1Pari(3) User Contributed Perl Documentation Pari(3)
2
6 Math::Pari - Perl interface to PARI.
7
9 use Math::Pari;
10 $a = PARI 2;
11 print $a**10000;
12 or
14 use Math::Pari qw(Mod);
16 $a = Mod(3,5);
17 print $a**10000;
18
20 This package is a Perl interface to famous library PARI for
21 numerical/scientific/number-theoretic calculations. It allows use of
22 most PARI functions as Perl functions, and (almost) seamless merging of
23 PARI and Perl data. In what follows we suppose prior knowledge of what
24 PARI is (see <ftp://megrez.math.u-bordeaux.fr/pub/pari>, or
25 Math::libPARI).
26
28 DEFAULT
29 By default the package exports functions PARI(), PARIcol(),
30 PARIvar(), PARImat(), PARImat_tr() and parse_as_gp() which convert
31 their argument(s) to a PARI object. (In fact PARI() is just an
32 alias for "new Math::Pari"). The function PARI() accepts following
33 data as its arguments
34 One integer Is converted to a PARI integer.
36 One float Is converted to a PARI float.
38 One string Is executed as a PARI expression (so should not
40 contain whitespace).
41 PARI object Is passed unchanged.
43 Reference to a Perl array
45 Each element is converted using the same rules,
46 PARI vector-row with these elements is returned.
47 Several of above The same as with a reference to array.
49 Conflicts of rules in PARI()
51 In deciding what rule of the above to apply the preference is given
52 to the uppermost choice of those available now. If none matches,
53 then the string rule is used. So PARI(1) returns integer,
54 "PARI(1.)" returns float, "PARI("1")" evaluates 1 as a PARI
55 expression (well, the result is the same as PARI(1), only slower).
56 Note that for Perl these data are synonymous, since Perl freely
58 converts between integers, float and strings. However, to PARI()
59 only what the argument is now is important. If $v is 1 in the Perl
60 world, "PARI($v)" may convert it to an integer, float, or to the
61 result of evaluating the PARI program 1 (all depending on how $v
62 was created and accessed in Perl).
63 This is a fundamental limitation of creating an interface between
65 two systems, both with polymorphic objects, but with subtly
66 different semantic of the flavors of these objects. In reality,
67 however, this is rarely a problem.
68 PARIcol(), PARImat() and PARImat_tr()
70 PARIcol() behaves in the same way as PARI() unless given several
71 arguments. In the latter case it returns a vector-column instead of
72 a vector-row.
73 PARImat() constructs a matrix out of the given arguments. It will
75 work if PARI() will construct a vector of vectors given the same
76 arguments. The internal vectors become columns of the matrix.
77 PARImat_tr() behaves similarly, but the internal vectors become
78 rows of the matrix.
79 Since PARI matrices are similar to vector-rows of vector-columns,
81 PARImat() is quicker, but PARImat_tr() better corresponds to the
82 PARI input and output forms of matrices:
83 print PARImat [[1,2], [3,4]]; # prints [1,3;2,4]
85 print PARImat_tr [[1,2], [3,4]]; # prints [1,2;3,4]
86 parse_as_gp()
88 Did you notice that when taking a string, PARI() requires that
89 there is no whitespace there (outside of string constants)? This
90 is exactly as the "PARI" library parses strings. However, to
91 simplify human interaction, the "gp" calculator allows whitespace,
92 comments, breaking into multiple lines, many independent
93 expressions (such as function definitions).
94 We do not include the corresponding C code from the calculator, but
96 provide a Perl clone. It supports whitespace, "\\"- and "/*
97 */"-comments, and, for multi-line arguments, it supports line
98 continuation via trailing "\", trailing binary ops, comma, opening
99 parenthesis/bracket; moreover, group of lines in "{}" are joined
100 into one line. (Additionally, "\q" and "\p" are recognized, as
101 well as trailing allocatemem(). "\e" is tolerated.)
