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Math::BigInt(3pm)      Perl Programmers Reference Guide      Math::BigInt(3pm)




NAME

       Math::BigInt - Arbitrary size integer/float math package


SYNOPSIS

         use Math::BigInt;

         # or make it faster: install (optional) Math::BigInt::GMP
         # and always use (it will fall back to pure Perl if the
         # GMP library is not installed):

         # will warn if Math::BigInt::GMP cannot be found
         use Math::BigInt lib => 'GMP';

         # to supress the warning use this:
         # use Math::BigInt try => 'GMP';

         my $str = '1234567890';
         my @values = (64,74,18);
         my $n = 1; my $sign = '-';

         # Number creation
         my $x = Math::BigInt->new($str);      # defaults to 0
         my $y = $x->copy();                   # make a true copy
         my $nan  = Math::BigInt->bnan();      # create a NotANumber
         my $zero = Math::BigInt->bzero();     # create a +0
         my $inf = Math::BigInt->binf();       # create a +inf
         my $inf = Math::BigInt->binf('-');    # create a -inf
         my $one = Math::BigInt->bone();       # create a +1
         my $mone = Math::BigInt->bone('-');   # create a -1

         my $pi = Math::BigInt->bpi();         # returns '3'
                                               # see Math::BigFloat::bpi()

         $h = Math::BigInt->new('0x123');      # from hexadecimal
         $b = Math::BigInt->new('0b101');      # from binary
         $o = Math::BigInt->from_oct('0101');  # from octal

         # Testing (don't modify their arguments)
         # (return true if the condition is met, otherwise false)

         $x->is_zero();        # if $x is +0
         $x->is_nan();         # if $x is NaN
         $x->is_one();         # if $x is +1
         $x->is_one('-');      # if $x is -1
         $x->is_odd();         # if $x is odd
         $x->is_even();        # if $x is even
         $x->is_pos();         # if $x >= 0
         $x->is_neg();         # if $x <  0
         $x->is_inf($sign);    # if $x is +inf, or -inf (sign is default '+')
         $x->is_int();         # if $x is an integer (not a float)

         # comparing and digit/sign extraction
         $x->bcmp($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. If you want to preserve
         # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
         # necessary when mixing $a = $b assignments with non-overloaded math.

         $x->bzero();          # set $x to 0
         $x->bnan();           # set $x to NaN
         $x->bone();           # set $x to +1
         $x->bone('-');        # set $x to -1
         $x->binf();           # set $x to inf
         $x->binf('-');        # set $x to -inf

         $x->bneg();           # negation
         $x->babs();           # absolute value
         $x->bnorm();          # normalize (no-op in BigInt)
         $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

         $x->bmuladd($y,$z);   # $x = $x * $y + $z

         $x->bmod($y);            # modulus (x % y)
         $x->bmodpow($exp,$mod);  # modular exponentation (($num**$exp) % $mod))
         $x->bmodinv($mod);       # the inverse of $x in the given modulus $mod

         $x->bpow($y);            # power of arguments (x ** y)
         $x->blsft($y);           # left shift in base 2
         $x->brsft($y);           # right shift in base 2
                                  # returns (quo,rem) or quo if in scalar context
         $x->blsft($y,$n);        # left shift by $y places in base $n
         $x->brsft($y,$n);        # right shift by $y places in base $n
                                  # returns (quo,rem) or quo if in scalar context

         $x->band($y);            # bitwise and
         $x->bior($y);            # bitwise inclusive or
         $x->bxor($y);            # bitwise exclusive or
         $x->bnot();              # bitwise not (two's complement)

         $x->bsqrt();             # calculate square-root
         $x->broot($y);           # $y'th root of $x (e.g. $y == 3 => cubic root)
         $x->bfac();              # factorial of $x (1*2*3*4*..$x)

         $x->bnok($y);            # x over y (binomial coefficient n over k)

         $x->blog();              # logarithm of $x to base e (Euler's number)
         $x->blog($base);         # logarithm of $x to base $base (f.i. 2)
         $x->bexp();              # calculate e ** $x where e is Euler's number

         $x->round($A,$P,$mode);  # round to accuracy or precision using mode $mode
         $x->bround($n);          # accuracy: preserve $n digits
         $x->bfround($n);         # $n > 0: round $nth digits,
                                  # $n < 0: round to the $nth digit after the
                                  # dot, no-op for BigInts

         # The following do not modify their arguments in BigInt (are no-ops),
         # but do so in BigFloat:

         $x->bfloor();            # return integer less or equal than $x
         $x->bceil();             # return integer greater or equal than $x

         # The following do not modify their arguments:

         # greatest common divisor (no OO style)
         my $gcd = Math::BigInt::bgcd(@values);
         # lowest common multiplicator (no OO style)
         my $lcm = Math::BigInt::blcm(@values);

         $x->length();            # return number of digits in number
         ($xl,$f) = $x->length(); # length of number and length of fraction part,
                                  # latter is always 0 digits long for BigInts

         $x->exponent();          # return exponent as BigInt
         $x->mantissa();          # return (signed) mantissa as BigInt
         $x->parts();             # return (mantissa,exponent) as BigInt
         $x->copy();              # make a true copy of $x (unlike $y = $x;)
         $x->as_int();            # return as BigInt (in BigInt: same as copy())
         $x->numify();            # return as scalar (might overflow!)

         # conversation to string (do not modify their argument)
         $x->bstr();              # normalized string (e.g. '3')
         $x->bsstr();             # norm. string in scientific notation (e.g. '3E0')
         $x->as_hex();            # as signed hexadecimal string with prefixed 0x
         $x->as_bin();            # as signed binary string with prefixed 0b
         $x->as_oct();            # as signed octal string with prefixed 0


         # precision and accuracy (see section about rounding for more)
         $x->precision();         # return P of $x (or global, if P of $x undef)
         $x->precision($n);       # set P of $x to $n
         $x->accuracy();          # return A of $x (or global, if A of $x undef)
         $x->accuracy($n);        # set A $x to $n

         # Global methods
         Math::BigInt->precision();    # get/set global P for all BigInt objects
         Math::BigInt->accuracy();     # get/set global A for all BigInt objects
         Math::BigInt->round_mode();   # get/set global round mode, one of
                                       # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
         Math::BigInt->config();       # return hash containing configuration


DESCRIPTION

       All operators (including 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.

       Input
         Input values to these routines may be any string, that looks like a
         number and results in an integer, including hexadecimal and binary
         numbers.

         Scalars holding numbers may also be passed, but note that non-integer
         numbers may already have lost precision due to the conversation to
         float. Quote your input if you want BigInt to see all the digits:

                 $x = Math::BigInt->new(12345678890123456789);   # bad
                 $x = Math::BigInt->new('12345678901234567890'); # good

         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.

