Catalogue of GP/PARI Functions: GP defaults
Author: Nils-Peter Skoruppa
Keywords: PARI, GP, DOC

### GP defaults

#### TeXstyle

The bits of this default allow `gp` to use less rigid TeX formatting commands in the logfile. This default is only taken into account when `log` = 3. The bits of `TeXstyle` have the following meaning

2: insert `\right` / `\left` pairs where appropriate.

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

The default value is `0`.

#### breakloop

If true, enables the "break loop" debugging mode, see Section [Label: se:break_loop].

The default value is `1` if we are running an interactive `gp` session, and `0` otherwise.

#### colors

This default is only usable if `gp` is running within certain color-capable terminals. For instance `rxvt`, `color_xterm` and modern versions of `xterm` under X Windows, or standard Linux/DOS text consoles. It causes `gp` to use a small palette of colors for its output. With xterms, the colormap used corresponds to the resources `Xterm*colorn` where n ranges from 0 to 15 (see the file `misc/color.dft` for an example). Accepted values for this default are strings `"a_1,...,a_k"` where k `<=` 7 and each a_i is either

* the keyword `no` (use the default color, usually black on transparent background)

* an integer between 0 and 15 corresponding to the aforementioned colormap

* 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

      default(colors, "9, 5, no, no, 4")

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 `colors` is set to `darkbg`, resp. `lightbg` (or any proper prefix: `d` is recognized as an abbreviation for `darkbg`). A bold variant of `darkbg`, called `boldfg`, is provided if you find the former too pale.

In the present version, this default is incompatible with PariEmacs. Changing it will just fail silently (the alternative would be to display escape sequences as is, since Emacs will refuse to interpret them). You must customize color highlighting from the PariEmacs side, see its documentation.

The default value is `""` (no colors).

#### compatible

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:

`compatible` = 0: no backward compatibility. In this mode, a very handy function, to be described in Section [Label: se:whatnow], is `whatnow`, which tells you what has become of your favourite functions, which `gp` suddenly can't seem to remember.

`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 behavior 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 `gp` old-timers. Thus, to encourage switching to `compatible` = 0, it is not possible to disable the warning.

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

`compatible` = 3: try to mimic exactly the former behavior. 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 before changing `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 `compatible` = 0.

Note that the default `new_galois_format` is another compatibility setting, which is completely independent of `compatible`.

The default value is `0`.

#### datadir

The name of directory containing the optional data files. For now, this includes the `elldata`, `galdata`, `galpol`, `seadata` packages.

The default value is \datadir (the location of installed precomputed data, can be specified via `Configure --datadir =` ).

#### debug

Debugging level. If it is non-zero, some extra messages may be printed, according to what is going on (see `\g`).

The default value is `0` (no debugging messages).

#### debugfiles

File usage debugging level. If it is non-zero, `gp` will print information on file descriptors in use, from PARI's point of view (see `\gf`).

The default value is `0` (no debugging messages).

#### debugmem

Memory debugging level. If it is non-zero, `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 `\gm`).

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).

The default value is `0` (no debugging messages).

#### echo

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

The default value is `0` (no echo).

#### factor_add_primes

This toggle is either 1 (on) or 0 (off). If on, the integer factorization machinery calls `addprimes` on primes factor that were difficult to find (larger than 2^24), 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 lets you recover the prime factors already found. The downside is that a huge `addprimes` table unrelated to the current computations will slow down arithmetic functions relying on integer factorization; one should then empty the table using `removeprimes`.

The defaut value is `0`.

#### factor_proven

This toggle is either 1 (on) or 0 (off). By default, the factors output by the integer factorization machinery are only pseudo-primes, not proven primes. If this toggle is set, a primality proof is done for each factor and all results depending on integer factorization are fully proven. This flag does not affect partial factorization when it is explicitly requested. It also does not affect the private table managed by `addprimes`: its entries are included as is in factorizations, without being tested for primality.

