd{"a"
1+1 "b"} is equivalent to \kbd{"a2b"}: three strings get created, the middle
one being the expansion of \kbd{1+1}, and these are then concatenated
according to the rule described above. Another tricky point here: assume you
did not assign a value to \kbd{aaa} in a GP expression before. Then typing
\kbd{aaa} by itself in a string context will actually produce the correct
output (i.e.~the string whose content is aaa), but in a fortuitous way. This
\kbd{aaa} gets expanded to the monomial of degree one in the variable
\kbd{aaa}, which is of course printed as \kbd{aaa}, and thus will expand to
the three letters you were expecting.

\misctitle{Warning:} expression involving strings are not handled in a
special way; even in string context, the largest possible expression is
evaluated, hence \kbd{print("a"[1])} is incorrect since \kbd{"a"} is not an
object whose first component can be extracted. On the other hand
\kbd{print("a", [1])} is correct (two distinct argument, each converted to a
string), and so is \kbd{print("a" 1)} (since \kbd{"a"1} is not a valid
expression, only \kbd{"a"} gets expanded, then \kbd{1}, and the result is
concatenated as explained above). In case of doubt, you can surround part of
your text by parenthesis to force immediate interpretation of a
subexpression: \kbd{print("a"([1]))} is another solution.

\subsec{Keywords}. Since there are cases where expansion is not desirable, we
now distinguish between ``Keywords'' and ``Strings''. String is what has been
described so far. Keywords are special relatives of Strings which are
automatically assumed to be quoted, whether you actually type in the quotes
or not. Thus expansion is never performed on them. They get concatenated,
though. The analyzer supplies automatically the quotes you have ``forgotten''
and treats Keywords just as normal strings otherwise. For instance, if you
type \kbd{"a"b+b} in Keyword context, you will get the string whose contents
are ab+b. In String context, on the other hand, you would get a2\kbd{*}b.

All GP functions have prototypes (described in Chapter~3 below) which
specify the types of arguments they expect: either generic PARI objects
(GEN), or strings, or keywords, or unevaluated expression sequences. In the
keyword case, only a very small set of words will actually be meaningful
(the \kbd{default} function is a prominent example).

\misctitle{Reference:} The arguments of the following functions are processed
in string context:

\settabs\+\indent&\cr
\+&\tet{Str}\cr
\+&\tet{addhelp} (second argument)\cr
\+&\tet{default} (second argument)\cr
\+&\tet{error}\cr
\+&\tet{extern}\cr
\+&\tet{plotstring} (second argument)\cr
\+&\tet{plotterm} (first argument)\cr
\+&\tet{read} and \tet{readvec}\cr
\+&\tet{system}\cr
\+&all the \tet{print}\var{xxx} functions\cr
\+&all the \tet{write}\var{xxx} functions\cr

\noindent The arguments of the following functions are processed as keywords:

\+&\tet{alias}\cr
\+&\tet{default} (first argument)\cr
\+&\tet{install} (all arguments but the last)\cr
\+&\tet{trap} (first argument)\cr
\+&\tet{type} (second argument)\cr
\+&\tet{whatnow}\cr

\subsec{Useful examples} The function \kbd{Str} converts its arguments into
strings and concatenate them. Coupled with \tet{eval}, it is very powerful.
The following example creates generic matrices\sidx{generic
matrix}\sidx{matrix}:
\bprog
? genmat(u,v,s="x") = matrix(u,v,i,j, eval( Str(s,i,j) ))
? genmat(2,3) + genmat(2,3,"m")
%1 =
[x11 + m11 x12 + m12 x13 + m13]
[x21 + m21 x22 + m22 x23 + m23]
@eprog\noindent

Two last examples: \kbd{hist(10,20)} returns all \idx{history} entries from
\kbd{\%10} to \kbd{\%20} neatly packed into a single
vector; \kbd{histlast(10)} returns the last $10$ history entries:
\bprog
  hist(a,b) = vector(b-a+1, i, eval(Str("%", a-1+i)))
  histlast(n) = vector(n, i, eval(Str("%", %#-i+1)))
@eprog

\section{Errors and error recovery}

\subsec{Errors}. There are two kind of errors: syntax errors, and errors
produced by functions in the PARI library. Both kinds are fatal to your
computation: \kbd{gp} will report the error, perform some cleanup (restore
variables modified while evaluating the erroneous command, close open files,
reclaim unused memory, etc.), and will output its usual prompt.

When reporting a syntax error, \kbd{gp} tries to give meaningful context by copying
the sentence it was trying to read (whitespace and comments stripped
out), indicating an error with a little caret like in
\bprog
? factor(x^2 - 1
  ***   expected character: ',' instead of: factor(x^2-1
                                                        ^
@eprog
\noindent possibly enlarged to a full arrow given enough trailing context
\bprog
? if (siN(x) < eps, do_something())
  ***   expected character: '=' instead of: if(siN(x)<eps,do_something())
                                                     ^--------------------
@eprog
\noindent
Error messages will often be mysterious, because \kbd{gp} cannot guess what
you were trying to do and the error usually occurs once \kbd{gp} has been
sidetracked. Let's have a look at the two messages above.

The first error is a missing parenthesis, but from \kbd{gp}'s point of view, you
might as well have intended to give further arguments to \kbd{factor} (this is
possible, and often useful, see the description of the function). Since
\kbd{gp}
did not see the closing parenthesis, it tried to read a second argument,
first looking for the comma that would separate it form the first. The error
occurred at this point. So \kbd{gp} tells you that it was expecting a comma and saw
a blank.

The second error is even weirder. It is a simple typo, \kbd{siN} instead of
\kbd{sin} and \kbd{gp} tells us that is was expecting an equal sign a few
characters later? What happens is this: \kbd{siN} is not a recognized
identifier, but from the context, it looks like a function (it is followed by
an open parenthesis), then we have an argument, then a closing parenthesis.
Then if \kbd{siN} were a known function we would evaluate it; but it is not,
so \kbd{gp} assumes that you were trying to \emph{define} it, as in
\bprog
? if (siN(x)=sin(x), ...)
@eprog
This is actually allowed (!) and defines the function \kbd{siN} as an alias for
\kbd{sin}. As any expression a function definition has a value, which is $0$,
hence the test is meaningful, and false, so nothing happens. (Admittedly this
doesn't look like a useful syntax but it can be interesting in other contexts
to let functions define other functions. Anyway, it is allowed by the
language definition.)  So \kbd{gp} tells you in good faith that to correctly define
a function, you need an equal sign between its name and its body.

Error messages from the library will usually be clearer since, by
definition, they answer a correctly worded query (otherwise \kbd{gp} would have
protested first). Also they have more mathematical content, which should be
easier to grasp than a parser's logic. For instance:
\bprog
? 1/0
  ***   division by zero
@eprog

\subsec{Error recovery}.\sidx{error recovery}\label{se:errorrec}

It is quite annoying to wait for some program to finish and find out the hard
way that there was a mistake in it (like the division by 0 above), sending
you back to the prompt. First you may lose some valuable intermediate data.
Also, correcting the error may not be ovious; you might have to change your
program, adding a number of extra statements and tests to try and narrow down
the problem.

A slightly different situation, still related to error recovery, is when you
you actually foresee that some error may occur, are unable to prevent it, but
quite capable of recovering from it, given the chance. Examples include lazy
factorization (cf.~\tet{addprimes}), where you knowingly use a pseudo prime
$N$ as if it were prime; you may then encounter an ``impossible'' situation,
but this would usually exhibit a factor of $N$, enabling you to refine the
factorization and go on. Or you might run an expensive computation at low
precision to guess the size of the output, hence the right precision to use.
You can then encounter errors like ``precision loss in truncation'', e.g when
trying to convert \kbd{1E1000}, known to $28$ digits of accuracy, to an
integer; or ``division by 0'', e.g inverting \kbd{0E1000} when all accuracy
has been lost, and no significant digit remains. It would be enough to
restart part of the computation at a slightly higher precision.

We now describe \tev{error trapping}, a useful mechanism which alleviates
much of the pain in the first situation, and provides a satisfactory way out
of the second one. Everything is handled via the \tet{trap} function whose
different modes we now describe.