         Hexadecimal (prefixed with "0x") and binary numbers (prefixed with
         "0b") are accepted, too. Please note that octal numbers are not
         recognized by new(), so the following will print "123":

                 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'

         To convert an octal number, use from_oct();

                 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'

         Currently, Math::BigInt::new() defaults to 0, while
         Math::BigInt::new('') results in 'NaN'. This might change in the
         future, so use always the following explicit forms to get a zero or
         NaN:

                 $zero = Math::BigInt->bzero();
                 $nan = Math::BigInt->bnan();

         "bnorm()" on a BigInt object is now effectively a no-op, since the
         numbers are always stored in normalized form. If passed a string,
         creates a BigInt object from the input.

       Output
         Output values are BigInt objects (normalized), except for the methods
         which return a string (see SYNOPSIS).

         Some routines ("is_odd()", "is_even()", "is_zero()", "is_one()",
         "is_nan()", etc.) return true or false, while others ("bcmp()",
         "bacmp()") return either undef (if NaN is involved), <0, 0 or >0 and
         are suited for sort.


METHODS

       Each of the methods below (except config(), accuracy() and precision())
       accepts three additional parameters. These arguments $A, $P and $R are
       "accuracy", "precision" and "round_mode". Please see the section about
       "ACCURACY and PRECISION" for more information.

       config()

               use Data::Dumper;

               print Dumper ( Math::BigInt->config() );
               print Math::BigInt->config()->{lib},"\n";

       Returns a hash containing the configuration, e.g. the version number,
       lib loaded etc. The following hash keys are currently filled in with
       the appropriate information.

               key             Description
                               Example
               ============================================================
               lib             Name of the low-level math library
                               Math::BigInt::Calc
               lib_version     Version of low-level math library (see 'lib')
                               0.30
               class           The class name of config() you just called
                               Math::BigInt
               upgrade         To which class math operations might be upgraded
                               Math::BigFloat
               downgrade       To which class math operations might be downgraded
                               undef
               precision       Global precision
                               undef
               accuracy        Global accuracy
                               undef
               round_mode      Global round mode
                               even
               version         version number of the class you used
                               1.61
               div_scale       Fallback accuracy for div
                               40
               trap_nan        If true, traps creation of NaN via croak()
                               1
               trap_inf        If true, traps creation of +inf/-inf via croak()
                               1

       The following values can be set by passing "config()" a reference to a
       hash:

               trap_inf trap_nan
               upgrade downgrade precision accuracy round_mode div_scale

       Example:

               $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );

       accuracy()

               $x->accuracy(5);                # local for $x
               CLASS->accuracy(5);             # global for all members of CLASS
                                               # Note: This also applies to new()!

               $A = $x->accuracy();            # read out accuracy that affects $x
               $A = CLASS->accuracy();         # read out global accuracy

       Set or get the global or local accuracy, aka how many significant
       digits the results have. If you set a global accuracy, then this also
       applies to new()!

       Warning! The accuracy sticks, e.g. once you created a number under the
       influence of "CLASS->accuracy($A)", all results from math operations
       with that number will also be rounded.

       In most cases, you should probably round the results explicitly using
       one of round(), bround() or bfround() or by passing the desired
       accuracy to the math operation as additional parameter:

               my $x = Math::BigInt->new(30000);
               my $y = Math::BigInt->new(7);
               print scalar $x->copy()->bdiv($y, 2);           # print 4300
               print scalar $x->copy()->bdiv($y)->bround(2);   # print 4300

       Please see the section about "ACCURACY AND PRECISION" for further
       details.

       Value must be greater than zero. Pass an undef value to disable it:

               $x->accuracy(undef);
               Math::BigInt->accuracy(undef);

       Returns the current accuracy. For "$x-"accuracy()> it will return
       either the local accuracy, or if not defined, the global. This means
       the return value represents the accuracy that will be in effect for $x:

               $y = Math::BigInt->new(1234567);        # unrounded
               print Math::BigInt->accuracy(4),"\n";   # set 4, print 4
               $x = Math::BigInt->new(123456);         # $x will be automatically rounded!
               print "$x $y\n";                        # '123500 1234567'
               print $x->accuracy(),"\n";              # will be 4
               print $y->accuracy(),"\n";              # also 4, since global is 4
               print Math::BigInt->accuracy(5),"\n";   # set to 5, print 5
               print $x->accuracy(),"\n";              # still 4
               print $y->accuracy(),"\n";              # 5, since global is 5

       Note: Works also for subclasses like Math::BigFloat. Each class has
       it's own globals separated from Math::BigInt, but it is possible to
       subclass Math::BigInt and make the globals of the subclass aliases to
       the ones from Math::BigInt.

       precision()

               $x->precision(-2);      # local for $x, round at the second digit right of the dot
               $x->precision(2);       # ditto, round at the second digit left of the dot

               CLASS->precision(5);    # Global for all members of CLASS
                                       # This also applies to new()!
               CLASS->precision(-5);   # ditto

               $P = CLASS->precision();        # read out global precision
               $P = $x->precision();           # read out precision that affects $x

       Note: You probably want to use accuracy() instead. With accuracy you
       set the number of digits each result should have, with precision you
       set the place where to round!

       "precision()" sets or gets the global or local precision, aka at which
       digit before or after the dot to round all results. A set global
       precision also applies to all newly created numbers!

       In Math::BigInt, passing a negative number precision has no effect
       since no numbers have digits after the dot. In Math::BigFloat, it will
       round all results to P digits after the dot.

       Please see the section about "ACCURACY AND PRECISION" for further
       details.

       Pass an undef value to disable it:

               $x->precision(undef);
               Math::BigInt->precision(undef);

       Returns the current precision. For "$x-"precision()> it will return
       either the local precision of $x, or if not defined, the global. This
       means the return value represents the prevision that will be in effect
       for $x:

               $y = Math::BigInt->new(1234567);        # unrounded
               print Math::BigInt->precision(4),"\n";  # set 4, print 4
               $x = Math::BigInt->new(123456);         # will be automatically rounded
               print $x;                               # print "120000"!

       Note: Works also for subclasses like Math::BigFloat. Each class has its
       own globals separated from Math::BigInt, but it is possible to subclass
       Math::BigInt and make the globals of the subclass aliases to the ones
       from Math::BigInt.

       brsft()

               $x->brsft($y,$n);

       Shifts $x right by $y in base $n. Default is base 2, used are usually
       10 and 2, but others work, too.