The defaut value is `0`.

#### format

Of the form x.n, where x (conversion style) is a letter in {`e`,`f`,`g`}, and n (precision) is an integer; this affects the way real numbers are printed:

* If the conversion style is `e`, real numbers are printed in scientific format, always with an explicit exponent, e.g. `3.3 E-5`.

* In style `f`, real numbers are generally printed in \idx{fixed floating point format} without exponent, e.g. `0.000033`. A large real number, whose integer part is not well defined (not enough significant digits), is printed in style `e`. For instance `10.^100` known to ten significant digits is always printed in style `e`.

* In style `g`, non-zero real numbers are printed in `f` format, except when their decimal exponent is < -4, in which case they are printed in `e` format. Real zeroes (of arbitrary exponent) are printed in `e` format.

The precision 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). For more powerful formatting possibilities, see `printf` and `Strprintf`.

The default value is `"g.28"` and `"g.38"` on 32-bit and 64-bit machines, respectively.

#### graphcolormap

A vector of colors, to be used by hi-res graphing routines. Its length is arbitrary, but it must contain at least 3 entries: the first 3 colors are used for background, frame/ticks and axes respectively. All colors in the colormap may be freely used in `plotcolor` calls.

A color is either given as in the default by character strings or by an RGB code. For valid character strings, see the standard `rgb.txt` file in X11 distributions, where we restrict to lowercase letters and remove all whitespace from color names. An RGB code is a vector with 3 integer entries between 0 and 255. For instance `[250, 235, 215]` and `"antiquewhite"` represent the same color. RGB codes are cryptic but often easier to generate.

The default value is [`"white"`, `"black"`, `"blue"`, `"violetred"`, `"red"`, `"green"`, `"grey"`, `"gainsboro"`].

#### graphcolors

Entries in the `graphcolormap` that will be used to plot multi-curves. The successive curves are drawn in colors

`graphcolormap[graphcolors[1]]`, `graphcolormap[graphcolors[2]]`, ...

cycling when the `graphcolors` list is exhausted.

The default value is `[4,5]`.

#### help

Name of the external help program to use from within `gp` when extended help is invoked, usually through a `??` or `???` request (see Section [Label: se:exthelp]), or `M-H` under readline (see Section [Label: se:readline]).

The default value is the path to the `gphelp` script we install.

#### histfile

Name of a file where `gp` will keep a history of all input commands (results are omitted). If this file exists when the value of `histfile` changes, it is read in and becomes part of the session history. Thus, setting this default in your gprc saves your readline history between sessions. Setting this default to the empty string `""` changes it to `< undefined >`

The default value is `< undefined >` (no history file).

#### histsize

`gp` keeps a history of the last `histsize` results computed so far, which you can recover using the `%` notation (see Section [Label: 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.

The default value is `5000`.

#### lines

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

The default value is `0`.

#### linewrap

If set to a positive value, `gp` wraps every single line after printing that many characters.

The default value is `0` (unset).

#### log(x)

Principal branch of the natural logarithm of x belongs to C^*, i.e. such that {Im(log}(x)) belongs to ]-Pi,Pi]. The branch cut lies along the negative real axis, continuous with quadrant 2, i.e. such that lim_{b\to 0^+} log (a+bi) = log a for a belongs to R^*. The result is complex (with imaginary part equal to Pi) if x belongs to R and x < 0. In general, the algorithm uses the formula log(x) ~ (Pi)/(2{agm}(1, 4/s)) - m log 2, if s = x 2^m is large enough. (The result is exact to B bits provided s > 2^{B/2}.) At low accuracies, the series expansion near 1 is used.

p-adic arguments are also accepted for x, with the convention that log(p) = 0. Hence in particular exp(log(x))/x is not in general equal to 1 but to a (p-1)-th root of unity (or ±1 if p = 2) times a power of p.