102 Keep in mind that this is just a convenience function, and no
104 attempt was performed to make it particularly quick. Moreover, the
105 PARI user functions (or maybe it is better to call them user
106 macros?) are currently not automatically importable into Perl, so
107 to access functions defined in parse_as_gp()' argument may be
108 awkward. (The temporary fix is to use a temporary convenience
109 function __wrap_PARI_macro():
110 parse_as_gp <<EOP;
112 add2(x) = x + 2
113 EOP
114 *add2 = Math::Pari::__wrap_PARI_macro 'add2';
115 print add2(17);
116 but keep in mind that the generated this way wrapper is also not
118 designed to be quick.)
119 With "TRUE" optional second argument, it would return an
121 unevaluated array of strings instead of the result of evaluation.
122 Special strings "\q" etc. are replaced by references to appropriate
123 (undocumented) Perl arrays.
124 With "TRUE" optional third argument, would "echo" the commands
126 (preceded by "? ") before execution. With "TRUE" optional fourth
127 argument (command counter), would "echo" the result too (preceded
128 by "%C = ", with "C" being the command counter, which is
129 incremented).
130 "use" with arguments
132 If arguments are specified in the "use Math::Pari" directive, the
133 PARI functions appearing as arguments are exported in the caller
134 context. In this case the function PARI() and friends is not
135 exported, so if you need them, you should include them into export
136 list explicitly, or include ":DEFAULT" tag:
137 use Math::Pari qw(factorint PARI);
139 use Math::Pari qw(:DEFAULT factorint);
140 or simply do it in two steps
142 use Math::Pari;
144 use Math::Pari 'factorint';
145 The other tags recognized are ":PARI", ":all", "prec=NUMBER",
147 number tags (e.g., ":4"), overloaded constants tags (":int",
148 ":float", ":hex") and section names tags. The number tags export
149 functions from the PARI library from the given class (except for
150 ":PARI", which exports all of the classes). Tag ":all" exports all
151 of the exportable symbols and ":PARI".
152 Giving "?" command to "gp" (PARI calculator) lists the following
154 classes (this list depends significantly on the version; so better
155 use symbolic names):
156 1: Standard monadic or dyadic OPERATORS
158 2: CONVERSIONS and similar elementary functions
159 3: TRANSCENDENTAL functions
160 4: NUMBER THEORETICAL functions
161 5: Functions related to ELLIPTIC CURVES
162 6: Functions related to general NUMBER FIELDS
163 7: POLYNOMIALS and power series
164 8: Vectors, matrices, LINEAR ALGEBRA and sets
165 9: SUMS, products, integrals and similar functions
166 10: GRAPHIC functions
167 11: PROGRAMMING under GP
168 One can use section names instead of number tags. Recognized names
170 are
171 :standard :conversions :transcendental :number :elliptic
173 :fields :polynomials :vectors :sums :graphic :programming
174 One can get the list of all of the functions accessible by
176 "Math::Pari", or the accessible functions from the given section
177 using listPari() function.
178 Starting from version 5.005 of Perl, three constant-overload tags
180 are supported: ":int", ":float", ":hex". If used, all the
181 integer/float/hex-or-octal-or-binary literals in Perl will be
182 automatically converted to became PARI objects. For example,
183 use Math::Pari ':int';
185 print 2**1000;
186 is equivalent to
188 print PARI(2)**PARI(1000);
190 (The support for this Perl feature is buggy before the Perl version
192 5.005_57 - unless Perl uses mymalloc options; you can check for
193 this with "perl -V:usemymalloc".) Note also that (at least with
194 some versions of Perl) one should enable ':float' for conversion of
195 long integer literals (Perl may consider them as floats, since they
196 won't fit into Perl integers); note that it is PARI which
197 determines which PARI subtype is assigned to each such literal:
198 use Math::Pari ':float', 'type_name';
200 print type_name 22222222222222222222222;
201 prints "t_INT".
203
205 Directly accessible from Perl
206 This package supports all the functions from the PARI library with a
207 signature which can be recognized by Math::Pari. This means that when
208 you update the PARI library, the newly added functions will we
209 available without any change to this package; only a recompile is
210 needed. In fact no recompile will be needed if you link libPARI
211 dynamically (you need to modify the Makefile manually to do this).