       Right shifting usually amounts to dividing $x by $n ** $y and
       truncating the result:

               $x = Math::BigInt->new(10);
               $x->brsft(1);                   # same as $x >> 1: 5
               $x = Math::BigInt->new(1234);
               $x->brsft(2,10);                # result 12

       There is one exception, and that is base 2 with negative $x:

               $x = Math::BigInt->new(-5);
               print $x->brsft(1);

       This will print -3, not -2 (as it would if you divide -5 by 2 and
       truncate the result).

       new()

               $x = Math::BigInt->new($str,$A,$P,$R);

       Creates a new BigInt object from a scalar or another BigInt object. The
       input is accepted as decimal, hex (with leading '0x') or binary (with
       leading '0b').

       See Input for more info on accepted input formats.

       from_oct()

               $x = Math::BigInt->from_oct("0775");    # input is octal

       from_hex()

               $x = Math::BigInt->from_hex("0xcafe");  # input is hexadecimal

       from_bin()

               $x = Math::BigInt->from_oct("0x10011"); # input is binary

       bnan()

               $x = Math::BigInt->bnan();

       Creates a new BigInt object representing NaN (Not A Number).  If used
       on an object, it will set it to NaN:

               $x->bnan();

       bzero()

               $x = Math::BigInt->bzero();

       Creates a new BigInt object representing zero.  If used on an object,
       it will set it to zero:

               $x->bzero();

       binf()

               $x = Math::BigInt->binf($sign);

       Creates a new BigInt object representing infinity. The optional
       argument is either '-' or '+', indicating whether you want infinity or
       minus infinity.  If used on an object, it will set it to infinity:

               $x->binf();
               $x->binf('-');

       bone()

               $x = Math::BigInt->binf($sign);

       Creates a new BigInt object representing one. The optional argument is
       either '-' or '+', indicating whether you want one or minus one.  If
       used on an object, it will set it to one:

               $x->bone();             # +1
               $x->bone('-');          # -1

       is_one()/is_zero()/is_nan()/is_inf()

               $x->is_zero();                  # true if arg is +0
               $x->is_nan();                   # true if arg is NaN
               $x->is_one();                   # true if arg is +1
               $x->is_one('-');                # true if arg is -1
               $x->is_inf();                   # true if +inf
               $x->is_inf('-');                # true if -inf (sign is default '+')

       These methods all test the BigInt for being one specific value and
       return true or false depending on the input. These are faster than
       doing something like:

               if ($x == 0)

       is_pos()/is_neg()/is_positive()/is_negative()

               $x->is_pos();                   # true if > 0
               $x->is_neg();                   # true if < 0

       The methods return true if the argument is positive or negative,
       respectively.  "NaN" is neither positive nor negative, while "+inf"
       counts as positive, and "-inf" is negative. A "zero" is neither
       positive nor negative.

       These methods are only testing the sign, and not the value.

       "is_positive()" and "is_negative()" are aliases to "is_pos()" and
       "is_neg()", respectively. "is_positive()" and "is_negative()" were
       introduced in v1.36, while "is_pos()" and "is_neg()" were only
       introduced in v1.68.

       is_odd()/is_even()/is_int()

               $x->is_odd();                   # true if odd, false for even
               $x->is_even();                  # true if even, false for odd
               $x->is_int();                   # true if $x is an integer

       The return true when the argument satisfies the condition. "NaN",
       "+inf", "-inf" are not integers and are neither odd nor even.

       In BigInt, all numbers except "NaN", "+inf" and "-inf" are integers.

       bcmp()

               $x->bcmp($y);

       Compares $x with $y and takes the sign into account.  Returns -1, 0, 1
       or undef.

       bacmp()

               $x->bacmp($y);

       Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.

       sign()

               $x->sign();

       Return the sign, of $x, meaning either "+", "-", "-inf", "+inf" or NaN.

       If you want $x to have a certain sign, use one of the following
       methods:

               $x->babs();             # '+'
               $x->babs()->bneg();     # '-'
               $x->bnan();             # 'NaN'
               $x->binf();             # '+inf'
               $x->binf('-');          # '-inf'

       digit()

               $x->digit($n);          # return the nth digit, counting from right

       If $n is negative, returns the digit counting from left.

       bneg()

               $x->bneg();

       Negate the number, e.g. change the sign between '+' and '-', or between
       '+inf' and '-inf', respectively. Does nothing for NaN or zero.

       babs()

               $x->babs();

       Set the number to its absolute value, e.g. change the sign from '-' to
       '+' and from '-inf' to '+inf', respectively. Does nothing for NaN or
       positive numbers.

       bnorm()

               $x->bnorm();                    # normalize (no-op)

       bnot()

               $x->bnot();

       Two's complement (bitwise not). This is equivalent to

               $x->binc()->bneg();

       but faster.

       binc()

               $x->binc();                     # increment x by 1

       bdec()

               $x->bdec();                     # decrement x by 1

       badd()

               $x->badd($y);                   # addition (add $y to $x)

       bsub()

               $x->bsub($y);                   # subtraction (subtract $y from $x)

       bmul()

               $x->bmul($y);                   # multiplication (multiply $x by $y)

       bmuladd()

               $x->bmuladd($y,$z);

       Multiply $x by $y, and then add $z to the result,

       This method was added in v1.87 of Math::BigInt (June 2007).

       bdiv()

               $x->bdiv($y);                   # divide, set $x to quotient
                                               # return (quo,rem) or quo if scalar

       bmod()

               $x->bmod($y);                   # modulus (x % y)

       bmodinv()

               num->bmodinv($mod);             # modular inverse

       Returns the inverse of $num in the given modulus $mod.  '"NaN"' is
       returned unless $num is relatively prime to $mod, i.e. unless
       "bgcd($num, $mod)==1".

       bmodpow()

               $num->bmodpow($exp,$mod);       # modular exponentation
                                               # ($num**$exp % $mod)

       Returns the value of $num taken to the power $exp in the modulus $mod
       using binary exponentation.  "bmodpow" is far superior to writing

               $num ** $exp % $mod

       because it is much faster - it reduces internal variables into the
       modulus whenever possible, so it operates on smaller numbers.

       "bmodpow" also supports negative exponents.

               bmodpow($num, -1, $mod)

       is exactly equivalent to

               bmodinv($num, $mod)

       bpow()

               $x->bpow($y);                   # power of arguments (x ** y)

       blog()

               $x->blog($base, $accuracy);     # logarithm of x to the base $base

       If $base is not defined, Euler's number (e) is used:

               print $x->blog(undef, 100);     # log(x) to 100 digits

       bexp()

               $x->bexp($accuracy);            # calculate e ** X

       Calculates the expression "e ** $x" where "e" is Euler's number.

       This method was added in v1.82 of Math::BigInt (April 2007).