\subsec{Break loop}.\label{se:breakloop}

A \tev{break loop} is a special debugging mode that you enter whenever an
error occurs, freezing the \kbd{gp} state, and preventing cleanup until you get out
of the loop. Any error: syntax error, library error, user error (from
\tet{error}), even user interrupts like \kbd{C-c} (Control-C). When a break
loop starts, a prompt is issued (\kbd{break>}). You can type in a \kbd{gp} command,
which is evaluated when you hit the \kbd{<Return>} key, and the result is
printed as during the main \kbd{gp} loop, except that no history of results is kept
Then the break loop prompt reappears and you can type further commands as
long as you do not exit the loop. If you are using readline, the history of
commands is kept, and line editing is available as usual. If you type in a
command that results in an error, you are sent back to the break loop prompt
(errors does \var{not} terminate the loop).

To get out of a break loop, you can use \tet{next}, \tet{break},
\tet{return}, or \kbd{C-d} (\kbd{EOF}), any of which will let \kbd{gp} perform its
usual cleanup, and send you back to the \kbd{gp} prompt. If the error is not fatal,
inputing an empty line, i.e hitting the \kbd{<Return>} key at the
\kbd{break>} prompt, will continue the temporarily interrupted computation.
An empty line has no effect in case of a fatal error, to ensure you do not
get out of the loop prematurely, thus losing most debugging data during the
cleanup (since user variables will be restored to their former values).

In current version \vers, an error is non-fatal if and only if it was
initiated by a \kbd{C-c} typed by the user.

Break loops are useful as a debugging tool to inspect the values of \kbd{gp}
variables to understand why an error occurred, or to change \kbd{gp}'s state in the
middle of a computation (increase debugging level, start storing results in a
logfile, set variables to different values\dots): hit \kbd{C-c}, type in your
modifications, then let the computation go on as explained above.

A break loop looks like this:
\bprog
? for(v = -2, 2, print(1/v))
-1/2
-1
  ***   division by zero in gdiv, gdivgs or ginv
  ***   Starting break loop (type 'break' to go back to GP):
  ***   for(v=-2,2,print(1/v))
                            ^--
break>
@eprog
\noindent So the standard error message is printed first, except now we always
have context, whether the error comes from the library or the parser. The
\kbd{break>} at the bottow is a prompt, and hitting \kbd{v} then
\kbd{<Return>}, we see:
\bprog
break> v
0
@eprog
\noindent explaining the problem. We could have typed any \kbd{gp} command, not
only the name of a variable, of course. There is no special set of commands
becoming available during a break loop, as they would in most debuggers.

\misctitle{Important Note:} upon startup, this mechanism is \emph{off}. Type
\kbd{trap()} (or include it in a script) to start trapping errors in this
way. By default, you will be sent back to the prompt.

\misctitle{Technical Note:} When you enter a break loop due to a PARI stack
overflow, the PARI stack is reset so that you can run commands (otherwise the
stack would immediately overflow again). Still, as explained above, you do
not lose the value of any \kbd{gp} variable in the process.

\subsec{Error handlers}. The break loop described above is a (sophisticated)
example of an \tev{error handler}: a function that is executed whenever an
error occurs, supposedly to try and recover. The break loop is quite a
satisfactory error handler, but it may not be adequate for some purposes, for
instance when \kbd{gp} runs in non-interactive mode, detached from a
terminal.

So, you can define a different error handler, to be used in place of the
break loop. This is the purpose of the \emph{second} argument of \kbd{trap}:
to specify an error handler. (We will discuss the first argument at the very
end.) For instance:
\bprog
? { trap( ,                \\@ note the comma: arg1 is omitted
      print(reorder);
      writebin("crash")) }
@eprog
\noindent After that, whenever an error occurs, the list of all user
variables is printed, and they are all saved in binary format in file
\kbd{crash}, ready for inspection. Of course break loops are no longer
available: the new handler has replaced the default one. Besides user-defined
handlers as above, there are two special handlers you can use in \kbd{trap},
which are

$\bullet$ \kbd{trap(, "")} (do-nothing handler): to disable the trapping
mechanism and let errors propagate, which is the default situation on
startup.

$\bullet$  \kbd{trap(, )} (omitted argument, default handler): to trap errors
by a break loop.

\subsec{Protecting code}. Finally \kbd{trap} can define a temporary handler
used within the scope of a code frament, protecting it from errors, by
providing replacement code should the trap be activated. The expression

  \kbd{trap}( , \var{recovery}, \var{statements})

\noindent evaluates and returns the value of \var{statements}, unless an
error occurs during the evaluation in which case the value of \var{recovery}
is returned. As in an if/else clause, with the difference that
\var{statements} has been partially evaluated, with possible side effects.
For instance one could define a fault tolerant inversion function as follows:
\bprog
? inv(x) = trap (, "oo", 1/x)
? for (i=-1,1, print(inv(i)))
-1
oo
1
@eprog
\noindent Protected codes can be nested without adverse effect, the last trap
seen being the first to spring.

\subsec{Trapping specific exceptions}. We have not yet seen the use of the
first argument of trap, which has been omitted in all previous examples. It
simply indicates that only errors of a specific type should be intercepted,
to be chosen among

\kbd{accurer}: accuracy problem

\kbd{gdiver}: division by 0

\kbd{invmoder}: impossible inverse modulo

\kbd{archer}: not available on this architecture or operating system

\kbd{typeer}: wrong type

\kbd{errpile}: the PARI stack overflows

\noindent Omitting the error name means we are trapping all errors. For
instance, the following can be used to check in a safe way whether
\tet{install} works correctly in your \kbd{gp}:
\bprog
broken_install() =
{
  trap(archer, return ("OS"),
    install(addii,GG)
  );
  trap(, "USE",
    if (addii(1,1) != 2, "BROKEN")
  )
}
@eprog
\noindent The function returns 0 if everything works (the omitted \var{else}
clause of the \kbd{if}), \kbd{OS} if the operating system does not support
\kbd{install}, \kbd{USE} if using an installed function triggers an error,
and \kbd{BROKEN} if the installed function did not behave as expected.

\section{Interfacing GP with other languages}
\noindent
The PARI library was meant to be interfaced with C programs. This specific
use will be dealt with extensively in Chapter~4. \kbd{gp} itself provides a
convenient, if simple-minded, interpreter, which enables you to execute
rather intricate scripts (see \secref{se:programming}).

Scripts, when properly written, tend to be shorter and clearer than C
programs, and are certainly easier to write, maintain or debug. You don't
need to deal with memory management, garbage collection, pointers,
declarations, and so on. Because of their intrinsic simplicity, they are more
robust as well. They are unfortunately somewhat slower. Thus their use will
remain complementary: it is suggested that you test and debug your algorithms
using scripts, before actually coding them in C for the sake of speed.
The GP2C compiler often eases this part.

Note also that the \kbd{install} command enables you to concentrate on
critical parts of your programs only (which can of course be written with the
help of other mathematical libraries than PARI!), and to efficiently import
foreign functions for use under \kbd{gp} (see~\secref{se:install}).

We are aware of four PARI-related public domain packages to embed PARI in
other languages. We \emph{neither endorse nor support} any of them; you might
want to give them a try if you are familiar with the languages they are based
on. The first is the \tet{Math::Pari} Perl module (see any CPAN mirror),
written by Ilya Zakharevich. The second is \tet{PariPython}%
\footnote{*}{see \kbd{http://www.fermigier.com/fermigier/PariPython/}},
%
by St\'efane Fermigier, which is no more maintained. Starting from
Fermigier's work, Wiliam Stein has embedded PARI into his Python-based
SAGE\footnote{**}{see \kbd{http://modular.fas.harvard.edu/sage/}}
system. Finally, Michael Stoll has integrated PARI into \tet{CLISP}
\footnote{***}{see \kbd{http://clisp.cons.org}}, which is a Common Lisp
implementation by Bruno Haible, Marcus Daniels and others; this interface has
been updated for pari-2 by Sam Steingold.