212 You can "reach" unsupported functions via going directly to PARI parser
214 using the string flavor of PARI() function, as in
215 3 + PARI('O(x^17)');
217 For some "unreachable" functions there is a special wrapper functions,
219 such as "O(variable,power)").
220 The following functions are specific to GP calculator, thus are not
222 available to Math::Pari in any way:
223 default error extern input print print1 printp printp1
225 printtex quit read system whatnow write write1 writetex
226 whatnow() function is useless, since Math::Pari does not support the
228 "compatibility" mode (with older PARI library). The functionality of
229 print(), write() and variants is available via automatic string
230 translation, and pari_print() function and its variants (see "Printout
231 functions").
232 default() is the only important function with functionality not
234 supported by the current interface. Note however, that four most
235 important default() actions are supported by allocatemem(),
236 setprimelimit(), setprecision() and setseriesprecision() functions.
237 (When called without arguments, these functions return the current
238 values.)
239 allocatemem($bytes) should not be called from inside Math::Pari
241 functions (such as forprimes()).
242 Arguments
244 Arguments to PARI functions are automatically converted to "long" or a
245 PARI object depending on the signature of the actual library function.
246 The arguments are forced into the given type, so even if "gp" rejects
247 your code similar to
248 func(2.5); # func() takes a long in C
250 arguing that a particular argument should be of "type T_INT" (i.e., a
252 Pari integer), the corresponding code will work in "Math::Pari", since
253 2.5 is silently converted to "long", per the function signature.
254 Return values
256 PARI functions return a PARI object or a Perl's integer depending on
257 what the actual library function returns.
258 Additional functions
260 Some PARI functions are available in "gp" (i.e., in "PARI" calculator)
261 via infix notation only. In "Math::Pari" these functions are available
262 in functional notations too. Some other convenience functions are also
263 made available.
264 Infix, prefix and postfix operations
266 are available under names
267 gneg, gadd, gsub, gmul, gdiv, gdivent, gmod, gpui,
269 gle, gge, glt, ggt, geq, gne, gegal, gor, gand,
270 gcmp, gcmp0, gcmp1, gcmp_1.
271 "gdivent" means euclidean quotient, "gpui" is power, "gegal"
273 checks whether two objects are equal, "gcmp" is applicable to two
274 real numbers only, "gcmp0", "gcmp1", "gcmp_1" compare with 0, 1
275 and -1 correspondingly (see PARI user manual for details, or
276 Math::libPARI). Note that all these functions are more readily
277 available via operator overloading, so instead of
278 gadd(gneg($x), $y)
280 one can write
282 -$x+$y
284 (as far as overloading may be triggered, see overload, so we
286 assume that at least one of $x or $y is a PARI object).
287 Conversion functions
289 pari2iv, pari2nv, pari2num, pari2pv, pari2bool
290 convert a PARI object to an integer, float, integer/float
292 (whatever is better), string, and a boolean value correspondingly.
293 Most the time you do not need these functions due to automatic
294 conversions.
295 Printout functions
297 pari_print, pari_pprint, pari_texprint
298 perform the same conversions to strings as their PARI
300 counterparts, but do not print the result. The difference of
301 pari_print() with pari2pv() is the number of significant digits
302 they output, and whitespace in the output. pari2pv(), which is
303 intended for "computer-readable strings", outputs as many digits
304 as is supported by the current precision of the number; while
305 pari_print(), which targets human-readable strings, takes into
306 account the currently specified output precision too.
