/ 
Math-1.33
/ Math::BigInt

NAME

Math::BigInt - Arbitrary size integer math package

SYNOPSIS

  use Math::BigInt;

  # Number creation	
  $x = Math::BigInt->new($str);	# defaults to 0
  $nan  = Math::BigInt->bnan(); # create a NotANumber
  $zero = Math::BigInt->bzero();# create a "+0"

  # Testing
  $x->is_zero();		# return whether arg is zero or not
  $x->is_nan();			# return whether arg is NaN or not
  $x->is_one();			# return true if arg is +1
  $x->is_one('-');		# return true if arg is -1
  $x->is_odd();			# return true if odd, false for even
  $x->is_even();		# return true if even, false for odd
  $x->bcmp($y);			# compare numbers (undef,<0,=0,>0)
  $x->bacmp($y);		# compare absolutely (undef,<0,=0,>0)
  $x->sign();			# return the sign, either +,- or NaN
  $x->digit($n);		# return the nth digit, counting from right
  $x->digit(-$n);		# return the nth digit, counting from left

  # The following all modify their first argument:

  # set 
  $x->bzero();			# set $x to 0
  $x->bnan();			# set $x to NaN

  $x->bneg();			# negation
  $x->babs();			# absolute value
  $x->bnorm();			# normalize (no-op)
  $x->bnot();			# two's complement (bit wise not)
  $x->binc();			# increment x by 1
  $x->bdec();			# decrement x by 1
  
  $x->badd($y);			# addition (add $y to $x)
  $x->bsub($y);			# subtraction (subtract $y from $x)
  $x->bmul($y);			# multiplication (multiply $x by $y)
  $x->bdiv($y);			# divide, set $x to quotient
				# return (quo,rem) or quo if scalar

  $x->bmod($y);			# modulus (x % y)
  $x->bpow($y);			# power of arguments (x ** y)
  $x->blsft($y);		# left shift
  $x->brsft($y);		# right shift 
  $x->blsft($y,$n);		# left shift, by base $n (like 10)
  $x->brsft($y,$n);		# right shift, by base $n (like 10)
  
  $x->band($y);			# bitwise and
  $x->bior($y);			# bitwise inclusive or
  $x->bxor($y);			# bitwise exclusive or
  $x->bnot();			# bitwise not (two's complement)

  $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
  $x->bround($N);               # accuracy: preserve $N digits
  $x->bfround($N);              # round to $Nth digit, no-op for BigInts

  $x->bsqrt();			# calculate square-root
  
  # The following do not modify their arguments:

  bgcd(@values);		# greatest common divisor
  blcm(@values);		# lowest common multiplicator
  
  $x->bstr();			# normalized string
  $x->bsstr();			# normalized string in scientific notation
  $x->length();			# return number of digits in number
  ($x,$f) = $x->length();	# length of number and length of fraction part

  $x->exponent();               # return exponent as BigInt
  $x->mantissa();               # return mantissa as BigInt
  $x->parts();                  # return (mantissa,exponent) as BigInt

DESCRIPTION

All operators (inlcuding basic math operations) are overloaded if you declare your big integers as

$i = new Math::BigInt '123_456_789_123_456_789';

Operations with overloaded operators preserve the arguments which is exactly what you expect.

Canonical notation

Big integer values are strings of the form /^[+-]\d+$/ with leading zeros suppressed.

'-0'                            canonical value '-0', normalized '0'
'   -123_123_123'               canonical value '-123123123'
'1_23_456_7890'                 canonical value '1234567890'
Input

Input values to these routines may be either Math::BigInt objects or strings of the form /^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/.

You can include one underscore between any two digits.

This means integer values like 1.01E2 or even 1000E-2 are also accepted. Non integer values result in NaN.

Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results in 'NaN'.

bnorm() on a BigInt object is now effectively a no-op, since the numbers are always stored in normalized form. On a string, it creates a BigInt object.