       See also blog().

       bnok()

               $x->bnok($y);              # x over y (binomial coefficient n over k)

       Calculates the binomial coefficient n over k, also called the "choose"
       function. The result is equivalent to:

               ( n )      n!
               | - |  = -------
               ( k )    k!(n-k)!

       This method was added in v1.84 of Math::BigInt (April 2007).

       bpi()

               print Math::BigInt->bpi(100), "\n";             # 3

       Returns PI truncated to an integer, with the argument being ignored.
       This means under BigInt this always returns 3.

       If upgrading is in effect, returns PI, rounded to N digits with the
       current rounding mode:

               use Math::BigFloat;
               use Math::BigInt upgrade => Math::BigFloat;
               print Math::BigInt->bpi(3), "\n";               # 3.14
               print Math::BigInt->bpi(100), "\n";             # 3.1415....

       This method was added in v1.87 of Math::BigInt (June 2007).

       bcos()

               my $x = Math::BigInt->new(1);
               print $x->bcos(100), "\n";

       Calculate the cosinus of $x, modifying $x in place.

       In BigInt, unless upgrading is in effect, the result is truncated to an
       integer.

       This method was added in v1.87 of Math::BigInt (June 2007).

       bsin()

               my $x = Math::BigInt->new(1);
               print $x->bsin(100), "\n";

       Calculate the sinus of $x, modifying $x in place.

       In BigInt, unless upgrading is in effect, the result is truncated to an
       integer.

       This method was added in v1.87 of Math::BigInt (June 2007).

       batan2()

               my $x = Math::BigInt->new(1);
               my $y = Math::BigInt->new(1);
               print $y->batan2($x), "\n";

       Calculate the arcus tangens of $y divided by $x, modifying $y in place.

       In BigInt, unless upgrading is in effect, the result is truncated to an
       integer.

       This method was added in v1.87 of Math::BigInt (June 2007).

       batan()

               my $x = Math::BigFloat->new(0.5);
               print $x->batan(100), "\n";

       Calculate the arcus tangens of $x, modifying $x in place.

       In BigInt, unless upgrading is in effect, the result is truncated to an
       integer.

       This method was added in v1.87 of Math::BigInt (June 2007).

       blsft()

               $x->blsft($y);          # left shift in base 2
               $x->blsft($y,$n);       # left shift, in base $n (like 10)

       brsft()

               $x->brsft($y);          # right shift in base 2
               $x->brsft($y,$n);       # right shift, in base $n (like 10)

       band()

               $x->band($y);                   # bitwise and

       bior()

               $x->bior($y);                   # bitwise inclusive or

       bxor()

               $x->bxor($y);                   # bitwise exclusive or

       bnot()

               $x->bnot();                     # bitwise not (two's complement)

       bsqrt()

               $x->bsqrt();                    # calculate square-root

       broot()

               $x->broot($N);

       Calculates the N'th root of $x.

       bfac()

               $x->bfac();                     # factorial of $x (1*2*3*4*..$x)

       round()

               $x->round($A,$P,$round_mode);

       Round $x to accuracy $A or precision $P using the round mode
       $round_mode.

       bround()

               $x->bround($N);               # accuracy: preserve $N digits

       bfround()

               $x->bfround($N);

       If N is > 0, rounds to the Nth digit from the left. If N < 0, rounds to
       the Nth digit after the dot. Since BigInts are integers, the case N < 0
       is a no-op for them.

       Examples:

               Input           N               Result
               ===================================================
               123456.123456   3               123500
               123456.123456   2               123450
               123456.123456   -2              123456.12
               123456.123456   -3              123456.123

       bfloor()

               $x->bfloor();

       Set $x to the integer less or equal than $x. This is a no-op in BigInt,
       but does change $x in BigFloat.

       bceil()

               $x->bceil();

       Set $x to the integer greater or equal than $x. This is a no-op in
       BigInt, but does change $x in BigFloat.

       bgcd()

               bgcd(@values);          # greatest common divisor (no OO style)

       blcm()

               blcm(@values);          # lowest common multiplicator (no OO style)

       head2 length()

               $x->length();
               ($xl,$fl) = $x->length();

       Returns the number of digits in the decimal representation of the
       number.  In list context, returns the length of the integer and
       fraction part. For BigInt's, the length of the fraction part will
       always be 0.

       exponent()

               $x->exponent();

       Return the exponent of $x as BigInt.

       mantissa()

               $x->mantissa();

       Return the signed mantissa of $x as BigInt.

       parts()

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

       copy()

               $x->copy();             # make a true copy of $x (unlike $y = $x;)

       as_int()/as_number()

               $x->as_int();

       Returns $x as a BigInt (truncated towards zero). In BigInt this is the
       same as "copy()".

       "as_number()" is an alias to this method. "as_number" was introduced in
       v1.22, while "as_int()" was only introduced in v1.68.

       bstr()

               $x->bstr();

       Returns a normalized string representation of $x.

       bsstr()

               $x->bsstr();            # normalized string in scientific notation

       as_hex()

               $x->as_hex();           # as signed hexadecimal string with prefixed 0x

       as_bin()

               $x->as_bin();           # as signed binary string with prefixed 0b

       as_oct()

               $x->as_oct();           # as signed octal string with prefixed 0

       numify()

               print $x->numify();

       This returns a normal Perl scalar from $x. It is used automatically
       whenever a scalar is needed, for instance in array index operations.

       This loses precision, to avoid this use as_int() instead.

       modify()

               $x->modify('bpowd');

       This method returns 0 if the object can be modified with the given
       peration, or 1 if not.

       This is used for instance by Math::BigInt::Constant.

       upgrade()/downgrade()

       Set/get the class for downgrade/upgrade operations. Thuis is used for
       instance by bignum. The defaults are '', thus the following operation
       will create a BigInt, not a BigFloat:

               my $i = Math::BigInt->new(123);
               my $f = Math::BigFloat->new('123.1');

               print $i + $f,"\n";                     # print 246

       div_scale()

       Set/get the number of digits for the default precision in divide
       operations.

       round_mode()

       Set/get the current round mode.


ACCURACY and PRECISION

       Since version v1.33, Math::BigInt and Math::BigFloat have full support
       for accuracy and precision based rounding, both automatically after
       every operation, as well as manually.

       This section describes the accuracy/precision handling in Math::Big* as
       it used to be and as it is now, complete with an explanation of all
       terms and abbreviations.

       Not yet implemented things (but with correct description) are marked
       with '!', things that need to be answered are marked with '?'.

       In the next paragraph follows a short description of terms used here
       (because these may differ from terms used by others people or
       documentation).

       During the rest of this document, the shortcuts A (for accuracy), P
       (for precision), F (fallback) and R (rounding mode) will be used.