These provide interfaces to \kbd{gp} functions for use in \kbd{perl},
\kbd{python} or \kbd{Lisp}\sidx{Perl}\sidx{Python}\sidx{Lisp} programs,
respectively.

\section{Defaults}\sidx{defaults}
\label{se:defaults}

\noindent There are many internal variables in \kbd{gp}, defining how the
system will behave in certain situations, unless a specific override has been
given. Most of them are a matter of basic customization (colors, prompt) and
will be set once and for all in your \idx{preferences file} (see
\secref{se:gprc}), but some of them are useful interactively (set timer on,
increase precision, etc.).

The function used to manipulate these values is called \kbd{default}, which
is described in \secref{se:default}. The basic syntax is

\kbd{default(\var{def}, \var{value})},

\noindent which sets the default \var{def} to \var{value}. In interactive
use, most of these can be abbreviated using historic \kbd{gp} metacommands
(mostly, starting with \b), which we shall describe in the next section.

Here we will only describe the available defaults and how they are used. Just
be aware that typing \kbd{default} by itself will list all of them, as well
as their current values (see \b{d}). Just after the default name, we give
between parentheses the initial value when \kbd{gp} starts (assuming you did
not tamper with it using command-line switches or a~\tet{gprc}).

\misctitle{Note:} the suffixes \kbd{k}, \kbd{M} or \kbd{G} can be appended to
a \var{value} which is a numeric argument, with the effect of multiplying it
by $10^3$, $10^6$ and $10^9$ respectively. Case is not taken into account
there, so for instance \kbd{30k} and \kbd{30K} both stand for $30000$. This
is mostly useful to modify or set the defaults \kbd{primelimit} or
\kbd{stacksize} which typically involve a lot of trailing zeroes.

\misctitle{(somewhat technical) Note:} As we will see in \secref{se:strings},
the second argument to \kbd{default} will be subject to string context
expansion, which means you can use run-time values. In other words, something
like
\bprog
  a = 3;
  default(logfile, "\var{some filename}" a ".log")
@eprog
logs the output in \kbd{\var{some filename}3.log}.

Some defaults will be expanded further when the values are used, after the
above expansion has been performed:

$\bullet$ \teb{time expansion}: the string is sent through the library
function \tet{strftime}. This means that \kbd{\%}\var{char} combinations have
a special meaning, usually related to the time and date. For instance,
\kbd{\%H} = hour (24-hour clock) and \kbd{\%M} = minute [00,59] (on a Unix
system, you can try \kbd{man strftime} at your shell prompt to get a complete
list). This is applied to \kbd{prompt}, \kbd{psfile}, and \kbd{logfile}. For
instance,

\kbd{default(prompt,"(\%H:\%M) ? ")}

\noindent
will prepend the time of day, in the form \kbd{(\var{hh}:\var{mm})}
to \kbd{gp}'s usual prompt.

$\bullet$ \teb{environment expansion}: When the string contains a sequence of
the form \kbd{\$\var{SOMEVAR}}, e.g.~\kbd{\$HOME}, the environment is
searched and if \var{SOMEVAR} is defined, the sequence is replaced by the
corresponding value. Also the \kbd{\til} symbol has the same meaning as in
many shells~--- \kbd{\til} by itself stands for your home directory, and
\kbd{\til{}user} is expanded to \kbd{user}'s home directory. This is applied
to all filenames\sidx{filename}. \label{se:envir}

\subsecidx{colors} (default \kbd{""}): this default is only usable if
\kbd{gp}\label{se:colors}
is running within certain color-capable terminals. For instance \kbd{rxvt},
\kbd{color\_xterm} and modern versions of \kbd{xterm} under X Windows, or
standard Linux/DOS text consoles. It causes \kbd{gp} to use a small palette of
colors for its output. With xterms, the colormap used corresponds to the
resources \kbd{Xterm*color$n$} where $n$ ranges from $0$ to $15$ (see the
file \kbd{misc/color.dft} for an example). Accepted values for this
default are strings \kbd{"$a_1$,\dots,$a_k$"} where $k\le7$ and each
$a_i$ is either

\noindent $\bullet$ the keyword \kbd{no} (use the default color, usually
black on transparent background)

\noindent $\bullet$ an integer between 0 and 15 corresponding to the
aforementioned colormap

\noindent $\bullet$ a triple $[c_0,c_1,c_2]$ where $c_0$ stands for foreground
color, $c_1$ for background color, and $c_2$ for attributes (0 is default, 1
is bold, 4 is underline).

The output objects thus affected are respectively error messages,
history numbers, prompt, input line, output, help messages, timer (that's
seven of them). If $k < 7$, the remaining $a_i$ are assumed to be $no$. For
instance
%
\bprog
default(colors, "9, 5, no, no, 4")
@eprog
\noindent
typesets error messages in color $9$, history numbers in color $5$, output in
color $4$, and does not affect the rest.

A set of default colors for dark (reverse video or PC console) and light
backgrounds respectively is activated when \kbd{colors} is set to
\kbd{darkbg}, resp.~\kbd{lightbg} (or any proper prefix: \kbd{d} is
recognized as an abbreviation for \kbd{darkbg}). A bold variant of
\kbd{darkbg}, called \kbd{boldfg}, is provided if you find the former too
pale.

\emacs In the present version, this default is incompatible with Emacs.
Changing it will just fail silently (the alternative would be to display
escape sequences as is, since Emacs will refuse to interpret them). On the
other hand, you can customize highlighting in your \kbd{.emacs} so as to mimic
exactly this behaviour. See \kbd{emacs/pariemacs.txt}.

\misctitle{Technical note:}
If you use an old \kbd{readline} library (version number less than 2.0),
you should do as in the example above and leave $a_3$ and $a_4$ (prompt
and input line) strictly alone. Since old versions of \kbd{readline} did
not handle escape characters correctly (or more accurately, treated them
in the only sensible way since they did not care to check all your terminal
capabilities: it just ignored them), changing them would result in many
annoying display bugs.

The specific thing to look for is to check the \kbd{readline.h} include
file, wherever your readline include files are, for the string
\kbd{RL\_PROMPT\_START\_IGNORE}. If it is there, you are safe. Another
sensible way is to make some experiments, and get a more recent
\kbd{readline} if yours doesn't work the way you would like it to. See
the file \kbd{misc/gprc.dft} for some examples.

\subsecidx{compatible} (default \kbd{0}): The GP function names and syntax
have changed tremendously between versions 1.xx and 2.00. To help you cope
with this we provide some kind of backward compatibility, depending on the
value of this default:

\quad \kbd{compatible} = 0: no backward compatibility. In this mode, a very
handy function, to be described in \secref{se:whatnow}, is \kbd{whatnow},
which tells you what has become of your favourite functions, which \kbd{gp}
suddenly can't seem to remember.

\quad \kbd{compatible} = 1: warn when using obsolete functions, but
otherwise accept them. The output uses the new conventions though, and
there may be subtle incompatibilities between the behaviour of former and
current functions, even when they share the same name (the current function
is used in such cases, of course!). We thought of this one as a transitory
help for \kbd{gp} old-timers. Thus, to encourage switching to \kbd{compatible}=0,
it is not possible to disable the warning.

\quad \kbd{compatible} = 2: use only the old function naming scheme (as
used up to version 1.39.15), but \emph{taking case into account}. Thus
\kbd{I} (${}=\sqrt{-1}$) is not the same as \kbd{i} (user variable, unbound
by default), and you won't get an error message using \kbd{i} as a loop
index as used to be the case.

\quad \kbd{compatible} = 3: try to mimic exactly the former behaviour. This
is not always possible when functions have changed in a fundamental way.
But these differences are usually for the better (they were meant to,
anyway), and will probably not be discovered by the casual user.

One adverse side effect is that any user functions and aliases that have
been defined \emph{before} changing \kbd{compatible} will get erased if this
change modifies the function list, i.e.~if you move between groups
$\{0,1\}$ and $\{2,3\}$ (variables are unaffected). We of course strongly
encourage you to try and get used to the setting \kbd{compatible}=0.

Note that the default \tet{new_galois_format} is another compatibility setting,
which is completely independent of \kbd{compatible}.