Output

Output values are BigInt objects (normalized), except for bstr(), which returns a string in normalized form. Some routines (is_odd(), is_even(), is_zero(), is_one(), is_nan()) return true or false, while others (bcmp(), bacmp()) return either undef, <0, 0 or >0 and are suited for sort.

Rounding

Only bround() is defined for BigInts, for further details of rounding see Math::BigFloat.

bfround ( +$scale ) rounds to the $scale'th place left from the '.'
bround ( +$scale ) preserves accuracy to $scale sighnificant digits counted from the left and paddes the number with zeros
bround ( -$scale ) preserves accuracy to $scale significant digits counted from the right and paddes the number with zeros.

bfround() does nothing in case of negative $scale. Both bround() and bfround() are a no-ops for a scale of 0.

All rounding functions take as a second parameter a rounding mode from one of the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.

The default is 'even'. By using Math::BigInt->round_mode($rnd_mode); you can get and set the default round mode for subsequent rounding.

The second parameter to the round functions than overrides the default temporarily.

Internals

Actual math is done in an internal format consisting of an array of elements of base 100000 digits with the least significant digit first. The sign /^[+-]$/ is stored separately. The string 'NaN' is used to represent the result when input arguments are not numbers, as well as the result of dividing by zero.

You sould neither care nor depend on the internal represantation, it might change without notice. Use only method calls like $x->sign(); instead relying on the internal hash keys like in $x->{sign};.

mantissa(), exponent() and parts()

mantissa() and exponent() return the said parts of the BigInt such that:

$m = $x->mantissa();
$e = $x->exponent();
$y = $m * ( 10 ** $e );
print "ok\n" if $x == $y;

($m,$e) = $x-parts()> is just a shortcut that gives you both of them in one go. Both the returned mantissa and exponent do have a sign.

Currently, for BigInts $e will be always 0, except for NaN where it will be NaN and for $x == 0, then it will be 1 (to be compatible with Math::BigFlaot's internal representation of a zero as 0E1).

$m will always be a copy of the original number. The relation between $e and $m might change in the future, but will be always equivalent in a numerical sense, e.g. $m might get minized.

EXAMPLES

use Math::BigInt qw(bstr bint);
$x = bstr("1234")                  	# string "1234"
$x = "$x";                         	# same as bstr()
$x = bneg("1234")                  	# Bigint "-1234"
$x = Math::BigInt->bneg("1234");   	# Bigint "-1234"
$x = Math::BigInt->babs("-12345"); 	# Bigint "12345"
$x = Math::BigInt->bnorm("-0 00"); 	# BigInt "0"
$x = bint(1) + bint(2);            	# BigInt "3"
$x = bint(1) + "2";                	# ditto (auto-BigIntify of "2")
$x = bint(1);                      	# BigInt "1"
$x = $x + 5 / 2;                   	# BigInt "3"
$x = $x ** 3;                      	# BigInt "27"
$x *= 2;                           	# BigInt "54"
$x = new Math::BigInt;             	# BigInt "0"
$x--;                              	# BigInt "-1"
$x = Math::BigInt->badd(4,5)		# BigInt "9"
$x = Math::BigInt::badd(4,5)		# BigInt "9"
print $x->bsstr();			# 9e+0

Autocreating constants

After use Math::BigInt ':constant' all the integer decimal constants in the given scope are converted to Math::BigInt. This conversion happens at compile time.

In particular

perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'

prints the integer value of 2**100. Note that without conversion of constants the expression 2**100 will be calculated as floating point number.

Please note that strings and floating point constants are not affected, so that

  	use Math::BigInt qw/:constant/;

	$x = 1234567890123456789012345678901234567890
		+ 123456789123456789;
	$x = '1234567890123456789012345678901234567890'
		+ '123456789123456789';

do both not work. You need a explicit Math::BigInt->new() around one of them.

PERFORMANCE

Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x must be made in the second case. For long numbers, the copy can eat up to 20% of the work (in case of addition/subtraction, less for multiplication/division). If $y is very small compared to $x, the form $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes more time then the actual addition.