The library syntax is `GEN glog(GEN x, long prec)`. For a `t_PADIC` x, the function `GEN Qp_log(GEN x)` is also available.

#### logfile

Name of the log file to be used when the `log` toggle is on. Environment and time expansion are performed.

The default value is `"pari.log"`.

#### nbthreads

Number of threads to use for parallel computing. The exact meaning an default depend on the `mt` engine used:

* `single`: not used (always one thread).

* `pthread`: number of threads (unlimited, default: number of core)

* `mpi`: number of MPI process to use (limited to the number allocated by `mpirun`, default: use all allocated process).

#### new_galois_format

This toggle is either 1 (on) or 0 (off). If on, the `polgalois` command will use a different, more consistent, naming scheme for Galois groups. This default is provided to ensure that scripts can control this behavior and do not break unexpectedly.

The defaut value is `0`. This value will change to 1 (set) in the next major version.

#### output

There are three possible values: 0 ( = raw), 1 ( = prettymatrix), or 3 ( = external prettyprint). This means that, independently of the default `format` for reals which we explained above, you can print results in three ways:

* 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.

* prettymatrix format: this is identical to raw format, except 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.

* external prettyprint: pipes all `gp` output in TeX format to an external prettyprinter, according to the value of `prettyprinter`. The default script (`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 `\a` and `\m` and `\B` respectively.

The default value is `1` (prettymatrix).

#### parisize

`gp`, and in fact any program using the PARI library, needs a stack in which to do its computations. `parisize` is the stack size, in bytes. It is strongly recommended you increase this default (using the `-s` command-line switch, or a `gprc`) if you can afford it. Don't increase it beyond the actual amount of RAM installed on your computer or `gp` will spend most of its time paging.

In case of emergency, you can use the `allocatemem` function to increase `parisize`, once the session is started.

The default value is 4M, resp. 8M on a 32-bit, resp. 64-bit machine.

#### path

This is a list of directories, separated by colons ':' (semicolons ';' in the DOS world, since colons are preempted for drive names). When asked to read a file whose name is not given by an absolute path (does not start with `/`, `./` or `../`), `gp` will look for it in these directories, in the order they were written in `path`. Here, as usual, `.` means the current directory, and `..` its immediate parent. Environment expansion is performed.

The default value is `".:~:~/gp"` on UNIX systems, `".;C:\;C:\GP"` on DOS, OS/2 and Windows, and `"."` otherwise.

#### prettyprinter

The name of an external prettyprinter to use when `output` is 3 (alternate prettyprinter). Note that the default `tex2mail` looks much nicer than the built-in "beautified format" (`output` = 2).

The default value is `"tex2mail -TeX -noindent -ragged -by_par"`.

#### primelimit

`gp` precomputes a list of all primes less than `primelimit` at initialization time, and can build fast sieves on demand to quickly iterate over primes up to the square of `primelimit`. These are used by many arithmetic functions, usually for trial division purposes. The maximal value is 2^{32} - 2049 (resp 2^{64} - 2049) on a 32-bit (resp. 64-bit) machine, but values beyond 10^8, allowing to iterate over primes up to 10^{16}, do not seem useful.

Since almost all arithmetic functions eventually require some table of prime numbers, PARI guarantees that the first 6547 primes, up to and including 65557, are precomputed, even if `primelimit` is 1.

This default is only used on startup: changing it will not recompute a new table.

Deprecated feature. `primelimit` was used in some situations by algebraic number theory functions using the `nf_PARTIALFACT` flag (`nfbasis`, `nfdisc`, `nfinit`,...): this assumes that all primes p > `primelimit` have a certain property (the equation order is p-maximal). This is never done by default, and must be explicitly set by the user of such functions. Nevertheless, these functions now provide a more flexible interface, and their use of the global default `primelimit` is deprecated.

Deprecated feature. `factor(N, 0)` was used to partially factor integers by removing all prime factors `<=` `primelimit`. Don't use this, supply an explicit bound: `factor(N, bound)`, which avoids relying on an unpredictable global variable.