       Precision P

       A fixed number of digits before (positive) or after (negative) the
       decimal point. For example, 123.45 has a precision of -2. 0 means an
       integer like 123 (or 120). A precision of 2 means two digits to the
       left of the decimal point are zero, so 123 with P = 1 becomes 120. Note
       that numbers with zeros before the decimal point may have different
       precisions, because 1200 can have p = 0, 1 or 2 (depending on what the
       inital value was). It could also have p < 0, when the digits after the
       decimal point are zero.

       The string output (of floating point numbers) will be padded with
       zeros:

               Initial value   P       A       Result          String
               ------------------------------------------------------------
               1234.01         -3              1000            1000
               1234            -2              1200            1200
               1234.5          -1              1230            1230
               1234.001        1               1234            1234.0
               1234.01         0               1234            1234
               1234.01         2               1234.01         1234.01
               1234.01         5               1234.01         1234.01000

       For BigInts, no padding occurs.

       Accuracy A

       Number of significant digits. Leading zeros are not counted. A number
       may have an accuracy greater than the non-zero digits when there are
       zeros in it or trailing zeros. For example, 123.456 has A of 6, 10203
       has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.

       The string output (of floating point numbers) will be padded with
       zeros:

               Initial value   P       A       Result          String
               ------------------------------------------------------------
               1234.01                 3       1230            1230
               1234.01                 6       1234.01         1234.01
               1234.1                  8       1234.1          1234.1000

       For BigInts, no padding occurs.

       Fallback F

       When both A and P are undefined, this is used as a fallback accuracy
       when dividing numbers.

       Rounding mode R

       When rounding a number, different 'styles' or 'kinds' of rounding are
       possible. (Note that random rounding, as in Math::Round, is not
       implemented.)

       'trunc'
         truncation invariably removes all digits following the rounding
         place, replacing them with zeros. Thus, 987.65 rounded to tens (P=1)
         becomes 980, and rounded to the fourth sigdig becomes 987.6 (A=4).
         123.456 rounded to the second place after the decimal point (P=-2)
         becomes 123.46.

         All other implemented styles of rounding attempt to round to the
         "nearest digit." If the digit D immediately to the right of the
         rounding place (skipping the decimal point) is greater than 5, the
         number is incremented at the rounding place (possibly causing a
         cascade of incrementation): e.g. when rounding to units, 0.9 rounds
         to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
         truncated at the rounding place: e.g. when rounding to units, 0.4
         rounds to 0, and -19.4 rounds to -19.

         However the results of other styles of rounding differ if the digit
         immediately to the right of the rounding place (skipping the decimal
         point) is 5 and if there are no digits, or no digits other than 0,
         after that 5. In such cases:

       'even'
         rounds the digit at the rounding place to 0, 2, 4, 6, or 8 if it is
         not already. E.g., when rounding to the first sigdig, 0.45 becomes
         0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.

       'odd'
         rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if it is
         not already. E.g., when rounding to the first sigdig, 0.45 becomes
         0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.

       '+inf'
         round to plus infinity, i.e. always round up. E.g., when rounding to
         the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5, and 0.4501
         also becomes 0.5.

       '-inf'
         round to minus infinity, i.e. always round down. E.g., when rounding
         to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6, but 0.4501
         becomes 0.5.

       'zero'
         round to zero, i.e. positive numbers down, negative ones up.  E.g.,
         when rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes
         -0.5, but 0.4501 becomes 0.5.

       'common'
         round up if the digit immediately to the right of the rounding place
         is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
         0.149 becomes 0.1.

       The handling of A & P in MBI/MBF (the old core code shipped with Perl
       versions <= 5.7.2) is like this:

       Precision
           * ffround($p) is able to round to $p number of digits after the decimal
             point
           * otherwise P is unused

       Accuracy (significant digits)
           * fround($a) rounds to $a significant digits
           * only fdiv() and fsqrt() take A as (optional) paramater
             + other operations simply create the same number (fneg etc), or more (fmul)
               of digits
             + rounding/truncating is only done when explicitly calling one of fround
               or ffround, and never for BigInt (not implemented)
           * fsqrt() simply hands its accuracy argument over to fdiv.
           * the documentation and the comment in the code indicate two different ways
             on how fdiv() determines the maximum number of digits it should calculate,
             and the actual code does yet another thing
             POD:
               max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
             Comment:
               result has at most max(scale, length(dividend), length(divisor)) digits
             Actual code:
               scale = max(scale, length(dividend)-1,length(divisor)-1);
               scale += length(divisor) - length(dividend);
             So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
             Actually, the 'difference' added to the scale is calculated from the
             number of "significant digits" in dividend and divisor, which is derived
             by looking at the length of the mantissa. Which is wrong, since it includes
             the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
             again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
             assumption that 124 has 3 significant digits, while 120/7 will get you
             '17', not '17.1' since 120 is thought to have 2 significant digits.
             The rounding after the division then uses the remainder and $y to determine
             wether it must round up or down.
          ?  I have no idea which is the right way. That's why I used a slightly more
          ?  simple scheme and tweaked the few failing testcases to match it.

       This is how it works now:

       Setting/Accessing
           * You can set the A global via C<< Math::BigInt->accuracy() >> or
             C<< Math::BigFloat->accuracy() >> or whatever class you are using.
           * You can also set P globally by using C<< Math::SomeClass->precision() >>
             likewise.
           * Globals are classwide, and not inherited by subclasses.
           * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
           * to undefine P, use C<< Math::SomeClass->precision(undef); >>
           * Setting C<< Math::SomeClass->accuracy() >> clears automatically
             C<< Math::SomeClass->precision() >>, and vice versa.
           * To be valid, A must be > 0, P can have any value.
           * If P is negative, this means round to the P'th place to the right of the
             decimal point; positive values mean to the left of the decimal point.
             P of 0 means round to integer.
           * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
           * to find out the current global P, use C<< Math::SomeClass->precision() >>
           * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
             setting of C<< $x >>.
           * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
             return eventually defined global A or P, when C<< $x >>'s A or P is not
             set.

       Creating numbers
           * When you create a number, you can give the desired A or P via:
             $x = Math::BigInt->new($number,$A,$P);
           * Only one of A or P can be defined, otherwise the result is NaN
           * If no A or P is give ($x = Math::BigInt->new($number) form), then the
             globals (if set) will be used. Thus changing the global defaults later on
             will not change the A or P of previously created numbers (i.e., A and P of
             $x will be what was in effect when $x was created)
           * If given undef for A and P, B<no> rounding will occur, and the globals will
             B<not> be used. This is used by subclasses to create numbers without
             suffering rounding in the parent. Thus a subclass is able to have its own
             globals enforced upon creation of a number by using
             C<< $x = Math::BigInt->new($number,undef,undef) >>:

                 use Math::BigInt::SomeSubclass;
                 use Math::BigInt;

                 Math::BigInt->accuracy(2);
                 Math::BigInt::SomeSubClass->accuracy(3);
                 $x = Math::BigInt::SomeSubClass->new(1234);

             $x is now 1230, and not 1200. A subclass might choose to implement
             this otherwise, e.g. falling back to the parent's A and P.

