Math::Pari - Perl interface to PARI.
use Math::Pari; $a = PARI 2; print $a**10000;
or
use Math::Pari qw(Mod); $a = Mod(3,5); print $a**10000;
This package is a Perl interface to famous library PARI for numerical/scientific/number-theoretic calculations. It allows use of most PARI functions as Perl functions, and (almost) seamless merging of PARI and Perl data. In what follows we suppose prior knowledge of what PARI is (see ftp://megrez.math.u-bordeaux.fr/pub/pari, or Math::libPARI).
By default the package exports functions PARI(), PARIcol(), PARIvar(), PARImat() and PARImat_tr() which convert their argument(s) to a PARI object. (In fact PARI() is just an alias for new Math::Pari
). The function PARI() accepts following data as its arguments
Is converted to a PARI integer.
Is converted to a PARI float.
Is executed as a PARI expression (so should not contain whitespace).
Is passed unchanged.
Each element is converted using the same rules, PARI vector-row with these elements is returned.
The same as with a reference to array.
In deciding what rule of the above to apply the preference is given to the uppermost choice of those available now. If none matches, then the string rule is used. So PARI(1)
returns integer, PARI(1.)
returns float, PARI("1")
evaluates 1
as a PARI expression (well, the result is the same as PARI(1)
, only slower).
Note that for Perl these data are synonymous, since Perl freely converts between integers, float and strings. However, to PARI() only what the argument is now is important. If $v is 1
in the Perl world, PARI($v)
may convert it to an integer, float, or to the result of evaluating the PARI program 1
(all depending on how $v was created and accessed in Perl).
This is a fundamental limitation of creating an interface between two systems, both with polymorphic objects, but with subtly different semantic of the flavors of these objects. In reality, however, this is rarely a problem.
PARIcol() behaves in the same way as PARI() unless given several arguments. In the latter case it returns a vector-column instead of a vector-row.
PARImat() constructs a matrix out of the given arguments. It will work if PARI() will construct a vector of vectors given the same arguments. The internal vectors become columns of the matrix. PARImat_tr() behaves similarly, but the internal vectors become rows of the matrix.
Since PARI matrices are similar to vector-rows of vector-columns, PARImat() is quicker, but PARImat_tr() better corresponds to the PARI input and output forms of matrices:
print PARImat [[1,2], [3,4]]; # prints [1,3;2,4] print PARImat_tr [[1,2], [3,4]]; # prints [1,2;3,4]
use
with argumentsIf arguments are specified in the use Math::Pari
directive, the PARI functions appearing as arguments are exported in the caller context. In this case the function PARI() and friends is not exported, so if you need them, you should include them into export list explicitly, or include :DEFAULT
tag:
use Math::Pari qw(factorint PARI); use Math::Pari qw(:DEFAULT factorint);
or simply do it in two steps
use Math::Pari; use Math::Pari 'factorint';
The other tags recognized are :PARI
, :all
, prec=NUMBER
, number tags (e.g., :4
), overloaded constants tags (:int
, :float
, :hex
) and section names tags. The number tags export functions from the PARI library from the given class (except for :PARI
, which exports all of the classes). Tag :all
exports all of the exportable symbols and :PARI
.
Giving ?
command to gp
(PARI calculator) lists the following classes:
1: Standard monadic or dyadic OPERATORS 2: CONVERSIONS and similar elementary functions 3: TRANSCENDENTAL functions 4: NUMBER THEORETICAL functions 5: Functions related to ELLIPTIC CURVES 6: Functions related to general NUMBER FIELDS 7: POLYNOMIALS and power series 8: Vectors, matrices, LINEAR ALGEBRA and sets 9: SUMS, products, integrals and similar functions 10: GRAPHIC functions 11: PROGRAMMING under GP
One can use section names instead of number tags. Recognized names are
:standard :conversions :transcendental :number :elliptic :fields :polynomials :vectors :sums :graphic :programming
One can get the list of all of the functions accessible by Math::Pari
, or the accessible functions from the given section using listPari() function.