The default value is `500k`.

#### prompt

A string that will be printed as prompt. Note that most usual escape sequences are available there: `\e` for Esc, `\n` for Newline,..., `\\` for `\`. Time expansion is performed.

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

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

S< >Caution: PariEmacs needs to know about the prompt pattern to separate your input from previous `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!). See PariEmacs's documentation.

The default value is `"? "`.

#### prompt_cont

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 `prompt` applies to `prompt_cont` as well.

The defaut value is `""`.

#### psfile

Name of the default file where `gp` is to dump its PostScript drawings (these are appended, so that no previous data are lost). Environment and time expansion are performed.

The default value is `"pari.ps"`.

#### A (too) short introduction to readline

In the following, `C-` stands for "the `Control` key combined with another" and the same for `M-` with the `Meta` key; generally `C-` combinations act on characters, while the `M-` ones operate on words. The `Meta` key might be called `Alt` on some keyboards, will display a black diamond on most others, and can safely be replaced by `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 `C-a`/`C-e` will take you to the start/end of the line, `M-b`/`M-f` move the cursor backward/forward by a word, etc. Just press the `< Return >` key at any point to send your command to `gp`.

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

Keeping the `M-` key down while you enter an integer (a minus sign meaning reverse behavior) gives an argument to your next readline command (for instance `M-- C-k` will kill text back to the start of line). If you prefer Vi--style editing, `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 `.inputrc` in your home directory. For instance (notice the embedding conditional in case you would want specific bindings for `gp`):

      $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"       #  can be annoying when copy-pasting!
        [: "\C-v[]\C-b"
      $endif

`C-x C-r` will re-read this init file, incorporating any changes made to it during the current session.

Note. By default, `(` and `[` are bound to the function `pari-matched-insert` which, if "electric parentheses" are enabled (default: off) will automatically insert the matching closure (respectively `)` and `]`). This behavior can be toggled on and off by giving the numeric argument -2 to `(` (`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 `M--0` / `M--1` to specifically switch it on or off).

Note. In some versions of readline (2.1 for instance), the `Alt` or `Meta` key can give funny results (output 8-bit accented characters for instance). If you do not want to fall back to the `Esc` combination, put the following two lines in your `.inputrc`:

        set convert-meta on
        set output-meta off

Command completion and online help. Hitting `< TAB >` will complete words for you. This mechanism is context-dependent: `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 `~`, we expect a user name, then a path to some file. Directly after `default(` has been typed, we would expect one of the `default` keywords. After `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, `gp` will provide it together with

* an ending comma if we are completing a default,

* a pair of parentheses if we are completing a function name. In that case hitting `< 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 `C-_`}.

Otherwise, hitting `< 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 `default(seriesprecision,10)`, just by hitting `def < TAB > se < TAB > 10`, which saves 18 keystrokes (out of 27).

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

#### realprecision

The number of significant digits used to convert exact inputs given to transcendental functions (see Section [Label: se:trans]), or to create absolute floating point constants (input as `1.0` or `Pi` for instance). Unless you tamper with the `format` default, this is also the number of significant digits used to print a `t_REAL` number; `format` will override this latter behaviour, and allow you to have a large internal precision while outputting few digits for instance.

Note that PARI's internal precision works on a word basis (by increments of 32 or 64 bits), hence may be a little larger than the number of decimal digits you expected. For instance to get 2 decimal digits you need one word of precision which, on a 64-bit machine, actually gives you 19 digits (19 < log_{10}(2^{64}) < 20). The value returned when typing `default(realprecision)` is the internal number of significant digits, not the number of printed digits:

      ? default(realprecision, 2)
            realprecision = 19 significant digits (2 digits displayed)
      ? default(realprecision)
      %1 = 19

The default value is `38`, resp. `28`, on a 64-bit, resp .32-bit, machine.