       Usage
           * If A or P are enabled/defined, they are used to round the result of each
             operation according to the rules below
           * Negative P is ignored in Math::BigInt, since BigInts never have digits
             after the decimal point
           * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
             Math::BigInt as globals does not tamper with the parts of a BigFloat.
             A flag is used to mark all Math::BigFloat numbers as 'never round'.

       Precedence
           * It only makes sense that a number has only one of A or P at a time.
             If you set either A or P on one object, or globally, the other one will
             be automatically cleared.
           * If two objects are involved in an operation, and one of them has A in
             effect, and the other P, this results in an error (NaN).
           * A takes precedence over P (Hint: A comes before P).
             If neither of them is defined, nothing is used, i.e. the result will have
             as many digits as it can (with an exception for fdiv/fsqrt) and will not
             be rounded.
           * There is another setting for fdiv() (and thus for fsqrt()). If neither of
             A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
             If either the dividend's or the divisor's mantissa has more digits than
             the value of F, the higher value will be used instead of F.
             This is to limit the digits (A) of the result (just consider what would
             happen with unlimited A and P in the case of 1/3 :-)
           * fdiv will calculate (at least) 4 more digits than required (determined by
             A, P or F), and, if F is not used, round the result
             (this will still fail in the case of a result like 0.12345000000001 with A
             or P of 5, but this can not be helped - or can it?)
           * Thus you can have the math done by on Math::Big* class in two modi:
             + never round (this is the default):
               This is done by setting A and P to undef. No math operation
               will round the result, with fdiv() and fsqrt() as exceptions to guard
               against overflows. You must explicitly call bround(), bfround() or
               round() (the latter with parameters).
               Note: Once you have rounded a number, the settings will 'stick' on it
               and 'infect' all other numbers engaged in math operations with it, since
               local settings have the highest precedence. So, to get SaferRound[tm],
               use a copy() before rounding like this:

                 $x = Math::BigFloat->new(12.34);
                 $y = Math::BigFloat->new(98.76);
                 $z = $x * $y;                           # 1218.6984
                 print $x->copy()->fround(3);            # 12.3 (but A is now 3!)
                 $z = $x * $y;                           # still 1218.6984, without
                                                         # copy would have been 1210!

             + round after each op:
               After each single operation (except for testing like is_zero()), the
               method round() is called and the result is rounded appropriately. By
               setting proper values for A and P, you can have all-the-same-A or
               all-the-same-P modes. For example, Math::Currency might set A to undef,
               and P to -2, globally.

          ?Maybe an extra option that forbids local A & P settings would be in order,
          ?so that intermediate rounding does not 'poison' further math?

       Overriding globals
           * you will be able to give A, P and R as an argument to all the calculation
             routines; the second parameter is A, the third one is P, and the fourth is
             R (shift right by one for binary operations like badd). P is used only if
             the first parameter (A) is undefined. These three parameters override the
             globals in the order detailed as follows, i.e. the first defined value
             wins:
             (local: per object, global: global default, parameter: argument to sub)
               + parameter A
               + parameter P
               + local A (if defined on both of the operands: smaller one is taken)
               + local P (if defined on both of the operands: bigger one is taken)
               + global A
               + global P
               + global F
           * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
             arguments (A and P) instead of one

       Local settings
           * You can set A or P locally by using C<< $x->accuracy() >> or
             C<< $x->precision() >>
             and thus force different A and P for different objects/numbers.
           * Setting A or P this way immediately rounds $x to the new value.
           * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.

       Rounding
           * the rounding routines will use the respective global or local settings.
             fround()/bround() is for accuracy rounding, while ffround()/bfround()
             is for precision
           * the two rounding functions take as the second parameter one of the
             following rounding modes (R):
             'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
           * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
             or by setting C<< $Math::SomeClass::round_mode >>
           * after each operation, C<< $result->round() >> is called, and the result may
             eventually be rounded (that is, if A or P were set either locally,
             globally or as parameter to the operation)
           * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
             this will round the number by using the appropriate rounding function
             and then normalize it.
           * rounding modifies the local settings of the number:

                 $x = Math::BigFloat->new(123.456);
                 $x->accuracy(5);
                 $x->bround(4);

             Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
             will be 4 from now on.

       Default values
           * R: 'even'
           * F: 40
           * A: undef
           * P: undef

       Remarks
           * The defaults are set up so that the new code gives the same results as
             the old code (except in a few cases on fdiv):
             + Both A and P are undefined and thus will not be used for rounding
               after each operation.
             + round() is thus a no-op, unless given extra parameters A and P


Infinity and Not a Number

       While BigInt has extensive handling of inf and NaN, certain quirks
       remain.

       oct()/hex()
         These perl routines currently (as of Perl v.5.8.6) cannot handle
         passed inf.

                 te@linux:~> perl -wle 'print 2 ** 3333'
                 inf
                 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
                 1
                 te@linux:~> perl -wle 'print oct(2 ** 3333)'
                 0
                 te@linux:~> perl -wle 'print hex(2 ** 3333)'
                 Illegal hexadecimal digit 'i' ignored at -e line 1.
                 0

         The same problems occur if you pass them Math::BigInt->binf()
         objects. Since overloading these routines is not possible, this
         cannot be fixed from BigInt.

       ==, !=, <, >, <=, >= with NaNs
         BigInt's bcmp() routine currently returns undef to signal that a NaN
         was involved in a comparison. However, the overload code turns that
         into either 1 or '' and thus operations like "NaN != NaN" might
         return wrong values.

       log(-inf)
         "log(-inf)" is highly weird. Since log(-x)=pi*i+log(x), then
         log(-inf)=pi*i+inf. However, since the imaginary part is finite, the
         real infinity "overshadows" it, so the number might as well just be
         infinity.  However, the result is a complex number, and since
         BigInt/BigFloat can only have real numbers as results, the result is
         NaN.

       exp(), cos(), sin(), atan2()
         These all might have problems handling infinity right.


INTERNALS

       The actual numbers are stored as unsigned big integers (with seperate
       sign).

       You should neither care about nor depend on the internal
       representation; it might change without notice. Use ONLY method calls
       like "$x->sign();" instead relying on the internal representation.