Starting from version 5.005 of Perl, three constant-overload tags are supported: :int
, :float
, :hex
. If used, all the integer/float/hex-or-octal-or-binary literals in Perl will be automatically converted to became PARI objects. For example,
use Math::Pari ':int'; print 2**1000;
is equivalent to
print PARI(2)**PARI(1000);
(The support for this Perl feature is buggy before the Perl version 5.005_57 - unless Perl uses mymalloc options; you can check for this with perl -V:usemymalloc
.) Note also that (at least with some versions of Perl) one should enable ':float'
for conversion of long integer literals (Perl may consider them as floats, since they won't fit into Perl integers); note that it is PARI which determines which PARI subtype is assigned to each such literal:
use Math::Pari ':float', 'type_name'; print type_name 22222222222222222222222;
prints t_INT
.
This package supports all the functions from the PARI library with a signature which can be recognized by Math::Pari. This means that when you update the PARI library, the newly added functions will we available without any change to this package; only a recompile is needed. In fact no recompile will be needed if you link libPARI dynamically (you need to modify the Makefile manually to do this).
You can "reach" unsupported functions via going directly to PARI parser using the string flavor of PARI() function, as in
3 + PARI('O(x^17)');
For some "unreachable" functions there is a special wrapper functions, such as O(variable,power)
).
The following functions are specific to GP calculator, thus are not available to Math::Pari in any way:
default error extern input print print1 printp printp1 printtex quit read system whatnow write write1 writetex
whatnow() function is useless, since Math::Pari does not support the "compatibility" mode (with older PARI library). The functionality of print(), write() and variants is available via automatic string translation, and pari_print() function and its variants (see "Printout functions").
default() is the only important function with functionality not supported by the current interface. Note however, that four most important default() actions are supported by allocatemem(), setprimelimit(), setprecision() and setseriesprecision() functions. (When called without arguments, these functions return the current values.)
allocatemem($bytes) should not be called from inside Math::Pari functions (such as forprimes()).
Arguments to PARI functions are automatically converted to long
or a PARI object depending on the signature of the actual library function. The arguments are forced into the given type, so even if gp
rejects your code similar to
func(2.5); # func() takes a long in C
arguing that a particular argument should be of type T_INT
(i.e., a Pari integer), the corresponding code will work in Math::Pari
, since 2.5 is silently converted to long
, per the function signature.
PARI functions return a PARI object or a Perl's integer depending on what the actual library function returns.
Some PARI functions are available in gp
(i.e., in PARI
calculator) via infix notation only. In Math::Pari
these functions are available in functional notations too. Some other convenience functions are also made available.
are available under names
gneg, gadd, gsub, gmul, gdiv, gdivent, gmod, gpui, gle, gge, glt, ggt, geq, gne, gegal, gor, gand, gcmp, gcmp0, gcmp1, gcmp_1.
gdivent
means euclidean quotient, gpui
is power, gegal
checks whether two objects are equal, gcmp
is applicable to two real numbers only, gcmp0
, gcmp1
, gcmp_1
compare with 0, 1 and -1 correspondingly (see PARI user manual for details, or Math::libPARI). Note that all these functions are more readily available via operator overloading, so instead of
gadd(gneg($x), $y)
one can write
-$x+$y
(as far as overloading may be triggered, see overload, so we assume that at least one of $x or $y is a PARI object).
pari2iv, pari2nv, pari2num, pari2pv, pari2bool
convert a PARI object to an integer, float, integer/float (whatever is better), string, and a boolean value correspondingly. Most the time you do not need these functions due to automatic conversions.
pari_print, pari_pprint, pari_texprint
perform the same conversions to strings as their PARI counterparts, but do not print the result. The difference of pari_print() with pari2pv() is the number of significant digits they output, and whitespace in the output. pari2pv(), which is intended for "computer-readable strings", outputs as many digits as is supported by the current precision of the number; while pari_print(), which targets human-readable strings, takes into account the currently specified output precision too.