307 Constant functions
309 Some mathematical constants appear as function without arguments
310 in PARI. These functions are available in Math::Pari too. If you
311 export them as in
312 use Math::Pari qw(:DEFAULT Pi I Euler);
314 they can be used as barewords in your program:
316 $x = Pi ** Euler;
318 Low-level functions
320 For convenience of low-level PARI programmers some low-level
321 functions are made available as well (all except type_name() and
322 changevalue() are not exportable):
323 typ($x)
325 lg($x)
326 lgef($x)
327 lgefint($x)
328 longword($x, $n)
329 type_name($x)
330 changevalue($name,$newvalue)
331 Here longword($x,$n) returns $n-th word in the memory
333 representation of $x (including non-code words). type_name()
334 differs from the PARI function type(): type() returns a PARI
335 object, while type_name() returns a Perl string. (PARI objects of
336 string type behave very non-intuitive w.r.t. string comparison
337 functions; remember that they are compared using lex() to the
338 results of evaluation of other argument of comparison!)
339 The function listPari($number) outputs a list of names of PARI
341 functions in the section $number. Use listPari(-1) to get the
342 list across all of the sections.
343 Uncompatible functions
345 O
346 Since implementing "O(7**6)" would be very tedious, we provide a
348 two-argument form "O(7,6)" instead (meaning the same as "O(7^6)"
349 in PARI). Note that with polynomials there is no problem like
350 this one, both "O($x,6)" and "O($x**6)" work.
351 ifact(n)
353 integer factorial functions, available from "gp" as "n!".
355 Looping functions
357 PARI has a big collection of functions which loops over some set. Such
358 a function takes two special arguments: loop variable, and the code to
359 execute in the loop.
360 The code can be either a string (which contains PARI code to execute -
362 thus should not contain whitespace), or a Perl code reference. The
363 loop variable can be a string giving the name of PARI variable (as in
364 fordiv(28, 'j', 'a=a+j+j^2');
366 or
368 $j= 'j';
370 fordiv(28, $j, 'a=a+j+j^2');
371 ), a PARI monomial (as in
373 $j = PARI 'j';
375 fordiv(28, $j, sub { $a += $j + $j**2 });
376 ), or a "delayed Math::Pari variable" (as in
378 $j = PARIvar 'j';
380 fordiv(28, $j, 'a=a+j+j^2');
381 ). If none of these applies, as in
383 my $j; # Have this in a separate statement
385 fordiv(28, $j, sub { $a += $j + $j**2 });
386 then during the execution of the "sub", Math::Pari would autogenerate a
388 PARI variable, and would put its value in $j; this value of $j is
389 temporary only, the old contents of $j is restored when fordiv()
390 returns.
391 Note that since you have no control over this name, you will not be
393 able to use this variable from your PARI code; e.g.,
394 $j = 7.8;
396 fordiv(28, $j, 'a=a+j+j^2');
397 will not make "j" mirror $j (unless you explicitly set up "j" to be a
399 no-argument PARI function mirroring $j, see "Accessing Perl functions
400 from PARI code").
401 Caveats. There are 2 flavors of the "code" arguments (string/"sub"),
403 and 4 types of the "variable" arguments
404 (string/monomial/"PARIvar"/other). However, not all 8 combinations
405 make sense. As we already explained, an "other" variable cannot work
406 with a "string" code.
407 Useless musing alert! Do not read the rest of this section! Do not use
409 "string" variables with "sub" code, and do not ask why!
410 Additionally, the following code will not do what you expect
412 $x = 0;
414 $j = PARI 'j';
415 fordiv(28, 'j', sub { $x += $j } ); # Use $j as a loop variable!
416 since the PARI function "fordiv" localizes the PARI variable "j" inside
418 the loop, but $j will still reference the old value; the old value is a
419 monomial, not the index of the loop (which is an integer each time
420 "sub" is called). The simplest workaround is not to use the above
421 syntax (i.e., not mixing literal loop variable with Perl loop code,
422 just using $j as the second argument to "fordiv" is enough):
423 $x = 0;
425 $j = PARI 'j';
426 fordiv(28, $j, sub { $x += $j } );
427 Alternately, one can make a delayed variable $j which will always
429 reference the same thing "j" references in PARI now by using "PARIvar"
430 constructor
431 $x = 0;
433 $j = PARIvar 'j';
434 fordiv(28, 'j', sub { $x += $j } );
435 (This problem is similar to
437 $ref = \$_; # $$ref is going to be old value even after
439 # localizing $_ in Perl's grep/map
440 not accessing localized values of $_ in the plain Perl.)
