% Copyright (c) 2000  The PARI Group
%
% This file is part of the PARI/GP documentation
%
% Permission is granted to copy, distribute and/or modify this document
% under the terms of the GNU General Public License
\chapter{Functions and Operations Available in PARI and GP}
\label{se:functions}

The functions and operators available in PARI and in the GP/PARI calculator
are numerous and ever-expanding. Here is a description of the ones available
in version \vers. It should be noted that many of these functions accept
quite different types as arguments, but others are more restricted. The list
of acceptable types will be given for each function or class of functions.
Except when stated otherwise, it is understood that a function or operation
which should make natural sense is legal.

On the other hand, many routines list explicit preconditions for some of their
arguments, e.g. $p$ is a prime number, or $q$ is a positive definite quadratic
form. For reasons of efficiency, all routines trust the user input and only
perform minimal sanity checks. When a precondition is not satisfied, any of the
following may occur: a regular exception is raised, the PARI stack overflows, a
\kbd{SIGSEGV} or \kbd{SIGBUS} signal is generated, or we enter an infinite
loop. The function can also quietly return a mathematically meaningless
result: junk in, junk out. In the following, we document the results
as \emph{undefined} in this case.

In this chapter, we will describe the functions according to a rough
classification. The general entry looks something like:

\key{foo}$(x,\{\fl=0\})$: short description.

The library syntax is \kbd{GEN foo(GEN x, long flag = 0)}.

\noindent
This means that the GP function \kbd{foo} has one mandatory argument $x$, and
an optional one, $\fl$, whose default value is 0. (The $\{\}$ should not be
typed, it is just a convenient notation that we will use throughout to denote
optional arguments.) That is, you can type \kbd{foo(x,2)}, or \kbd{foo(x)},
which is then understood to mean \kbd{foo(x,0)}. As well, a comma or closing
parenthesis, where an optional argument should have been, signals to GP it
should use the default. Thus, the syntax \kbd{foo(x,)} is also accepted as a
synonym for our last expression. When a function has more than one optional
argument, the argument list is filled with user supplied values, in order.
When none are left, the defaults are used instead. Thus, assuming that
\kbd{foo}'s prototype had been
$$\hbox{%
\key{foo}$(\{x=1\},\{y=2\},\{z=3\})$,%
}$$
typing in \kbd{foo(6,4)} would give
you \kbd{foo(6,4,3)}. In the rare case when you want to set some far away
argument, and leave the defaults in between as they stand, you can use the
``empty arg'' trick alluded to above: \kbd{foo(6,,1)} would yield
\kbd{foo(6,2,1)}. By the way, \kbd{foo()} by itself yields
\kbd{foo(1,2,3)} as was to be expected.

In this rather special case of a function having no mandatory argument, you
can even omit the $()$: a standalone \kbd{foo} would be enough (though we
do not recommend it for your scripts, for the sake of clarity). In defining
GP syntax, we strove to put optional arguments at the end of the argument
list (of course, since they would not make sense otherwise), and in order of
decreasing usefulness so that, most of the time, you will be able to ignore
them.

Finally, an optional argument (between braces) followed by a star, like
$\{\var{x}\}*$, means that any number of such arguments (possibly none) can
be given. This is in particular used by the various \kbd{print} routines.

\misctitle{Flags} A \tev{flag} is an argument which, rather than conveying
actual information to the routine, instructs it to change its default
behavior, e.g.~return more or less information. All such
flags are optional, and will be called \fl\ in the function descriptions to
follow. There are two different kind of flags

\item generic: all valid values for the flag are individually
described (``If \fl\ is equal to $1$, then\dots'').

\item binary:\sidx{binary flag} use customary binary notation as a
compact way to represent many toggles with just one integer. Let
$(p_0,\dots,p_n)$ be a list of switches (i.e.~of properties which take either
the value $0$ or~$1$), the number $2^3 + 2^5 = 40$ means that $p_3$ and $p_5$
are set (that is, set to $1$), and none of the others are (that is, they
are set to $0$). This is announced as ``The binary digits of $\fl$ mean 1:
$p_0$, 2: $p_1$, 4: $p_2$'', and so on, using the available consecutive
powers of~$2$.

\misctitle{Mnemonics for binary flags} Numeric flags as mentioned above are
obscure, error-prone, and quite rigid: should the authors want to adopt a new
flag numbering scheme, it would break backward compatibility. The only
advantage of explicit numeric values is that they are fast to type, so their
use is only advised when using the calculator \kbd{gp}.

As an alternative, one can replace a binary flag by a character string
containing symbolic identifiers (mnemonics). In the function description,
mnemonics corresponding to the various toggles are given after each of them.
They can be negated by prepending \kbd{no\_} to the mnemonic, or by removing
such a prefix. These toggles are grouped together using any punctuation
character (such as ',' or ';'). For instance (taken from description of
$\tet{ploth}(X=a,b,\var{expr},\{\fl=0\},\{n=0\})$)

\centerline{Binary digits of flags mean: $1=\kbd{Parametric}$,
$2=\kbd{Recursive}$, \dots}

\noindent so that, instead of $1$, one could use the mnemonic
\kbd{"Parametric; no\_Recursive"}, or simply \kbd{"Parametric"} since
\kbd{Recursive} is unset by default (default value of $\fl$ is $0$,
i.e.~everything unset). People used to the bit-or notation in languages like
C may also use the form \kbd{"Parametric | no\_Recursive"}.

\misctitle{Pointers} \varsidx{pointer} If a parameter in the function
prototype is prefixed with a \& sign, as in

\key{foo}$(x,\&e)$

\noindent it means that, besides the normal return value, the function may
assign a value to $e$ as a side effect. When passing the argument, the \&
sign has to be typed in explicitly. As of version \vers, this \tev{pointer}
argument is optional for all documented functions, hence the \& will always
appear between brackets as in \kbd{Z\_issquare}$(x,\{\&e\})$.

\misctitle{About library programming}
The \var{library} function \kbd{foo}, as defined at the beginning of this
section, is seen to have two mandatory arguments, $x$ and \fl: no function
seen in the present chapter has been implemented so as to accept a variable
number of arguments, so all arguments are mandatory when programming with the
library (usually, variants are provided corresponding to the various flag values).
We include an \kbd{= default value} token in the prototype to signal how a missing
argument should be encoded. Most of the time, it will be a \kbd{NULL} pointer, or
-1 for a variable number. Refer to the \emph{User's Guide to the PARI library}
for general background and details.