\subsecidx{datadir} (default: the location of installed precomputed data):
the name of directory containing the optional data files. For now,
only the \kbd{galdata} and \kbd{elldata} packages.

\subsecidx{debug} (default \kbd{0}): debugging level. If it is non-zero,
some extra messages may be printed (some of it in French), according to
what is going on (see~\b{g}).

\subsecidx{debugfiles} (default \kbd{0}): file usage debugging level. If it
is non-zero, \kbd{gp} will print information on file descriptors in use, from
PARI's point of view (see~\b{gf}).

\subsecidx{debugmem} (default \kbd{0}): memory debugging level. If it is
non-zero, \kbd{gp} will regularly print information on memory usage. If it's
greater than 2, it will indicate any important garbage collecting and the
function it is taking place in (see~\b{gm}).

\noindent {\bf Important Note:} As it noticeably slows down the performance,
the first functionality (memory usage) is disabled if you're not running a
version compiled for debugging (see Appendix~A).

\subsecidx{echo} (default \kbd{0}): this is a toggle, which can be either 1
(on) or 0 (off). When \kbd{echo} mode is on, each command is reprinted before
being executed. This can be useful when reading a file with the \b{r} or
\kbd{read} commands. For example, it is turned on at the beginning of the
test files used to check whether \kbd{gp} has been built correctly (see
\b{e}).

\subsec{factor\_add\_primes}\kbdsidx{factor_add_primes} (default \kbd{0}): if this is set, the integer
factorization machinery will call \kbd{addprimes} on primes
factor that were difficult to find, so they are automatically tried first in
other factorizations. If a routine is performing (or has performed) a
factorization and is interrupted by an error or via Control-C, this let you
recover the prime factors already found.

\subsecidx{format} (default \kbd{"g0.28"} and \kbd{"g0.38"} on 32-bit and
64-bit machines, respectively): of the form x$m.n$, where x is a letter in
$\{\kbd{e},\kbd{f},\kbd{g}\}$, and $n$, $m$ are integers. If x is \kbd{f},
real numbers will be printed in \idx{fixed floating point format} with no
explicit exponent (e.g.~\kbd{0.000033}), unless their integer part is not
defined (not enough significant digits); if the letter is \kbd{e}, they
will be printed in \idx{scientific format}, always with an explicit
exponent (e.g.~\kbd{3.3e-5}). If the letter is \kbd{g}, real numbers will
be printed in \kbd{f} format, except when their absolute value is less than
$2^{-32}$ or they are real zeroes (of arbitrary exponent), in which case
they are printed in \kbd{e} format.\label{se:format}

The number $n$ is the number of significant digits printed for real
numbers, except if $n<0$ where all the significant digits will be printed
(initial default 28, or 38 for 64-bit machines), and the number $m$ is the
number of characters to be used for printing integers, but is ignored if
equal to 0 (which is the default). This is a feeble attempt at formatting.

\subsecidxunix{help} (default: the location of the \kbd{gphelp} script): the
name of the external help program which will be used from within \kbd{gp} when
extended help is invoked, usually through a \kbd{??} or \kbd{???} request
(see \secref{se:exthelp}), or \kbd{M-H} under readline (see
\secref{se:readline}).

\subsecidx{histsize} (default \kbd{5000}): \kbd{gp} keeps a history of the last
\kbd{histsize} results computed so far, which you can recover using the
\kbd{\%} notation (see \secref{se:history}). When this number is exceeded,
the oldest values are erased. Tampering with this default is the only way to
get rid of the ones you do not need anymore.

\subsecidx{lines} (default \kbd{0}): if set to a positive value, \kbd{gp} prints at
most that many lines from each result, terminating the last line shown with
\kbd{[+++]} if further material has been suppressed. The various \kbd{print}
commands (see \secref{se:gp_program}) are unaffected, so you can always type
\kbd{print(\%)}, \b{a}, or \b{b} to view the full result. If the actual
screen width cannot be determined, a ``line'' is assumed to be 80 characters
long.

\subsecidx{log} (default \kbd{0}): this can be either 0 (off) or 1, 2, 3
(on, see below for the various modes). When logging mode is turned on,
\kbd{gp}
opens a log file, whose exact name is determined by the \kbd{logfile}
default. Subsequently, all the commands and results will be written to that
file (see \b{l}). In case a file with this precise name already existed, it
will not be erased: your data will be \emph{appended} at the end.

The specific positive values of \kbd{log} have the following meaning

1: plain logfile

2: emit color codes to the logfile (if \kbd{colors} is set).

3: write LaTeX output to the logfile (can be further customized using
\tet{TeXstyle}).

\subsecidx{logfile} (default \kbd{"pari.log"}): name of the log file to be
used when the \kbd{log} toggle is on. Environment and time expansion are
performed.

\subsec{new\_galois\_format}\kbdsidx{new_galois_format} (default \kbd{0}): if
this is set, the \tet{polgalois} command will use a different, more
consistent, naming scheme for Galois groups. This default is provided to
ensure that scripts can control this behaviour and do not break unexpectedly.
Note that the default value of $0$ (unset) will change to $1$ (set) in the
next major version.

\subsecidx{output} (default \kbd{1}): there are four possible values: 0
(=~\var{raw}), 1 (=~\var{prettymatrix}), 2 (=~\var{prettyprint}), or 3
(=~\var{external prettyprint}). This
means that, independently of the default \kbd{format} for reals which we
explained above, you can print results in four ways: either in \tev{raw
format}, i.e.~a format which is equivalent to what you input, including
explicit multiplication signs, and everything typed on a line instead of
two dimensional boxes. This can have several advantages, for instance it
allows you to pick the result with a mouse or an editor, and to paste it
somewhere else.\label{se:output}

The second format is the \tev{prettymatrix format}. The only difference to
raw format is that matrices are printed as boxes instead of horizontally.
This is prettier, but takes more space and cannot be used for input. Column
vectors are still printed horizontally.

The third format is the \tev{prettyprint format}, or beautified format. In
the present version \vers, this is not beautiful at all.

The fourth format is \tev{external prettyprint}, which pipes
all \kbd{gp} output in TeX format to an external prettyprinter, according to the
value of \tet{prettyprinter}. The default script (\tet{tex2mail}) converts
its input to readable two-dimensional text.

Independently of the setting of this default, an object can be printed
in any of the three formats at any time using the commands \b{a}, \b{m}
and~\b{b} respectively (see below).

\subsecidx{parisize}\label{se:parisize} (default 4M, resp.~8M on a 32-bit,
resp.~64-bit machine): \kbd{gp}, and in fact any program using the PARI
library, needs a \tev{stack} in which to do its computations. \kbd{parisize}
is the stack size, in bytes. It is strongly recommended you increase this
default (using the \kbd{-s} command-line switch, or a \tet{gprc}) if you can
afford it. Don't increase it beyond the actual amount of RAM installed on
your computer or \kbd{gp} will spend most of its time paging.

In case of emergency, you can use the \tet{allocatemem} function to
increase \kbd{parisize}, once the session is started.

\subsecidx{path} (default \kbd{".:\til:\til/gp"} on UNIX systems,
\kbd{".;C:\bs;C:\bs GP} on DOS, OS/2 and Windows, and \kbd{"."} otherwise):
This is a list of directories, separated by colons ':' (semicolons ';' in the
DOS world, since colons are pre-empted for drive names). When asked to read a
file whose name does not contain \kbd{/} (i.e.~no explicit path was given),
\kbd{gp} will look for it in these directories, in the order they were written in
\kbd{path}. Here, as usual, '.' means the current directory, and '$.\,.$' its
immediate parent. Environment expansion is performed.

\subsecidxunix{prettyprinter} (default \kbd{"tex2mail -TeX -noindent
-ragged -by\_par"}) the name of an external prettyprinter to use when
\kbd{output} is~3 (\var{alternate prettyprinter}). Note that the default
\tet{tex2mail} looks much nicer than the built-in ``beautified
format'' ($\kbd{output} = 2$).

\subsecidx{primelimit} (default \kbd{500k}): \kbd{gp} precomputes a list of
all primes less than \kbd{primelimit} at initialization time. These are used
by many arithmetical functions. If you don't plan to invoke any of them, you
can just set this to 1. The maximal value is a little less than $2^32$
(resp $2^64$) on a 32-bit (resp.~64-bit) machine.