442 Another possible quirk is that
444 fordiv(28, my $j, sub { $a += $j + $j**2 });
446 will not work too - by a different reason. "my" declarations change
448 the meaning of $j only after the end of the current statement; thus $j
449 inside "sub" will access a different variable $j (typically a non-
450 lexical, global variable $j) than one you declared on this line.
451 Accessing Perl functions from PARI code
453 Use the same name inside PARI code:
454 sub counter { $i += shift; }
456 $i = 145;
457 PARI 'k=5' ;
458 fordiv(28, 'j', 'k=k+counter(j)');
459 print PARI('k'), "\n";
460 prints
462 984
464 Due to a difference in the semantic of variable-number-of-parameters-
466 functions between PARI and Perl, if the Perl subroutine takes a
467 variable number of arguments (via "@" in the prototype or a missing
468 prototype), up to 6 arguments are supported when this function is
469 called from PARI. If called from PARI with fewer arguments, the rest
470 of arguments will be set to be integers "PARI 0".
471 Note also that no direct import of Perl variables is available yet (but
473 you can write a function wrapper for this):
474 sub getv () {$v}
476 There is an unsupported (and undocumented ;-) function for explicitly
478 importing Perl functions into PARI, possibly with a different name, and
479 possibly with explicitly specifying number of arguments.
480
482 Functions from PARI library may take as arguments and/or return values
483 the objects of C type "GEN". In Perl these data are encapsulated into
484 special kind of Perl variables: PARI objects. You can check for a
485 variable $obj to be a PARI object using
486 ref $obj and $obj->isa('Math::Pari');
488 Most the time you do not need this due to automatic conversions and
490 overloading.
491
493 If very lazy, one can code in Perl the same way one does it in PARI.
494 Variables in PARI are denoted by barewords, as in "x", and in the
495 default configuration (no warnings, no strict) Perl allows the same -
496 up to some extent. Do not do this, since there are many surprising
497 problems.
498 Some bareletters denote Perl operators, like "q", "x", "y", "s". This
500 can lead to errors in Perl parsing your expression. E.g.,
501 print sin(tan(t))-tan(sin(t))-asin(atan(t))+atan(asin(t));
503 may parse OK after "use Math::Pari qw(sin tan asin atan)". Why?
505 After importing, the word "sin" will denote the PARI function sin(),
507 not Perl operator sin(). The difference is subtle: the PARI function
508 implicitly forces its arguments to be converted PARI objects; it gets
509 't' as the argument, which is a string, thus is converted to what "t"
510 denotes in PARI - a monomial. While the Perl operator sin() grants
511 overloading (i.e., it will call PARI function sin() if the argument is
512 a PARI object), it does not force its argument; given 't' as argument,
513 it converts it to what sin() understands, a float (producing 0.), so
514 will give 0. as the answer.
515 However
517 print sin(tan(y))-tan(sin(y))-asin(atan(y))+atan(asin(y));
519 would not compile. You should avoid lower-case barewords used as PARI
521 variables, e.g., do
522 $y = PARI 'y';
524 print sin(tan($y))-tan(sin($y))-asin(atan($y))+atan(asin($y));
525 to get
527 -1/18*y^9+26/4725*y^11-41/1296*y^13+328721/16372125*y^15+O(y^16)
529 (BTW, it is a very good exercise to get the leading term by hand).
531 Well, the same advice again: do not use barewords anywhere in your
533 program!
534
536 Whenever an arithmetic operation includes at least one PARI object, the
537 other arguments are converted to a PARI object and the corresponding
538 PARI library functions is used to implement the operation. Currently
539 the following arithmetic operations are overloaded:
540 unary -
542 + - * / % ** abs cos sin exp log sqrt
543 << >>
544 <= == => < > != <=>
545 le eq ge lt gt ne cmp
546 | & ^ ~
547 Numeric comparison operations are converted to "gcmp" and friends,
549 string comparisons compare in lexicographical order using "lex".