Some mathematical constants appear as function without arguments in PARI. These functions are available in Math::Pari too. If you export them as in
use Math::Pari qw(:DEFAULT Pi I Euler);
they can be used as barewords in your program:
$x = Pi ** Euler;
For convenience of low-level PARI programmers some low-level functions are made available as well (all except type_name() and changevalue() are not exportable):
typ($x) lg($x) lgef($x) lgefint($x) longword($x, $n) type_name($x) changevalue($name,$newvalue)
Here longword($x,$n) returns $n
-th word in the memory representation of $x (including non-code words). type_name() differs from the PARI function type(): type() returns a PARI object, while type_name() returns a Perl string. (PARI objects of string type behave very non-intuitive w.r.t. string comparison functions; remember that they are compared using lex() to the results of evaluation of other argument of comparison!)
The function listPari($number) outputs a list of names of PARI functions in the section $number. Use listPari(-1) to get the list across all of the sections.
O
Since implementing O(7**6)
would be very tedious, we provide a two-argument form O(7,6)
instead (meaning the same as O(7^6)
in PARI). Note that with polynomials there is no problem like this one, both O($x,6)
and O($x**6)
work.
ifact(n)
integer factorial functions, available from gp
as n!
.
PARI has a big collection of functions which loops over some set. Such a function takes two special arguments: loop variable, and the code to execute in the loop.
The code can be either a string (which contains PARI code to execute - thus should not contain whitespace), or a Perl code reference. The loop variable can be a string giving the name of PARI variable (as in
fordiv(28, 'j', 'a=a+j+j^2');
or
$j= 'j'; fordiv(28, $j, 'a=a+j+j^2');
), a PARI monomial (as in
$j = PARI 'j'; fordiv(28, $j, sub { $a += $j + $j**2 });
), or a "delayed Math::Pari variable" (as in
$j = PARIvar 'j'; fordiv(28, $j, 'a=a+j+j^2');
). If none of these applies, as in
my $j; # Have this in a separate statement fordiv(28, $j, sub { $a += $j + $j**2 });
then during the execution of the sub
, Math::Pari would autogenerate a PARI variable, and would put its value in $j; this value of $j is temporary only, the old contents of $j is restored when fordiv() returns.
Note that since you have no control over this name, you will not be able to use this variable from your PARI code; e.g.,
$j = 7.8; fordiv(28, $j, 'a=a+j+j^2');
will not make j
mirror $j (unless you explicitly set up j
to be a no-argument PARI function mirroring $j, see "Accessing Perl functions from PARI code").
Caveats. There are 2 flavors of the "code" arguments (string/sub
), and 4 types of the "variable" arguments (string/monomial/PARIvar
/other). However, not all 8 combinations make sense. As we already explained, an "other" variable cannot work with a "string" code.
Useless musing alert! Do not read the rest of this section! Do not use "string" variables with sub
code, and do not ask why!
Additionally, the following code will not do what you expect
$x = 0; $j = PARI 'j'; fordiv(28, 'j', sub { $x += $j } ); # Use $j as a loop variable!
since the PARI function fordiv
localizes the PARI variable j
inside the loop, but $j will still reference the old value; the old value is a monomial, not the index of the loop (which is an integer each time sub
is called). The simplest workaround is not to use the above syntax (i.e., not mixing literal loop variable with Perl loop code, just using $j as the second argument to fordiv
is enough):
$x = 0; $j = PARI 'j'; fordiv(28, $j, sub { $x += $j } );
Alternately, one can make a delayed variable $j which will always reference the same thing j
references in PARI now by using PARIvar
constructor
$x = 0; $j = PARIvar 'j'; fordiv(28, 'j', sub { $x += $j } );
(This problem is similar to
$ref = \$_; # $$ref is going to be old value even after # localizing $_ in Perl's grep/map
not accessing localized values of $_ in the plain Perl.)
Another possible quirk is that
fordiv(28, my $j, sub { $a += $j + $j**2 });
will not work too - by a different reason. my
declarations change the meaning of $j only after the end of the current statement; thus $j inside sub
will access a different variable $j (typically a non-lexical, global variable $j) than one you declared on this line.