\subsecidx{prompt} (default \kbd{"? "}): a string that will be printed as
prompt. Note that most usual escape sequences are available there: \b{e} for
Esc, \b{n} for Newline, \dots, \kbd{\bs\bs} for \kbd{\bs}. Time expansion is
performed.

This string is sent through the library function \tet{strftime} (on a
Unix system, you can try \kbd{man strftime} at your shell prompt). This means
that \kbd{\%} constructs have a special meaning, usually related to the time
and date. For instance, \kbd{\%H} = hour (24-hour clock) and \kbd{\%M} =
minute [00,59] (use \kbd{\%\%} to get a real \kbd{\%}).

If you use \kbd{readline}, escape sequences in your prompt will result in
display bugs. If you have a relatively recent \kbd{readline} (see the comment
at the end of \secref{se:colors}), you can brace them with special sequences
(\kbd{\bs[} and \kbd{\bs]}), and you will be safe. If these just result in
extra spaces in your prompt, then you'll have to get a more recent
\kbd{readline}. See the file \kbd{misc/gprc.dft} for an example.

\emacs {\bf Caution}: Emacs needs to know about the prompt pattern to
separate your input from previous \kbd{gp} results, without ambiguity. It is
not a trivial problem to adapt automatically this regular expression to an
arbitrary prompt (which can be self-modifying!). Thus, in this version \vers,
Emacs relies on the prompt being the default one. So, do not tamper with the
\kbd{prompt} variable \emph{unless} you modify it simultaneously in your
\kbd{.emacs} file (see \kbd{emacs/pariemacs.txt} and \kbd{misc/gprc.dft} for
examples).

\subsec{prompt\_cont}\kbdsidx{prompt_cont} (default \kbd{""}): a string that will be printed
to prompt for continuation lines (e.g. in between braces, or after a
line-terminating backslash). Everything that applies to \kbd{prompt}
applies to \kbd{prompt\_cont} as well.

\subsecidx{psfile} (default \kbd{"pari.ps"}): name of the default file where
\kbd{gp} is to dump its PostScript drawings (these are appended, so that no
previous data are lost). Environment and time expansion are performed.

\subsecidx{readline} (default \kbd{1}): switches readline line-editing
facilities on and off. This may be useful if you are running \kbd{gp} in a Sun
\tet{cmdtool}, which interacts badly with readline. Of course, until readline
is switched on again, advanced editing features like automatic completion
and editing history are not available.

% leave the long line for gphelp (expects ':' on the first line)
\subsecidx{realprecision} (default \kbd{28} and \kbd{38} on 32-bit and 64-bit machines respectively): the number of significant digits and, at the same
time, the number of printed digits of real numbers (see~\b{p}). Note that
PARI internal precision works on a word basis (32 or 64 bits), hence may not
coincide with the number of decimal digits you input. For instance to get 2
decimal digits you need one word of precision which, on a 32-bit machine,
actually gives you 9 digits ($9 < \log_{10}(2^{32}) < 10$):

\bprog
? default(realprecision, 2)
      realprecision = 9 significant digits (2 digits displayed)
@eprog

\subsecidx{secure} (default \kbd{0}): this is a toggle which can be either 1
(on) or 0 (off). If on, the \tet{system} and \tet{extern} command are
disabled. These two commands are potentially dangerous when you execute
foreign scripts since they let \kbd{gp} execute arbitrary UNIX commands.
\kbd{gp} will
ask for confirmation before letting you (or a script) unset this toggle.

\subsecidx{seriesprecision} (default \kbd{16}): number of significant terms
when converting a polynomial or rational function to a power series
(see~\b{ps}).

\subsecidx{simplify} (default \kbd{1}): this is a toggle which can be either
1 (on) or 0 (off). When the PARI library computes something, the type of the
result is not always the simplest possible. The only type conversions which
the PARI library does automatically are rational numbers to integers (when
they are of type \typ{FRAC} and equal to integers), and similarly rational
functions to polynomials (when they are of type \typ{RFRAC} and equal to
polynomials). This feature is useful in many cases, and saves time, but can
be annoying at times. Hence you can disable this and, whenever you feel like
it, use the function \kbd{simplify} (see Chapter 3) which allows you to
simplify objects to the simplest possible types recursively (see~\b{y}).
\sidx{automatic simplification}

\subsecidx{strictmatch} (default \kbd{1}): this is a toggle which can be
either 1 (on) or 0 (off). If on, unused characters after a sequence has been
processed will produce an error. Otherwise just a warning is printed. This
can be useful when you're not sure how many parentheses you have to close after
complicated nested loops.

\subsecidx{TeXstyle} (default \kbd{0}): the bits of this default allow
\kbd{gp} to use less rigid TeX formatting commands in the logfile. This
default is only taken into account when $\kbd{log} = 3$. The bits of
\kbd{TeXstyle} have the following meaning

2: insert \kbd{\bs right} / \kbd{\bs left} pairs where appropriate.

4: insert discretionary breaks in polynomials, to enhance the probability of
a good line break.

\subsecidx{timer} (default \kbd{0}): this is a toggle which can be either 1
(on) or 0 (off). If on, every instruction sequence (anything ended by a
newline in your input) is timed, to some accuracy depending on the hardware
and operating system. The time measured is the user \idx{CPU time},
\emph{not} including the time for printing the results (see \kbd{\#} and
\kbd{\#\#}).

\section{Simple metacommands}\label{se:meta}

\noindent
Simple metacommands are meant as shortcuts and should not be used in GP
scripts (see \secref{se:programming}). Beware that these, as all of \kbd{gp} input,
are \emph{case sensitive}. For example, \b{Q} is not identical to \b{q}. In
the following list, braces are used to denote optional arguments, with their
default values when applicable, e.g.~$\{n=0\}$ means that if $n$ is not
there, it is assumed to be~$0$. Whitespace (or spaces) between the
metacommand and its arguments and within arguments is optional. (This can
cause problems only with \b{w}, when you insist on having a filename whose
first character is a digit, and with \b{r} or \b{w}, if the filename itself
contains a space. In such cases, just use the underlying \tet{read} or
\tet{write} function; see~\secref{se:write}).

\subseckbd{?} $\{\var{command}\}$: \kbd{gp} on-line help interface.
If you type \kbd{?$n$} where $n$ is a number from 1 to 11, you will get the
list of functions in Section $3.n$ of the manual (the list of sections being
obtained by simply typing \kbd{?}). \label{se:exthelp}

These names are in general not informative enough. More details can be
obtained by typing \kbd{?\var{function}}, which gives a short explanation of
the function's calling convention and effects. Of course, to have complete
information, read Chapter 3 of this manual (the source code is at your
disposal as well, though a trifle less readable).

If the line before the copyright message indicates that extended help is
available (this means \kbd{perl} is present on your system and the PARI
distribution was correctly installed), you can add more \kbd{?} signs for
extended functionalities:

\kbd{??~\var{keyword}} yields the functions description as it stands in this
manual, usually in Chapter~2 or~3. If you're not satisfied with the default
chapter chosen, you can impose a given chapter by ending the keyword with
\kbd{@} followed by the chapter number, e.g.~\kbd{??~Hello@2} will look in
Chapter~2 for section heading \kbd{Hello} (which doesn't exist, by the way).

All operators (e.g.~\kbd{+}, \kbd{\&\&}, etc.) are accepted by this
extended help, as well as a few other keywords describing key \kbd{gp} concepts,
e.g.~\kbd{readline} (the line editor), \kbd{integer}, \kbd{nf} (``number
field'' as used in most algebraic number theory computations), \kbd{ell}
(elliptic curves), etc.

In case of conflicts between function and default names (e.g \tet{log},
\tet{simplify}), the function has higher priority. To get the default help,
use 
\bprog
  ?? default(log)
  ?? default(simplify)
@eprog

\kbd{???~\var{pattern}} produces a list of sections in Chapter~3 of the
manual related to your query. As before, if \var{pattern} ends by \kbd{@}
followed by a chapter number, that chapter is searched instead; you also
have the option to append a simple \kbd{@} (without a chapter number) to
browse through the whole manual.