550 Additionally, whenever a PARI object appears in a situation that
552 requires integer, numeric, boolean or string data, it is converted to
553 the corresponding type. Boolean conversion is subject to usual PARI
554 pitfalls related to imprecise zeros (see documentation of "gcmp0" in
555 PARI reference).
556 For details on overloading, see overload.
558 Note that a check for equality is subject to same pitfalls as in PARI
560 due to imprecise values. PARI may also refuse to compare data of
561 different types for equality if it thinks this may lead to
562 counterintuitive results.
563 Note also that in PARI the numeric ordering is not defined for some
565 types of PARI objects. For string comparison operations we use PARI-
566 lexicographical ordering.
567
569 Perl
570 In the versions of perl earlier than 5.003 overloading used a different
571 interface, so you may need to convert "use overload" line to %OVERLOAD,
572 or, better, upgrade.
573 PARI
575 Starting from version 2.0, this module comes without a PARI library
576 included.
577 For the source of PARI library see
579 <ftp://megrez.math.u-bordeaux.fr/pub/pari>.
580
582 Note that the PARI notations should be used in the string arguments to
583 PARI() function, while the Perl notations should be used otherwise.
584 "^" Power is denoted by "**" in Perl.
586 "\" and "\/"
588 There are no such operators in Perl, use the word forms
589 "gdivent(x,y)" and "gdivround(x,y)" instead.
590 "~" There is no postfix "~" Perl operator. Use mattranspose() instead.
592 "'" There is no postfix "'" Perl operator. Use deriv() instead.
594 "!" There is no postfix "!" Perl operator. Use factorial()/ifact()
596 instead (returning a real or an integer correspondingly).
597 big integers
599 Perl converts big literal integers to doubles if they could not be
600 put into C integers (the particular flavor can be found in the
601 output of "perl -V" in newer version of Perl, look for
602 "ivtype"/"ivsize"). If you want to input such an integer, use
603 while ($x < PARI('12345678901234567890')) ...
605 instead of
607 while ($x < 12345678901234567890) ...
609 Why? Because conversion to double leads to precision loss
611 (typically above 1e15, see perlnumber), and you will get something
612 like 12345678901234567168 otherwise.
613 Starting from version 5.005 of Perl, if the tag ":int" is used on
615 the 'use Math::Pari' line, all of the integer literals in Perl will
616 be automatically converted to became PARI objects. E.g.,
617 use Math::Pari ':int';
619 print 2**1000;
620 is equivalent to
622 print PARI(2)**PARI(1000);
624 Similarly, large integer literals do not lose precision.
626 This directive is lexically scoped. There is a similar tag ":hex"
628 which affects hexadecimal, octal and binary constants. One may
629 also need to use tag ":float" for auto-conversion of large integer
630 literals which Perl considers as floating point literals (see
631 ""use" with arguments" for details).
632 doubles
634 Doubles in Perl are typically of precision approximately 15 digits
635 (see perlnumber). When you use them as arguments to PARI
636 functions, they are converted to PARI real variables, and due to
637 intermediate 15-digits-to-binary conversion of Perl variables the
638 result may be different than with the PARI many-digits-to-binary
639 conversion. E.g., "PARI(0.01)" and "PARI('0.01')" differ at 19-th
640 place, as
641 setprecision(38);
643 print pari_print(0.01), "\n",
644 pari_print('0.01'), "\n";
645 shows.
647 Note that setprecision() changes the output format of pari_print()
649 and friends, as well as the default internal precision. The
650 generic PARI===>string conversion does not take into account the
651 output format, thus
652 setprecision(38);
654 print PARI(0.01), "\n",
655 PARI('0.01'), "\n",
656 pari_print(0.01), "\n";
657 will print all the lines with different number of digits after the
659 point: the first one with 22, since the double 0.01 was converted
660 to a low-precision PARI object, the second one with 41, since
661 internal form for precision 38 requires that many digits for
662 representation, and the last one with 39 to have 38 significant
663 digits.