Use the same name inside PARI code:
sub counter { $i += shift; } $i = 145; PARI 'k=5' ; fordiv(28, 'j', 'k=k+counter(j)'); print PARI('k'), "\n";
prints
984
Due to a difference in the semantic of variable-number-of-parameters-functions between PARI and Perl, if the Perl subroutine takes a variable number of arguments (via @
in the prototype or a missing prototype), up to 6 arguments are supported when this function is called from PARI. If called from PARI with fewer arguments, the rest of arguments will be set to be integers PARI 0
.
Note also that no direct import of Perl variables is available yet (but you can write a function wrapper for this):
sub getv () {$v}
There is an unsupported (and undocumented ;-) function for explicitly importing Perl functions into PARI, possibly with a different name, and possibly with explicitly specifying number of arguments.
Functions from PARI library may take as arguments and/or return values the objects of C type GEN
. In Perl these data are encapsulated into special kind of Perl variables: PARI objects. You can check for a variable $obj
to be a PARI object using
ref $obj and $obj->isa('Math::Pari');
Most the time you do not need this due to automatic conversions and overloading.
If very lazy, one can code in Perl the same way one does it in PARI. Variables in PARI are denoted by barewords, as in x
, and in the default configuration (no warnings, no strict) Perl allows the same - up to some extent. Do not do this, since there are many surprising problems.
Some bareletters denote Perl operators, like q
, x
, y
, s
. This can lead to errors in Perl parsing your expression. E.g.,
print sin(tan(t))-tan(sin(t))-asin(atan(t))+atan(asin(t));
may parse OK after use Math::Pari qw(sin tan asin atan)
. Why?
After importing, the word sin
will denote the PARI function sin(), not Perl operator sin(). The difference is subtle: the PARI function implicitly forces its arguments to be converted PARI objects; it gets 't'
as the argument, which is a string, thus is converted to what t
denotes in PARI - a monomial. While the Perl operator sin() grants overloading (i.e., it will call PARI function sin() if the argument is a PARI object), it does not force its argument; given 't'
as argument, it converts it to what sin() understands, a float (producing 0.
), so will give 0.
as the answer.
However
print sin(tan(y))-tan(sin(y))-asin(atan(y))+atan(asin(y));
would not compile. You should avoid lower-case barewords used as PARI variables, e.g., do
$y = PARI 'y'; print sin(tan($y))-tan(sin($y))-asin(atan($y))+atan(asin($y));
to get
-1/18*y^9+26/4725*y^11-41/1296*y^13+328721/16372125*y^15+O(y^16)
(BTW, it is a very good exercise to get the leading term by hand).
Well, the same advice again: do not use barewords anywhere in your program!
Whenever an arithmetic operation includes at least one PARI object, the other arguments are converted to a PARI object and the corresponding PARI library functions is used to implement the operation. Currently the following arithmetic operations are overloaded:
unary - + - * / % ** abs cos sin exp log sqrt << >> <= == => < > != <=> le eq ge lt gt ne cmp | & ^ ~
Numeric comparison operations are converted to gcmp
and friends, string comparisons compare in lexicographical order using lex
.
Additionally, whenever a PARI object appears in a situation that requires integer, numeric, boolean or string data, it is converted to the corresponding type. Boolean conversion is subject to usual PARI pitfalls related to imprecise zeros (see documentation of gcmp0
in PARI reference).
For details on overloading, see overload.
Note that a check for equality is subject to same pitfalls as in PARI due to imprecise values. PARI may also refuse to compare data of different types for equality if it thinks this may lead to counterintuitive results.
Note also that in PARI the numeric ordering is not defined for some types of PARI objects. For string comparison operations we use PARI-lexicographical ordering.
In the versions of perl earlier than 5.003 overloading used a different interface, so you may need to convert use overload
line to %OVERLOAD
, or, better, upgrade.
Starting from version 2.0, this module comes without a PARI library included.
For the source of PARI library see ftp://megrez.math.u-bordeaux.fr/pub/pari.
Note that the PARI notations should be used in the string arguments to PARI() function, while the Perl notations should be used otherwise.
^
Power is denoted by **
in Perl.
\
and \/
There are no such operators in Perl, use the word forms gdivent(x,y)
and gdivround(x,y)
instead.
~
There is no postfix ~
Perl operator. Use mattranspose() instead.