If your query contains dangerous characters (e.g \kbd{?} or blanks) it is
advisable to enclose it within double quotes, as for GP strings (e.g
\kbd{???~"elliptic curve"}).

Note that extended help is much more powerful than the short help, since
it knows about operators as well: you can type \kbd{??~*} or
\kbd{??~\&\&}, whereas a single \kbd{?} would just yield a not too helpful

\kbd{*** unknown identifier.}

\noindent message. Also, you can ask for extended help on section
number~$n$ in Chapter~3, just by typing \kbd{??~$n$} (where \kbd{?$n$} would
yield merely a list of functions). Finally, a few key concepts in \kbd{gp} are
documented in this way: metacommands (e.g \kbd{??~"??"}), defaults (e.g
\kbd{??~psfile}) and type names (e.g \typ{INT} or \kbd{integer}), as well as
various miscellaneous keywords such as \kbd{edit} (short summary of line
editor commands), \kbd{operator}, \kbd{member}, \kbd{"user defined"},
\kbd{nf}, \kbd{ell}, \dots

Last but not least: \kbd{??} without argument will open a \kbd{dvi}
previewer (\kbd{xdvi} by default, \kbd{\$GPXDVI} if it is defined in your
environment) containing the full user's manual. \kbd{??tutorial} and
\kbd{??refcard} do the same with the \idx{tutorial} and \idx{reference card}
respectively.

\misctitle{Technical note:} these functionalities are provided by an
external \kbd{perl} script that you are free to use outside any \kbd{gp} session
(and modify to your liking, if you are perl-knowledgeable). It is called
\tet{gphelp}, lies in the \kbd{doc} subdirectory of your distribution
(just make sure you run \kbd{Configure} first, see Appendix~A) and is
really two programs in one. The one which is used from within \kbd{gp} is
\kbd{gphelp} which runs \TeX\ on a selected part of this manual, then opens
a previewer. \kbd{gphelp -detex} is a text mode equivalent, which looks
often nicer especially on a colour-capable terminal (see
\kbd{misc/gprc.dft} for examples). The default \kbd{help} selects which
help program will be used from within \kbd{gp}. You are welcome to improve this
help script, or write new ones (and we would like to know about it
so that we may include them in future distributions). By the way, outside
of \kbd{gp} you can give more than one keyword as argument to \kbd{gphelp}.

\subseckbd{/*...*/}: comment. Everything between the stars is ignored by
\kbd{gp}. These comments can span any number of lines.

\subseckbd{\bs\bs}: one-line comment. The rest of the line
is ignored by \kbd{gp}.

\subsec{\b{a}} $\{n\}$: prints the object number $n$ ($\%n$)
in raw format. If the number $n$ is omitted, print the latest computed object
($\%$). \label{se:history}

\subsec{\b{b}} $\{n\}$: Same as \b{a}, in prettyprint (i.e.~beautified)
format.

\subsec{\b{c}}:\sidx{available commands} prints the list of all available
hardcoded functions under \kbd{gp}, not including operators written as special
symbols (see \secref{se:operators}). More information can be obtained using
the \kbd{?} metacommand (see above). For user-defined functions / member
functions, see \b{u} and \b{um}.

\subsec{\b{d}}: prints the \idx{defaults} as described in the
previous section (shortcut for \kbd{default()}, see \secref{se:default}).

\subsec{\b{e}} $\{n\}$: switches the \tet{echo} mode on (1) or off (0). If
$n$ is explicitly given, set echo to $n$.

\subsec{\b{g}} $\{n\}$: sets the debugging level \tet{debug} to the
non-negative integer $n$.

\subsec{\b{gf}} $\{n\}$: sets the file usage debugging level \tet{debugfiles}
to the non-negative integer $n$.

\subsec{\b{gm}} $\{n\}$: sets the memory debugging level \tet{debugmem}
to the non-negative integer $n$.

\subsec{\b{h}} $\{m$\kbd{-}$n\}$: outputs some debugging info about the
hashtable. If the argument is a number $n$, outputs the contents of cell
$n$. Ranges can be given in the form $m$\kbd{-}$n$ (from cell $m$ to cell
$n$, \$ = last cell). If a function name is given instead of a number or
range, outputs info on the internal structure of the hash cell this
function occupies (a \kbd{struct entree} in C). If the range is reduced to
a dash ('\kbd{-}'), outputs statistics about hash cell usage.

\subsec{\b{l}} $\{$\var{logfile}$\}$: switches \tet{log} mode on and off.
If a \var{logfile} argument is given, change the default logfile name to
\var{logfile} and switch log mode on.

\subsec{\b{m}}: as \b{a}, but using prettymatrix format.

\subsec{\b{o}} $\{n\}$: sets \tet{output} mode to $n$ ($0$: raw, $1$:
prettymatrix, $2$: prettyprint, $3$: external prettyprint).

\subsec{\b{p}} $\{n\}$: sets \tet{realprecision} to $n$ decimal
digits. Prints its current value if $n$ is omitted.

\subsec{\b{ps}} $\{n\}$: sets \tet{seriesprecision} to $n$ significant terms.
Prints its current value if $n$ is omitted.

\subsec{\b{q}}: quits the \kbd{gp} session and returns to the system.
Shortcut for the function \tet{quit} (see \secref{se:quit}).

\subsec{\b{r}} $\{$\var{filename}$\}$: \idx{read}s into \kbd{gp} all the
commands contained in the named file as if they had been typed from the
keyboard, one line after the other. Can be used in combination with the \b{w}
command (see below). Related but not equivalent to the function \kbd{read}
(see \secref{se:read}); in particular, if the file contains more than one
line of input, there will be one history entry for each of them, whereas
\kbd{read} would only record the last one. If \var{filename} is omitted,
re-read the previously used input file (fails if no file has ever been
successfully read in the current session). If a \kbd{gp} \tet{binary file}
(see \secref{se:writebin}) is read using this command, it is silently loaded,
without cluttering the history.

Assuming \kbd{gp} figures how to decompress files on your machine, this
command accepts compressed files in \tet{compress}ed (\kbd{.Z}) or
\tet{gzip}ped (\kbd{.gz} or \kbd{.z}) format. They will be uncompressed on
the fly as \kbd{gp} reads them, without changing the files themselves.

\subsec{\b{s}}: prints the state of the PARI \tev{stack} and \tev{heap}.
This is used primarily as a debugging device for PARI.

\subsec{\b{t}}: prints the \idx{internal longword format} of all the PARI
types. The detailed bit or byte format of the initial codeword(s) is
explained in Chapter~4, but its knowledge is not necessary for a \kbd{gp} user.

\subsec{\b{u}}: prints the definitions of all user-defined functions.

\subsec{\b{um}}: prints the definitions of all user-defined member functions.

\subsec{\b{v}}: prints the \idx{version number} and implementation architecture
(680x0, Sparc, Alpha, other) of the \kbd{gp} executable you are using. In library
mode, you can use instead the two character strings \kbd{PARIVERSION} and
\kbd{PARIINFO}, which correspond to the first two lines printed by \kbd{gp} just
before the Copyright message.

\subsec{\b{w}} $\{n\}$ $\{$\var{filename}$\}$: writes the object number
$n$ ( $\%n$ ) into the named file, in raw format. If the number $n$ is
omitted, writes the latest computed object ( $\%$ ). If \var{filename} is
omitted, appends to \kbd{logfile} (the GP function \tet{write} is a trifle more
powerful, as you can have arbitrary filenames).

\subsec{\b{x}}: prints the complete tree with addresses and contents (in
hexadecimal) of the \idx{internal representation} of the latest computed
object in \kbd{gp}. As for \b{s}, this is used primarily as a debugging device for
PARI, and the format should be self-explanatory (a $*$ before an object --
typically a modulus -- means the corresponding component is out of stack).
However, used on a PARI integer, it can be used as a
decimal$\rightarrow$hexadecimal converter.

\subsec{\b{y}} $\{n\}$: switches \kbd{simplify} on (1) or off (0). If $n$
is explicitly given, set simplify to $n$.