664 Starting from version 5.005 of Perl, if the tag ":float" is used on
666 the "use Math::Pari" line, all the float literals in Perl will be
667 automatically converted to became PARI objects. E.g.,
668 use Math::Pari ':float';
670 print atan(1.);
671 is equivalent to
673 print atan(PARI('1.'));
675 Similarly, large float literals do not lose precision.
677 This directive is lexically scoped.
679 array base
681 Arrays are 1-based in PARI, are 0-based in Perl. So while array
682 access is possible in Perl, you need to use different indices:
683 $nf = PARI 'nf'; # assume that PARI variable nf contains a number field
685 $a = PARI('nf[7]');
686 $b = $nf->[6];
687 Now $a and $b contain the same value.
689 matrices
691 Note that "PARImat([[...],...,[...])" constructor creates a matrix
692 with specified columns, while in PARI the command "[1,2,3;4,5,6]"
693 creates a matrix with specified rows. Use a convenience function
694 PARImat_tr() which will transpose a matrix created by PARImat() to
695 use the same order of elements as in PARI.
696 builtin perl functions
698 Some PARI functions, like "length" and "eval", are Perl
699 (semi-)reserved words. To reach these functions, one should either
700 import them:
701 use Math::Pari qw(length eval);
703 or call them with prefix (like &length) or the full name (like
705 "Math::Pari::length").
706
708 If you have Term::Gnuplot Perl module installed, you may use high-
709 resolution graphic primitives of PARI. Before the usage you need to
710 establish a link between Math::Pari and Term::Gnuplot by calling
711 link_gnuplot(). You can change the output filehandle by calling
712 set_plot_fh(), and output terminal by calling plotterm(), as in
713 use Math::Pari qw(:graphic asin);
715 link_gnuplot(); # automatically loads Term::Gnuplot
717 plotterm('emtex');
718 plot_outfile_set('out.tex'); # better do after plotterm()
719 ploth($x, .5, .999, sub {asin $x});
720 close FH or die;
721
723 libPARI documentation is included, see Math::libPARI. It is converted
724 from Chapter 3 of PARI/GP documentation by the gphelp script of
725 GP/PARI.
726
728 No environment variables are used.
729
731 · A few of PARI functions are available indirectly only.
732 · Using overloading constants with the Perl versions below 5.005_57
734 could lead to segfaults (at least without "-D usemymalloc"), as
735 in:
736 use Math::Pari ':int';
738 for ( $i = 0; $i < 10 ; $i++ ) { print "$i\n" }
739 · It may be possible that conversion of a Perl value which has both
741 the integer slot and the floating slot set may create a PARI
742 integer, even if the actual value is not an integer.
743 · problems with refcounting of array elements and Mod().
745 Workaround: make the modulus live longer than the result of Mod().
747 Until Perl version 5.6.1, one should exercise a special care so
748 that the modulus goes out of scope on a different statement than
749 the result:
750 { my $modulus = 125;
752 { my $res = Mod(34, $modulus);
753 print $res;
754 }
755 $fake = 1; # A (fake) statement here is required
756 }
757 Here $res is destructed before the "$fake = 1" statement, $modulus
759 is destructed before the first statement after the provided block.
760 However, if you remove the "$fake = 1" statement, both these
761 variables are destructed on the first statement after the provided
762 block (and in a wrong order!).
763 In 5.6.1 declaring $modulus before $res is all that is needed to
765 circumvent the same problem:
766 { my $modulus = 125;
768 my $res = Mod(34, $modulus);
769 print $res;
770 } # destruction will happen in a correct order.
771 Access to array elements may result in similar problems. Hard to
773 fix since in PARI the data is not refcounted.
774 · Legacy implementations of dynalinking require the code of DLL to
776 be compiled to be "position independent" code (PIC). This slows
777 down the execution, while allowing sharing the loaded copy of the
778 DLL between different processes. [On contemporary architectures
779 the same effect is allowed without the position-independent hack.]
780 Currently, PARI assembler files are not position-independent.
782 When compiled for the dynamic linking on legacy systems, this
783 creates a DLL which cannot be shared between processes. Some
784 legacy systems are reported to recognize this situation, and load
785 the DLL as a non-shared module. However, there may be systems
786 (are there?) on which this can cause some "problems".