       MATH LIBRARY

       Math with the numbers is done (by default) by a module called
       "Math::BigInt::Calc". This is equivalent to saying:

               use Math::BigInt lib => 'Calc';

       You can change this by using:

               use Math::BigInt lib => 'BitVect';

       The following would first try to find Math::BigInt::Foo, then
       Math::BigInt::Bar, and when this also fails, revert to
       Math::BigInt::Calc:

               use Math::BigInt lib => 'Foo,Math::BigInt::Bar';

       Since Math::BigInt::GMP is in almost all cases faster than Calc
       (especially in math involving really big numbers, where it is much
       faster), and there is no penalty if Math::BigInt::GMP is not installed,
       it is a good idea to always use the following:

               use Math::BigInt lib => 'GMP';

       Different low-level libraries use different formats to store the
       numbers. You should NOT depend on the number having a specific format
       internally.

       See the respective math library module documentation for further
       details.

       SIGN

       The sign is either '+', '-', 'NaN', '+inf' or '-inf'.

       A sign of 'NaN' is used to represent the result when input arguments
       are not numbers or as a result of 0/0. '+inf' and '-inf' represent plus
       respectively minus infinity. You will get '+inf' when dividing a
       positive number by 0, and '-inf' when dividing any negative number by
       0.

       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 have a sign.

       Currently, for BigInts $e is always 0, except +inf and -inf, where it
       is "+inf"; and for NaN, where it is "NaN"; and for "$x == 0", where it
       is 1 (to be compatible with Math::BigFloat's internal representation of
       a zero as 0E1).

       $m is currently just a copy of the original number. The relation
       between $e and $m will stay always the same, though their real values
       might change.


EXAMPLES

         use Math::BigInt;

         sub bint { Math::BigInt->new(shift); }

         $x = Math::BigInt->bstr("1234")       # string "1234"
         $x = "$x";                            # same as bstr()
         $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 = Math::BigInt->new(0);            # BigInt "0"
         $x--;                                 # BigInt "-1"
         $x = Math::BigInt->badd(4,5)          # BigInt "9"
         print $x->bsstr();                    # 9e+0

       Examples for rounding:

         use Math::BigFloat;
         use Test;

         $x = Math::BigFloat->new(123.4567);
         $y = Math::BigFloat->new(123.456789);
         Math::BigFloat->accuracy(4);          # no more A than 4

         ok ($x->copy()->fround(),123.4);      # even rounding
         print $x->copy()->fround(),"\n";      # 123.4
         Math::BigFloat->round_mode('odd');    # round to odd
         print $x->copy()->fround(),"\n";      # 123.5
         Math::BigFloat->accuracy(5);          # no more A than 5
         Math::BigFloat->round_mode('odd');    # round to odd
         print $x->copy()->fround(),"\n";      # 123.46
         $y = $x->copy()->fround(4),"\n";      # A = 4: 123.4
         print "$y, ",$y->accuracy(),"\n";     # 123.4, 4

         Math::BigFloat->accuracy(undef);      # A not important now
         Math::BigFloat->precision(2);         # P important
         print $x->copy()->bnorm(),"\n";       # 123.46
         print $x->copy()->fround(),"\n";      # 123.46

       Examples for converting:

         my $x = Math::BigInt->new('0b1'.'01' x 123);
         print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";


Autocreating constants

       After "use Math::BigInt ':constant'" all the integer decimal,
       hexadecimal and binary 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 perl scalar.

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

               use Math::BigInt qw/:constant/;

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

       do not work. You need an explicit Math::BigInt->new() around one of the
       operands. You should also quote large constants to protect loss of
       precision:

               use Math::BigInt;

               $x = Math::BigInt->new('1234567889123456789123456789123456789');

       Without the quotes Perl would convert the large number to a floating
       point constant at compile time and then hand the result to BigInt,
       which results in an truncated result or a NaN.

       This also applies to integers that look like floating point constants:

               use Math::BigInt ':constant';

               print ref(123e2),"\n";
               print ref(123.2e2),"\n";

       will print nothing but newlines. Use either bignum or Math::BigFloat to
       get this to work.


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 the 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 technique called copy-on-write, the cost of copying with
       overload could be minimized or even completely avoided. A test
       implementation of COW did show performance gains for overloaded math,
       but introduced a performance loss due to a constant overhead for all
       other operations. So Math::BigInt does currently not COW.

       The rewritten version of this module (vs. v0.01) is slower on certain
       operations, like "new()", "bstr()" and "numify()". The reason are that
       it does now more work and handles much more cases. The time spent in
       these operations is usually gained in the other math operations so that
       code on the average should get (much) faster. If they don't, please
       contact the author.

       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.  These optimizations were done on purpose.

       If you find the Calc module to slow, try to install any of the
       replacement modules and see if they help you.

       Alternative math libraries

       You can use an alternative library to drive Math::BigInt via:

               use Math::BigInt lib => 'Module';

       See "MATH LIBRARY" for more information.

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

       SUBCLASSING



Subclassing Math::BigInt

       The basic design of Math::BigInt allows simple subclasses with very
       little work, as long as a few simple rules are followed:

       o The public API must remain consistent, i.e. if a sub-class is
         overloading addition, the sub-class must use the same name, in this
         case badd(). The reason for this is that Math::BigInt is optimized to
         call the object methods directly.

       o The private object hash keys like "$x-"{sign}> may not be changed,
         but additional keys can be added, like "$x-"{_custom}>.

       o Accessor functions are available for all existing object hash keys
         and should be used instead of directly accessing the internal hash
         keys. The reason for this is that Math::BigInt itself has a pluggable
         interface which permits it to support different storage methods.

       More complex sub-classes may have to replicate more of the logic
       internal of Math::BigInt if they need to change more basic behaviors. A
       subclass that needs to merely change the output only needs to overload
       "bstr()".

       All other object methods and overloaded functions can be directly
       inherited from the parent class.

       At the very minimum, any subclass will need to provide its own "new()"
       and can store additional hash keys in the object. There are also some
       package globals that must be defined, e.g.:

         # Globals
         $accuracy = undef;
         $precision = -2;       # round to 2 decimal places
         $round_mode = 'even';
         $div_scale = 40;

       Additionally, you might want to provide the following two globals to
       allow auto-upgrading and auto-downgrading to work correctly:

         $upgrade = undef;
         $downgrade = undef;

       This allows Math::BigInt to correctly retrieve package globals from the
       subclass, like $SubClass::precision.  See t/Math/BigInt/Subclass.pm or
       t/Math/BigFloat/SubClass.pm completely functional subclass examples.

       Don't forget to

               use overload;

       in your subclass to automatically inherit the overloading from the
       parent. If you like, you can change part of the overloading, look at
       Math::String for an example.