With a technic called copy-on-write the cost of copying with overload could be minimized or even completely avoided. This is currently not implemented.

The new version of this module is slower on new(), bstr() and numify(). Some operations may be slower for small numbers, but are significantly faster for big numbers. Other operations are now constant (O(1), like bneg(), babs() etc), instead of O(N) and thus nearly always take much less time.

For more benchmark results see http://bloodgate.com/perl/benchmarks.html

BUGS

:constant and eval()

Under Perl prior to 5.6.0 having an use Math::BigInt ':constant'; and eval() in your code will crash with "Out of memory". This is probably an overload/exporter bug. You can workaround by not having eval() and ':constant' at the same time or upgrade your Perl.

CAVEATS

Some things might not work as you expect them. Below is documented what is known to be troublesome:

stringify, bstr(), bsstr() and 'cmp'

Both stringify and bstr() now drop the leading '+'. The old code would return '+3', the new returns '3'. This is to be consistent with Perl and to make cmp (especially with overloading) to work as you expect. It also solves problems with Test.pm, it's ok() uses 'eq' internally.

Mark said, when asked about to drop the '+' altogether, or make only cmp work:

I agree (with the first alternative), don't add the '+' on positive
numbers.  It's not as important anymore with the new internal 
form for numbers.  It made doing things like abs and neg easier,
but those have to be done differently now anyway.

So, the following examples will now work all as expected:

	use Test;
        BEGIN { plan tests => 1 }
	use Math::BigInt;

	my $x = new Math::BigInt 3*3;
	my $y = new Math::BigInt 3*3;

	ok ($x,3*3);
	print "$x eq 9" if $x eq $y;
	print "$x eq 9" if $x eq '9';
	print "$x eq 9" if $x eq 3*3;

Additionally, the following still works:

print "$x == 9" if $x == $y;
print "$x == 9" if $x == 9;
print "$x == 9" if $x == 3*3;

There is now a bsstr() method to get the string in scientific notation aka 1e+2 instead of 100. Be advised that overloaded 'eq' always uses bstr() for comparisation, but Perl will represent some numbers as 100 and others as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:

	use Test;
        BEGIN { plan tests => 3 }
	use Math::BigInt;

	$x = Math::BigInt->new('1e56'); $y = 1e56;
	ok ($x,$y);			# will fail
	ok ($x->bsstr(),$y);		# okay
	$y = Math::BigInt->new($y);
	ok ($x,$y);			# okay
int()

int() will return (at least for Perl v5.7.1 and up) another BigInt, not a Perl scalar:

$x = Math::BigInt->new(123);
$y = int($x);				# BigInt 123
$x = Math::BigFloat->new(123.45);
$y = int($x);				# BigInt 123

In all Perl versions you can use as_number() for the same effect:

$x = Math::BigFloat->new(123.45);
$y = $x->as_number();			# BigInt 123

This also works for other subclasses, like Math::String.

bdiv

The following will probably not do what you expect:

print $c->bdiv(10000),"\n";

It prints both quotient and reminder since print calls bdiv() in list context. Also, bdiv() will modify $c, so be carefull. You probably want to use

print $c / 10000,"\n";
print scalar $c->bdiv(10000),"\n";  # or if you want to modify $c

instead.

The quotient is always the greatest integer less than or equal to the real-valued quotient of the two operands, and the remainder (when it is nonzero) always has the same sign as the second operand; so, for example,

1 / 4   => ( 0, 1)
1 / -4  => (-1,-3)
-3 / 4  => (-1, 1)
-3 / -4 => ( 0,-3)

As a consequence, the behavior of the operator % agrees with the behavior of Perl's built-in % operator (as documented in the perlop manpage), and the equation

$x == ($x / $y) * $y + ($x % $y)

holds true for any $x and $y, which justifies calling the two return values of bdiv() the quotient and remainder.