\subseckbd{\#}: switches the \kbd{timer} on or off.

\subseckbd{\#\#}: prints the time taken by the latest computation.
Useful when you forgot to turn on the \kbd{timer}.


\section{The preferences file}\sidx{startup}\sidx{preferences file}
\label{se:gprc}

This file, called \tet{gprc} in the sequel, is used to modify or extend
\kbd{gp}
default behaviour, in all \kbd{gp} sessions: e.g customize \kbd{default} values or
load common user functions and aliases. \kbd{gp} opens the \kbd{gprc} file and
processes the commands in there, \emph{before} doing anything else,
e.g.~creating the PARI stack. If the file does not exist or cannot be read,
\kbd{gp} will proceed to the initialization phase at once, eventually emitting a
prompt. If any explicit command line switches are given, they override the
values read from the preferences file.

\subsec{Syntax}. The syntax in the \kbd{gprc} file (and valid in this file
only) is simple-minded, but should be sufficient for most purposes. The file
is read line by line; as usual, white space is ignored unless surrounded by
quotes and the standard multiline constructions using braces, \kbd{\bs}, or
\kbd{=} are available (multiline comments between \kbd{/*~\dots~*/} are also
recognized).

\subsubsec{Preprocessor:}
Two types of lines are first dealt with by a preprocessor:

$\bullet$ comments are removed. This applies to all text surrounded by
\kbd{/*~\dots~*/} as well as to everything following \kbd{\bs\bs} on a given
line.

$\bullet$ lines starting with \kbd{\#if} \var{boolean} are treated as
comments if \var{boolean} evaluates to \kbd{false}, and read normally
otherwise. The condition can be negated using either \kbd{\#if not} (or
\kbd{\#if !}). If the rest of the current line is empty, the test applies to
the next line (same behaviour as \kbd{=} under \kbd{gp}). Only three tests can be
performed:

\kbd{EMACS}: \kbd{true} if \kbd{gp} is running in an Emacs or TeXmacs shell (see
\secref{se:emacs}).

\kbd{READL}: \kbd{true} if \kbd{gp} is compiled with \kbd{readline} support (see
\secref{se:readline}).

\kbd{VERSION} \var{op} \var{number}: where \var{op} is in the set
$\{ \kbd{>}, \kbd{<}, \kbd{<=}, \kbd{>=} \}$, and \var{number} is a PARI
version number of the form \var{Major}.\var{Minor}.\var{patch}, where the
last two components can be omitted (i.e.~$1$ is understood as versio $1.0.0$).
This is \kbd{true} if \kbd{gp}'s version number satisfies the required
inequality.

\subsubsec{Commands:}
After the preprocessing the remaining lines are executed as
sequence of expressions (as usual, separated by \kbd{;} if necessary). Only
two kinds of expressions are recognized:

$\bullet$ \var{default} \kbd{=} \var{value}, where \var{default} is one of
the available defaults (see \secref{se:defaults}), which will be set to
\var{value} on actual startup. Don't forget the quotes around strings
(e.g.~for \kbd{prompt} or \kbd{help}).

$\bullet$ \kbd{read "\var{some\_GP\_file}"} where \kbd{\var{some\_GP\_file}}
is a regular GP script this time, which will be read just before \kbd{gp} prompts
you for commands, but after initializing the defaults. In particular, file
input is delayed until the \kbd{gprc} has been fully loaded. This is the
right place to input files containing \kbd{alias} commands, or your favorite
macros.

\noindent For instance you could set your prompt in the following portable way:
\bprog
\\ self modifying prompt looking like @com\hbox{\rm(18:03) \key{gp}\kbd{ >}}
prompt   = "(%H:%M) \e[1mgp\e[m > "

\\ readline wants non-printing characters to be braced between ^A/^B pairs
#if READL prompt = "(%H:%M) ^A\e[1m^Bgp^A\e[m^B > "

\\ escape sequences not supported under emacs
#if EMACS prompt = "(%H:%M) gp > "
@eprog

\noindent Note that any of the last two lines could be broken in the
following way
\bprog
#if EMACS
  prompt = "(%H:%M) gp > "
@eprog
\noindent since the preprocessor directive applies to the next line if the
current one is empty.

A sample \kbd{gprc} file called \kbd{misc/gprc.dft} is provided in the
standard distribution. It is a good idea to have a look at it and customize
it to your needs. Since this file does not use multiline constructs, here is
one (note the terminating \kbd{;} to separate the expressions):
\bprog
#if VERSION > 2.2.3
{
  read "my_scripts";     \\ syntax errors in older versions
  new_galois_format = 1; \\ default introduced in 2.2.4
}
#if ! EMACS
{
  colors = "9, 5, no, no, 4, 1, 2";
  help   = "gphelp -detex -ch 4 -cb 0 -cu 2";
}
@eprog

\subsec{Where is it?}
When \kbd{gp} is started, it looks for a customization file, or \kbd{gprc} in the
following places (in this order, only the first one found will be loaded):

\noindent$\bullet$ On the Macintosh (only), \kbd{gp} looks in the directory
which contains the \kbd{gp} executable itself for a file called \kbd{gprc}.

\noindent$\bullet$ \kbd{gp} checks whether the environment variable
\tet{GPRC} is set. Under DOS, you can set it in \kbd{AUTOEXEC.BAT}. On Unix,
this can be done with something like: \smallskip

\settabs\+\indent&\kbd{GPRC=/my/dir/anyname; export GPRC}\quad&\cr

\+&\kbd{GPRC=/my/dir/anyname; export GPRC}\quad&in \kbd{sh} syntax
(for instance in your \kbd{.profile}),\cr

\+&\kbd{setenv GPRC /my/dir/anyname} &in \kbd{csh} syntax
(in your \kbd{.login} or \kbd{.cshrc} file).\cr

\noindent If so, the file named by \kbd{\$GPRC} is the \kbd{gprc}.

\noindent$\bullet$ If \kbd{GPRC} is not set, and if the environment variable
\kbd{HOME} is defined, \kbd{gp} then tries

\kbd{\$HOME/.gprc} on a Unix system

\kbd{\$HOME\bs\_$\,$gprc} on a DOS, OS/2, or Windows system.

\noindent$\bullet$ If \kbd{HOME} also leaves us clueless, we try

\strut\kbd{\til/.gprc} on a Unix system (where as usual \kbd{\til} stands for
your home directory), or

\kbd{\b{\_}$\,$gprc} on a DOS, OS/2, or Windows system.

\noindent$\bullet$ Finally, if no gprc was found among the user files
mentioned above we look for \kbd{/etc/gprc} (\kbd{\bs etc\bs gprc})
for a system-wide gprc file (you will need root privileges to set up such a
file yourself).

Note that on Unix systems, the \kbd{gprc}'s default name starts with a '.' and
thus is hidden to regular \kbd{ls} commands; you need to type \kbd{ls -a} to
list it.


\section{Using GNU Emacs}
\label{se:emacs}

If GNU \idx{Emacs} is installed on your machine, it is possible to use
\kbd{gp} as
a subprocess in \idx{Emacs}. To use this, you should include in your
\kbd{.emacs} file the following lines:
\bprog
(autoload 'gp-mode "pari" nil t)
(autoload 'gp-script-mode "pari" nil t)
(autoload 'gp "pari" nil t)
(autoload 'gpman "pari" nil t)
(setq auto-mode-alist
  (cons '("\\.gp$" . gp-script-mode) auto-mode-alist))
@eprog
\noindent which autoloads functions from \kbd{pari.el}. See also
\kbd{pariemacs.txt}. These files are included in the PARI distribution and
are installed at the same time as \kbd{gp}.

Once this is done, under GNU Emacs if you type \kbd{M-x gp} (where as usual
\kbd{M} is the \kbd{Meta} key, i.e.~Escape, or on SUN keyboards, the Left
key), a special shell will be started, which in particular launches \kbd{gp} with
the default stack size, prime limit and input buffer size. If you type
instead \kbd{C-u M-x gp}, you will be asked for the name of the \kbd{gp}
executable, the stack size and the prime limit before the execution of
\kbd{gp}
begins. If for any of these you simply type return, the default value will
be used. On UNIX machines it will be the place you told \kbd{Configure}
(usually \kbd{/usr/local/bin/gp}) for the executable, \kbd{10M} for the
stack and \kbd{500k} for the prime limit.