#### recover

This toggle is either 1 (on) or 0 (off). If you change this to 0, any error becomes fatal and causes the gp interpreter to exit immediately. Can be useful in batch job scripts.

The default value is `1`.

#### secure

This toggle is either 1 (on) or 0 (off). If on, the `system` and `extern` command are disabled. These two commands are potentially dangerous when you execute foreign scripts since they let `gp` execute arbitrary UNIX commands. `gp` will ask for confirmation before letting you (or a script) unset this toggle.

The default value is `0`.

#### seriesprecision

Number of significant terms when converting a polynomial or rational function to a power series (see `\ps`).

The default value is `16`.

#### simplify(x)

This function simplifies x as much as it can. Specifically, a complex or quadratic number whose imaginary part is the integer 0 (i.e. not `Mod(0,2)` or `0.E-28`) is converted to its real part, and a polynomial of degree 0 is converted to its constant term. Simplifications occur recursively.

This function is especially useful before using arithmetic functions, which expect integer arguments:

      ? x = 2 + y - y
      %1 = 2
      ? isprime(x)
        ***   at top-level: isprime(x)
        ***                 ^----------
        *** isprime: not an integer argument in an arithmetic function
      ? type(x)
      %2 = "t_POL"
      ? type(simplify(x))
      %3 = "t_INT"

Note that GP results are simplified as above before they are stored in the history. (Unless you disable automatic simplification with `\y`, that is.) In particular

      ? type(%1)
      %4 = "t_INT"

The library syntax is `GEN simplify(GEN x)`.

#### sopath

This is a list of directories, separated by colons ':' (semicolons ';' in the DOS world, since colons are preempted for drive names). When asked to `install` an external symbol from a shared library whose name is not given by an absolute path (does not start with `/`, `./` or `../`), `gp` will look for it in these directories, in the order they were written in `sopath`. Here, as usual, `.` means the current directory, and `..` its immediate parent. Environment expansion is performed.

The default value is `""`, corresponding to an empty list of directories: `install` will use the library name as input (and look in the current directory if the name is not an absolute path).

#### strictargs

This toggle is either 1 (on) or 0 (off). If on, all arguments to new user functions are mandatory unless the function supplies an explicit default value. Otherwise arguments have the default value 0.

In this example,

        fun(a,b=2)=a+b

`a` is mandatory, while `b` is optionnal. If `strictargs` is on:

      ? fun()
       ***   at top-level: fun()
       ***                 ^-----
       ***   in function fun: a,b=2
       ***                    ^-----
       ***   missing mandatory argument 'a' in user function.

This applies to functions defined while `strictargs` is on. Changing `strictargs` does not affect the behavior of previously defined functions.

The default value is `0`.

#### strictmatch

This toggle is 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 are unsure how many parentheses you have to close after complicated nested loops. Please do not use this; find a decent text-editor instead.

The default value is `1`.

#### threadsize

In parallel mode, each threads needs its own private stack in which to do its computations, see `parisize`. This value determines the size in bytes of the stacks of each threads, so the total memory allocated will be `parisize`+`nbthreads` x `threadsize`.

If set to 0, the value used is the same as `parisize`.

The default value is 0.

#### timer

This toggle is either 1 (on) or 0 (off). Every instruction sequence in the gp calculator (anything ended by a newline in your input) is timed, to some accuracy depending on the hardware and operating system. When `timer` is on, each such timing is printed immediately before the output as follows:

      ? factor(2^2^7+1)
      time = 108 ms.     \\ this line omitted if 'timer' is 0
      %1 =
      [     59649589127497217 1]

      [5704689200685129054721 1]

(See also `#` and `##`.)

The time measured is the user CPU time, not including the time for printing the results. If the time is negligible ( < 1 ms.), nothing is printed: in particular, no timing should be printed when defining a user function or an alias, or installing a symbol from the library.

The default value is `0` (off).