Perl's 'use integer;' changes the behaviour of % and / for scalars, but will not change BigInt's way to do things. This is because under 'use integer' Perl will do what the underlying C thinks is right and this is different for each system. If you need BigInt's behaving exactly like Perl's 'use integer', bug the author to implement it ;)

Modifying and =

Beware of:

$x = Math::BigFloat->new(5);
$y = $x;

It will not do what you think, e.g. making a copy of $x. Instead it just makes a second reference to the same object and stores it in $y. Thus anything that modifies $x will modify $y, and vice versa.

$x->bmul(2);
print "$x, $y\n";       # prints '10, 10'

If you want a true copy of $x, use:

$y = $x->copy();

See also the documentation in overload regarding =.

bpow

bpow() (and the rounding functions) now modifies the first argument and return it, unlike the old code which left it alone and only returned the result. This is to be consistent with badd() etc. The first three will modify $x, the last one won't:

print bpow($x,$i),"\n"; 	# modify $x
print $x->bpow($i),"\n"; 	# ditto
print $x **= $i,"\n";		# the same
print $x ** $i,"\n";		# leave $x alone

The form $x **= $y is faster than $x = $x ** $y;, though.

Overloading -$x

The following:

$x = -$x;

is slower than

$x->bneg();

since overload calls sub($x,0,1); instead of neg($x). The first variant needs to preserve $x since it does not know that it later will get overwritten. This makes a copy of $x and takes O(N). But $x->bneg() is O(1).

With Copy-On-Write, this issue will be gone. Stay tuned...

Mixing different object types

In Perl you will get a floating point value if you do one of the following:

$float = 5.0 + 2;
$float = 2 + 5.0;
$float = 5 / 2;

With overloaded math, only the first two variants will result in a BigFloat:

use Math::BigInt;
use Math::BigFloat;

$mbf = Math::BigFloat->new(5);
$mbi2 = Math::BigInteger->new(5);
$mbi = Math::BigInteger->new(2);

				# what actually gets called:
$float = $mbf + $mbi;		# $mbf->badd()
$float = $mbf / $mbi;		# $mbf->bdiv()
$integer = $mbi + $mbf;		# $mbi->badd()
$integer = $mbi2 / $mbi;	# $mbi2->bdiv()
$integer = $mbi2 / $mbf;	# $mbi2->bdiv()

This is because math with overloaded operators follows the first (dominating) operand, this one's operation is called and returns thus the result. So, Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether the result should be a Math::BigFloat or the second operant is one.

To get a Math::BigFloat you either need to call the operation manually, make sure the operands are already of the proper type or casted to that type via Math::BigFloat->new():

$float = Math::BigFloat->new($mbi2) / $mbi;	# = 2.5

Beware of simple "casting" the entire expression, this would only convert the already computed result:

$float = Math::BigFloat->new($mbi2 / $mbi);	# = 2.0 thus wrong!

Beware of the order of more complicated expressions like:

$integer = ($mbi2 + $mbi) / $mbf;		# int / float => int
$integer = $mbi2 / Math::BigFloat->new($mbi);	# ditto

If in doubt, break the expression into simpler terms, or cast all operands to the desired resulting type.

Scalar values are a bit different, since:

$float = 2 + $mbf;
$float = $mbf + 2;

will both result in the proper type due to the way the overloaded math works.

This section also applies to other overloaded math packages, like Math::String.

bsqrt()

bsqrt() works only good if the result is an big integer, e.g. the square root of 144 is 12, but from 12 the square root is 3, regardless of rounding mode.

If you want a better approximation of the square root, then use:

$x = Math::BigFloat->new(12);
$Math::BigFloat::precision = 0;
Math::BigFloat->round_mode('even');
print $x->copy->bsqrt(),"\n";		# 4

$Math::BigFloat::precision = 2;
print $x->bsqrt(),"\n";			# 3.46
print $x->bsqrt(3),"\n";		# 3.464

LICENSE

This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.

AUTHORS

Original code by Mark Biggar, overloaded interface by Ilya Zakharevich. Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.