UPGRADING

       When used like this:

               use Math::BigInt upgrade => 'Foo::Bar';

       certain operations will 'upgrade' their calculation and thus the result
       to the class Foo::Bar. Usually this is used in conjunction with
       Math::BigFloat:

               use Math::BigInt upgrade => 'Math::BigFloat';

       As a shortcut, you can use the module "bignum":

               use bignum;

       Also good for oneliners:

               perl -Mbignum -le 'print 2 ** 255'

       This makes it possible to mix arguments of different classes (as in 2.5
       + 2) as well es preserve accuracy (as in sqrt(3)).

       Beware: This feature is not fully implemented yet.

       Auto-upgrade

       The following methods upgrade themselves unconditionally; that is if
       upgrade is in effect, they will always hand up their work:

       bsqrt()
       div()
       blog()
       bexp()

       Beware: This list is not complete.

       All other methods upgrade themselves only when one (or all) of their
       arguments are of the class mentioned in $upgrade (This might change in
       later versions to a more sophisticated scheme):


EXPORTS

       "Math::BigInt" exports nothing by default, but can export the following
       methods:

               bgcd
               blcm


CAVEATS

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

       bstr(), bsstr() and 'cmp'
        Both "bstr()" and "bsstr()" as well as automated stringify via
        overload 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", because its "ok()" uses 'eq' internally.

        Mark Biggar 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 comparison, but Perl will represent some
        numbers as 100 and others as 1e+308. If in doubt, convert both
        arguments to Math::BigInt before comparing them as strings:

                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

        Alternatively, simple use "<=>" for comparisons, this will get it
        always right. There is not yet a way to get a number automatically
        represented as a string that matches exactly the way Perl represents
        it.

        See also the section about "Infinity and Not a Number" for problems in
        comparing NaNs.

       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()" or "as_int" for the
        same effect:

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

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

        If you want a real Perl scalar, use "numify()":

                $y = $x->numify();                      # 123 as scalar

        This is seldom necessary, though, because this is done automatically,
        like when you access an array:

                $z = $array[$x];                        # does work automatically

       length
        The following will probably not do what you expect:

                $c = Math::BigInt->new(123);
                print $c->length(),"\n";                # prints 30

        It prints both the number of digits in the number and in the fraction
        part since print calls "length()" in list context. Use something like:

                print scalar $c->length(),"\n";         # prints 3

       bdiv
        The following will probably not do what you expect:

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

        It prints both quotient and remainder since print calls "bdiv()" in
        list context. Also, "bdiv()" will modify $c, so be careful. 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)
                -11 / 2  => (-5,1)
                 11 /-2  => (-5,-1)

        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. The only exception to
        this rule are when $y == 0 and $x is negative, then the remainder will
        also be negative. See below under "infinity handling" for the
        reasoning behind this.

        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 ;)

       infinity handling
        Here are some examples that explain the reasons why certain results
        occur while handling infinity:

        The following table shows the result of the division and the
        remainder, so that the equation above holds true. Some "ordinary"
        cases are strewn in to show more clearly the reasoning:

                A /  B  =   C,     R so that C *    B +    R =    A
             =========================================================
                5 /   8 =   0,     5         0 *    8 +    5 =    5
                0 /   8 =   0,     0         0 *    8 +    0 =    0
                0 / inf =   0,     0         0 *  inf +    0 =    0
                0 /-inf =   0,     0         0 * -inf +    0 =    0
                5 / inf =   0,     5         0 *  inf +    5 =    5
                5 /-inf =   0,     5         0 * -inf +    5 =    5
                -5/ inf =   0,    -5         0 *  inf +   -5 =   -5
                -5/-inf =   0,    -5         0 * -inf +   -5 =   -5
               inf/   5 =  inf,    0       inf *    5 +    0 =  inf
              -inf/   5 = -inf,    0      -inf *    5 +    0 = -inf
               inf/  -5 = -inf,    0      -inf *   -5 +    0 =  inf
              -inf/  -5 =  inf,    0       inf *   -5 +    0 = -inf
                 5/   5 =    1,    0         1 *    5 +    0 =    5
                -5/  -5 =    1,    0         1 *   -5 +    0 =   -5
               inf/ inf =    1,    0         1 *  inf +    0 =  inf
              -inf/-inf =    1,    0         1 * -inf +    0 = -inf
               inf/-inf =   -1,    0        -1 * -inf +    0 =  inf
              -inf/ inf =   -1,    0         1 * -inf +    0 = -inf
                 8/   0 =  inf,    8       inf *    0 +    8 =    8
               inf/   0 =  inf,  inf       inf *    0 +  inf =  inf
                 0/   0 =  NaN

        These cases below violate the "remainder has the sign of the second of
        the two arguments", since they wouldn't match up otherwise.

                A /  B  =   C,     R so that C *    B +    R =    A
             ========================================================
              -inf/   0 = -inf, -inf      -inf *    0 +  inf = -inf
                -8/   0 = -inf,   -8      -inf *    0 +    8 = -8

       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 (except overloaded operators) will
        modify $y, and vice versa.  Or in other words, "=" is only safe if you
        modify your BigInts only via overloaded math. As soon as you use a
        method call it breaks:

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

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

                $y = $x->copy();

        You can also chain the calls like this, this will make first a copy
        and then multiply it by 2:

                $y = $x->copy()->bmul(2);

        See also the documentation for overload.pm regarding "=".

       bpow
        "bpow()" (and the rounding functions) now modifies the first argument
        and returns 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).

       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, and the operation of that 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 also 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.

        One solution to you problem might be autoupgrading|upgrading. See the
        pragmas bignum, bigint and bigrat for an easy way to do this.

       bsqrt()
        "bsqrt()" works only good if the result is a big integer, e.g. the
        square root of 144 is 12, but from 12 the square root is 3, regardless
        of rounding mode. The reason is that the result is always truncated to
        an integer.

        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

       brsft()
        For negative numbers in base see also brsft.


LICENSE

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


SEE ALSO

       Math::BigFloat(3), Math::BigRat(3) and Math::Big(3) as well as
       Math::BigInt::BitVect(3), Math::BigInt::Pari(3) and 
       Math::BigInt::GMP(3).

       The pragmas bignum(3), bigint(3) and bigrat(3) also might be of interest
       because they solve the autoupgrading/downgrading issue, at least partly.

       The package at <http://search.cpan.org/search?mode=mod-
       ule&query=Math%3A%3ABigInt> contains more documentation including a
       full version history, testcases, empty subclass files and benchmarks.


AUTHORS

       Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
       Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 -
       2006 and still at it in 2007.

       Many people contributed in one or more ways to the final beast, see the
       file CREDITS for an (incomplete) list. If you miss your name, please
       drop me a mail. Thank you!



perl v5.10.0                      2007-12-18                 Math::BigInt(3pm)

Mac OS X 10.6 - Generated Thu Sep 17 20:13:36 CDT 2009
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