\smallskip
  You can then work as usual under \kbd{gp}, but with two notable advantages (which
don't really matter if readline is available to you, see below). First and
foremost, you have at your disposal all the facilities of a text editor like
Emacs, in particular for correcting or copying blocks. Second, you can have
an on-line help which is much more complete than what you obtain by typing
\kbd{?name}. This is done by typing \kbd{M-?}. In the minibuffer, Emacs asks
what function you want to describe, and after your reply you obtain the
description which is in the users manual, including the description of
functions (such as \kbd{\bs}, \kbd{\%}) which use special symbols.

  This help system can also be menu-driven, by using the command
\kbd{M-\char`\\ c} which opens a help menu window which enables you to choose
the category of commands for which you want an explanation.

  Nevertheless, if extended help is available on your system (see
\secref{se:exthelp}), you should use it instead of the above, since it's
nicer (it ran through \TeX) and understands many more keywords.

  Finally you can use command completion in the following way. After the
prompt, type the first few letters of the command, then \kbd{<TAB>} where
\kbd{<TAB>} is the TAB key. If there exists a unique command starting with
the letters you have typed, the command name will be completed. If not,
either the list of commands starting with the letters you typed will be
displayed in a separate window (which you can then kill by typing as usual
\kbd{C-x 1} or by typing in more letters), or ``no match found'' will be
displayed in the Emacs command line. If your \kbd{gp} was linked with the readline
library, read the section on completion in the section below (the paragraph
on online help is not relevant).

Note that if for some reason the session crashes (due to a bug in your
program or in the PARI system), you will usually stay under Emacs, but the
\kbd{gp} buffer will be killed. To recover it, simply type again \kbd{M-x gp}
(or \kbd{C-u M-x gp}), and a new session of \kbd{gp} will be started after
the old one, so you can recover what you have typed. Note that this will of
course \emph{not} work if for some reason you kill Emacs and start a new
session.

\smallskip You also have at your disposal a few other commands and many
possible customizations (colours, prompt). Read the file
\kbd{emacs/pariemacs.txt} in standard distribution for details.


\section{Using readline} \sidx{line editor}\sidx{completion}

  Thanks to the initial help of Ilya Zakharevich, there is a possibility of
line editing and command name completion outside of an Emacs buffer \emph{if}
you have compiled \kbd{gp} with the GNU \idx{readline} library. If you do not
have Emacs available, or cannot stand using it, we really advise you to make
sure you get this very useful library before configuring or compiling
\kbd{gp}. In fact, with \kbd{readline}, even line editing becomes \emph{more}
powerful outside an Emacs buffer!

\subsec{A (too) short introduction to readline}: \label{se:readline} The
basics are as follows (read the readline user manual~!), assume that \kbd{C-}
stands for ``the \kbd{Control} key combined with another'' and the same for
\kbd{M-} with the \kbd{Meta} key (generally \kbd{C-} combinations act on
characters, while the \kbd{M-} ones operate on words). The \kbd{Meta} key
might be called \kbd{Alt} on some keyboards, will display a black diamond on
most others, and can safely be replaced by \kbd{Esc} in any case. Typing any
ordinary key inserts text where the cursor stands, the arrow keys enabling
you to move in the line. There are many more movement commands, which will be
familiar to the Emacs user, for instance \kbd{C-a}/\kbd{C-e} will take you to
the start/end of the line, \kbd{M-b}/\kbd{M-f} move the cursor
backward/forward by a word, etc. Just press the \kbd{<Return>} key at any
point to send your command to \kbd{gp}.

  All the commands you type in are stored in a history (with multiline
commands being saved as single concatenated lines). The Up and Down arrows
(or \kbd{C-p}/\kbd{C-n}) will move you through it, \kbd{M-<}/\kbd{M->}
sending you to the start/end of the history. \kbd{C-r}/\kbd{C-s} will start
an incremental backward/forward search. You can kill text (\kbd{C-k} kills
till the end of line, \kbd{M-d} to the end of current word) which you can
then yank back using the \kbd{C-y} key (\kbd{M-y} will rotate the kill-ring).
\kbd{C-\_} will undo your last changes incrementally (\kbd{M-r} undoes all
changes made to the current line). \kbd{C-t} and \kbd{M-t} will transpose the
character (word) preceding the cursor and the one under the cursor.

  Keeping the \kbd{M-} key down while you enter an integer (a minus sign
meaning reverse behaviour) gives an argument to your next readline command
(for instance \kbd{M-- C-k} will kill text back to the start of line). If you
prefer \idx{Vi}--style editing, \kbd{M-C-j} will toggle you to Vi mode.

  Of course you can change all these default bindings. For that you need to
create a file named \kbd{.inputrc} in your home directory. For instance
(notice the embedding conditional in case you would want specific bindings
for \kbd{gp}):
%
\bprog
$if Pari-GP
  set show-all-if-ambiguous
  "\C-h": backward-delete-char
  "\e\C-h": backward-kill-word
  "\C-xd": dump-functions
  (: "\C-v()\C-b"       #@com can be annoying when copy-pasting !
  [: "\C-v[]\C-b"
$endif
@eprog
\noindent\kbd{C-x C-r} will re-read this init file, incorporating any
changes made to it during the current session.

\misctitle{Note:} By default, \kbd{(} and \kbd{[} are bound to the function
\kbd{pari-matched-insert} which, if ``electric parentheses'' are enabled
(default: off) will automatically insert the matching closure (respectively
\kbd{)} and \kbd{]}). This behaviour can be toggled on and off by giving
the numeric argument $-2$ to \kbd{(} (\kbd{M--2(}), which is useful if you
want, e.g to copy-paste some text into the calculator. If you do not want a
toggle, you can use \kbd{M--0} / \kbd{M--1} to specifically switch it on or
off).

\misctitle{Note:} In some versions of readline (2.1 for instance), the
\kbd{Alt} or \kbd{Meta} key can give funny results (output 8-bit accented
characters for instance). If you do not want to fall back to the \kbd{Esc}
combination, put the following two lines in your \kbd{.inputrc}:
%
\bprog
  set convert-meta on
  set output-meta off
@eprog

% don't remove this leading space (prevent gphelp from splitting the entry)
 \subsec{Command completion and online help}. As in the Emacs shell,
\kbd{<TAB>} will complete words for you. But, under readline, this mechanism
will be context-dependent: \kbd{gp} will strive to only give you meaningful
completions in a given context (it will fail sometimes, but only under rare
and restricted conditions).

  For instance, shortly after a \kbd{\til}, we expect a user name, then a
path to some file. Directly after \kbd{default(} has been typed, we would
expect one of the \kbd{default} keywords. After \kbd{whatnow(} , we expect
the name of an old function, which may well have disappeared from this
version. After a '.', we expect a member keyword. And generally of course, we
expect any GP symbol which may be found in the hashing lists: functions (both
yours and GP's), and variables.

  If, at any time, only one completion is meaningful, \kbd{gp} will provide it
together with

$\bullet$ an ending comma if we are completing a default,

$\bullet$ a pair of parentheses if we are completing a function name. In
that case hitting \kbd{<TAB>} again will provide the argument list as given
by the online help\footnote{*}{recall that you can always undo the effect
of the preceding keys by hitting \kbd{C-\_}}.

Otherwise, hitting \kbd{<TAB>} once more will give you the list of possible
completions. Just experiment with this mechanism as often as possible,
you will probably find it very convenient. For instance, you can obtain
\kbd{default(seriesprecision,10)}, just by hitting \kbd{def<TAB>se<TAB>10},
which saves 18 keystrokes (out of 27).

  Hitting \kbd{M-h} will give you the usual short online help concerning the
word directly beneath the cursor, \kbd{M-H} will yield the extended help
corresponding to the \kbd{help} default program (usually opens a \idx{dvi}
previewer, or runs a primitive tex-to-ASCII program). None of these disturb
the line you were editing.
\vfill\eject