'
There is no postfix '
Perl operator. Use deriv() instead.
!
There is no postfix !
Perl operator. Use factorial()/ifact() instead (returning a real or an integer correspondingly).
Perl converts big literal integers to doubles if they could not be put into C integers (the particular flavor can be found in the output of perl -V
in newer version of Perl, look for ivtype
/ivsize
). If you want to input such an integer, use
while ($x < PARI('12345678901234567890')) ...
instead of
while ($x < 12345678901234567890) ...
Why? Because conversion to double leads to precision loss (typically above 1e15, see perlnumber), and you will get something like 12345678901234567168 otherwise.
Starting from version 5.005 of Perl, if the tag :int
is used on the 'use Math::Pari' line, all of the integer literals in Perl will be automatically converted to became PARI objects. E.g.,
use Math::Pari ':int'; print 2**1000;
is equivalent to
print PARI(2)**PARI(1000);
Similarly, large integer literals do not lose precision.
This directive is lexically scoped. There is a similar tag :hex
which affects hexadecimal, octal and binary constants. One may also need to use tag :float
for auto-conversion of large integer literals which Perl considers as floating point literals (see "use
with arguments" for details).
Doubles in Perl are typically of precision approximately 15 digits (see perlnumber). When you use them as arguments to PARI functions, they are converted to PARI real variables, and due to intermediate 15-digits-to-binary conversion of Perl variables the result may be different than with the PARI many-digits-to-binary conversion. E.g., PARI(0.01)
and PARI('0.01')
differ at 19-th place, as
setprecision(38); print pari_print(0.01), "\n", pari_print('0.01'), "\n";
shows.
Note that setprecision() changes the output format of pari_print() and friends, as well as the default internal precision. The generic PARI===>string conversion does not take into account the output format, thus
setprecision(38); print PARI(0.01), "\n", PARI('0.01'), "\n", pari_print(0.01), "\n";
will print all the lines with different number of digits after the point: the first one with 22, since the double 0.01 was converted to a low-precision PARI object, the second one with 41, since internal form for precision 38 requires that many digits for representation, and the last one with 39 to have 38 significant digits.
Starting from version 5.005 of Perl, if the tag :float
is used on the use Math::Pari
line, all the float literals in Perl will be automatically converted to became PARI objects. E.g.,
use Math::Pari ':float'; print atan(1.);
is equivalent to
print atan(PARI('1.'));
Similarly, large float literals do not lose precision.
This directive is lexically scoped.
Arrays are 1-based in PARI, are 0-based in Perl. So while array access is possible in Perl, you need to use different indices:
$nf = PARI 'nf'; # assume that PARI variable nf contains a number field $a = PARI('nf[7]'); $b = $nf->[6];
Now $a and $b contain the same value.
Note that PARImat([[...],...,[...])
constructor creates a matrix with specified columns, while in PARI the command [1,2,3;4,5,6]
creates a matrix with specified rows. Use a convenience function PARImat_tr() which will transpose a matrix created by PARImat() to use the same order of elements as in PARI.
Some PARI functions, like length
and eval
, are Perl (semi-)reserved words. To reach these functions, one should either import them:
use Math::Pari qw(length eval);
or call them with prefix (like &length
) or the full name (like Math::Pari::length
).
If you have Term::Gnuplot Perl module installed, you may use high-resolution graphic primitives of PARI. Before the usage you need to establish a link between Math::Pari and Term::Gnuplot by calling link_gnuplot(). You can change the output filehandle by calling set_plot_fh(), and output terminal by calling plotterm(), as in
use Math::Pari qw(:graphic asin); open FH, '>out.tex' or die; link_gnuplot(); # automatically loads Term::Gnuplot set_plot_fh(\*FH); plotterm('emtex'); ploth($x, .5, .999, sub {asin $x}); close FH or die;
libPARI documentation is included, see Math::libPARI. It is converted from Chapter 3 of PARI/GP documentation by the gphelp script of GP/PARI.
No environment variables are used.