787 Summary: if the dynaloading on your system requires some kind of
789 "-fPIC" flag, using "assembler" compiles (anything but
790 "machine=none") *may* force you to do a static build (i.e.,
791 creation of a custom Perl executable with
792 perl Makefile.PL static
794 make perl
795 make test_static
796 ).
798 · isprime() is a misnomer before PARI version 2.3!
800 In older versions of PARI, the one-argument variant of the
802 function isprime() is actually checking for probable primes.
803 Moreover, it has certain problems.
804 POSSIBLE WORKAROUND (not needed for newer PARI): before version
806 2.3 of PARI, to get probability of misdetecting a prime below
807 1e-12, call isprime() twice; below 1e-18, call it 3 times; etc.
808 (The algorithm is probabilistic, and the implementation is such
809 that the result depends on which calls to isprime() were performed
810 ealier.)
811 The problems: first, while the default algorithm (before version
813 2.3) gives practically acceptable results in non-adversarial
814 situations, the worst-case behaviour is significantly worse than
815 the average behaviour. The algorithm is looking for so-called
816 "witnesses" (with up to 10 tries) among random integers; usually,
817 witnesses are abundant. However, there are non-prime numbers for
818 which the fraction of witnesses is close to the theoretical
819 minimum, 0.75; with 10 random tries, the probability of missing a
820 witness for such numbers is close to 1e-6. (The known worst-case
821 numbers M have phi(M)/4 non-witnesses, with M=P(2P-1), prime P,
822 2P-1 and 4|P+1; the proportion of such numbers near K is expected
823 to be const/sqrt(K)log(K)^2. Note that numbers which have more
824 than about 5% non-witnesses may also be candidates for false
825 positives. Conjecturally, they are of the form (aD+1)(bD+1) with
826 a<b, ab <= const, prime aD+1, and bD+1, and D not divisible by
827 high power of 2 (above a=1, b=2 and D is odd); the proportion of
828 such numbers may have a similar asymptotic const/sqrt(K)log(K)^2.)
829 Second, the random number generator is "reset to known state" when
831 PARI library is initialized. That means that the behaviour is
832 actually predictable if one knows which calls to isprime() are
833 performed; an adversary can find non-primes M which will trigger a
834 false positive exactly on the Nth call to isprime(M) (for
835 particular values of N). With enough computing resources, one can
836 find non-primes M for which N is relatively small (with M about
837 1e9, one can achieve N as low as 1000). Compare with similar (but
838 less abundant) examples for simpler algorithm, Carmichael numbers
839 <http://en.wikipedia.org/wiki/Carmichael_numbers>; see also
840 numbers with big proportion of non-witnesses
841 <http://oeis.org/A090659> and numbers with many non-witnesses
842 <http://oeis.org/A141768>, and the conjecture about proportion
843 <http://web.archive.org/web/*/http://www.ma.iup.edu/MAA/proceedings/vol1/higgins.pdf>.
844 See the discussion of isprime()
846 <https://rt.cpan.org/Public/Bug/Display.html?id=93652>.
847
849 When Math::Pari is loaded, it examines variables $Math::Pari::initmem
850 and $Math::Pari::initprimes. They specify up to which number the
851 initial list of primes should be precalculated, and how large should be
852 the arena for PARI calculations (in bytes). (These values have safe
853 defaults.)
854 Since setting these values before loading requires either a "BEGIN"
856 block, or postponing the loading ("use" vs. "require"), it may be more
857 convenient to set them via Math::PariInit:
858 use Math::PariInit qw( primes=12000000 stack=1e8 );
860 "use Math::PariInit" also accepts arbitrary Math::Pari import
862 directives, see Math::PariInit.
863 These values may be changed at runtime too, via allocatemem() and
865 setprimelimit(), with performance penalties for
866 recalculation/reallocation.
867
869 Ilya Zakharevich, ilyaz@cpan.org
870perl v5.30.1 2020-01-30 Pari(3)