-D usemymalloc
), as in:
use Math::Pari ':int'; for ( $i = 0; $i < 10 ; $i++ ) { print "$i\n" }
Workaround: make the modulus live longer than the result of Mod(). Until Perl version 5.6.1
, one should exercise a special care so that the modulus goes out of scope on a different statement than the result:
{ my $modulus = 125; { my $res = Mod(34, $modulus); print $res; } $fake = 1; # A (fake) statement here is required }
Here $res is destructed before the $fake = 1
statement, $modulus is destructed before the first statement after the provided block. However, if you remove the $fake = 1
statement, both these variables are destructed on the first statement after the provided block (and in a wrong order!).
In 5.6.1
declaring $modulus before $res is all that is needed to circumvent the same problem:
{ my $modulus = 125; my $res = Mod(34, $modulus); print $res; } # destruction will happen in a correct order.
Access to array elements may result in similar problems. Hard to fix since in PARI the data is not refcounted.
Currently, PARI assembler files are not position-independent. When compiled for the dynamic linking on legacy systems, this creates a DLL which cannot be shared between processes. Some legacy systems are reported to recognize this situation, and load the DLL as a non-shared module. However, there may be systems (are there?) on which this can cause some "problems".
Summary: if the dynaloading on your system requires some kind of -fPIC
flag, using "assembler" compiles (anything but machine=none
) *may* force you to do a static build (i.e., creation of a custom Perl executable with
perl Makefile.PL static make perl make test_static
).
In older versions of PARI, the one-argument variant of the function isprime() is actually checking for probable primes. Moreover, it has certain problems.
POSSIBLE WORKAROUND (not needed for newer PARI): before version 2.3 of PARI, to get probability of misdetecting a prime below 1e-12, call isprime() twice; below 1e-18, call it 3 times; etc. (The algorithm is probabilistic, and the implementation is such that the result depends on which calls to isprime() were performed ealier.)
The problems: first, while the default algorithm (before version 2.3) gives practically acceptable results in non-adversarial situations, the worst-case behaviour is significantly worse than the average behaviour. The algorithm is looking for so-called "witnesses" (with up to 10 tries) among random integers; usually, witnesses are abundant. However, there are non-prime numbers for which the fraction of witnesses is close to the theoretical mininum, 0.75; with 10 random tries, the probability of missing a witness for such numbers is close to 1e-6. (The known worst-case numbers M have phi(M)/4 non-witnesses, with M=P(2P-1), prime P, 2P-1 and 4|P+1; the proportion of such numbers near K is expected to be const/sqrt(K)log(K)^2. Note that numbers which have more than about 5% non-witnesses may also be candidates for false positives. Conjecturally, they are of the form (aD+1)(bD+1) with a<b, ab <= const, prime aD+1, and bD+1, and D not divisible by high power of 2 (above a=1, b=2 and D is odd); the proportion of such numbers may have a similar asymptotic const/sqrt(K)log(K)^2.)
Second, the random number generator is "reset to known state" when PARI library is initialized. That means that the behaviour is actually predictable if one knows which calls to isprime() are performed; an adversary can find non-primes M which will trigger a false positive exactly on the Nth call to isprime(M) (for particular values of N). With enough computing resources, one can find non-primes M for which N is relatively small (with M about 1e9, one can achieve N as low as 1000). Compare with similar (but less abundant) examples for simpler algorithm, Carmichael numbers; see also numbers with big proportion of non-witnesses and numbers with many non-witnesses, and the conjecture about proportion.
When Math::Pari is loaded, it examines variables $Math::Pari::initmem and $Math::Pari::initprimes. They specify up to which number the initial list of primes should be precalculated, and how large should be the arena for PARI calculations (in bytes). (These values have safe defaults.)
Since setting these values before loading requires either a BEGIN
block, or postponing the loading (use
vs. require
), it may be more convenient to set them via Math::PariInit:
use Math::PariInit qw( primes=12000000 stack=1e8 );
use Math::PariInit
also accepts arbitrary Math::Pari import directives, see Math::PariInit.
These values may be changed at runtime too, via allocatemem() and setprimelimit(), with performance penalties for recalculation/reallocation.
Ilya Zakharevich, ilyaz@cpan.org