Math::BigFloat

(source, CPAN)

version 1.998

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NAME
SYNOPSIS
DESCRIPTION
METHODS
- accuracy
- precision()
- bexp()
- bnok()
- bpi()
- bcos()
- bsin()
- batan2()
- batan()
- bmuladd()
Autocreating constants
- Math library
- Using Math::BigInt::Lite
EXPORTS
BUGS
CAVEATS
SEE ALSO
LICENSE
AUTHORS

#NAME

Math::BigFloat - Arbitrary size floating point math package

#SYNOPSIS

use Math::BigFloat;

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

my $pi = Math::BigFloat->bpi(100);     # PI to 100 digits

# the following examples compute their result to 100 digits accuracy:
my $cos  = Math::BigFloat->new(1)->bcos(100);        # cosinus(1)
my $sin  = Math::BigFloat->new(1)->bsin(100);        # sinus(1)
my $atan = Math::BigFloat->new(1)->batan(100);       # arcus tangens(1)

my $atan2 = Math::BigFloat->new(  1 )->batan2( 1 ,100); # batan(1)
my $atan2 = Math::BigFloat->new(  1 )->batan2( 8 ,100); # batan(1/8)
my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2)

# Testing
$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_odd();           # true if odd, false for even
$x->is_even();          # true if even, false for odd
$x->is_pos();           # true if >= 0
$x->is_neg();           # true if <  0
$x->is_inf(sign);       # true if +inf, or -inf (default is '+')

$x->bcmp($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 pre-
# serve $x, use $z = $x->copy()->bXXX($y); See under L</CAVEATS> for
# necessary when mixing $a = $b assignments with non-overloaded math.

# set 
$x->bzero();            # set $i to 0
$x->bnan();             # set $i 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)
$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)
$x->bmodpow($exp,$mod); # modular exponentiation (($num**$exp) % $mod))
$x->blsft($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->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

$x->band($y);           # bit-wise and
$x->bior($y);           # bit-wise inclusive or
$x->bxor($y);           # bit-wise exclusive or
$x->bnot();             # bit-wise 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)

$x->bround($N);         # accuracy: preserve $N digits
$x->bfround($N);        # precision: round to the $Nth digit

$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:

bgcd(@values);          # greatest common divisor
blcm(@values);          # lowest common multiplicator

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

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

$x->length();           # number of digits (w/o sign and '.')
($l,$f) = $x->length(); # number of digits, and length of fraction

$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

# these get/set the appropriate global value for all BigFloat objects
Math::BigFloat->precision();   # Precision
Math::BigFloat->accuracy();    # Accuracy
Math::BigFloat->round_mode();  # rounding mode

#DESCRIPTION

All operators (including basic math operations) are overloaded if you declare your big floating point numbers as

$i = new Math::BigFloat '12_3.456_789_123_456_789E-2';

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

#Canonical notation

Input to these routines are either BigFloat objects, or strings of the following four forms:

/^[+-]\d+$/
/^[+-]\d+\.\d*$/
/^[+-]\d+E[+-]?\d+$/
/^[+-]\d*\.\d+E[+-]?\d+$/

all with optional leading and trailing zeros and/or spaces. Additionally, numbers are allowed to have an underscore between any two digits.

Empty strings as well as other illegal numbers results in 'NaN'.

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

#Output

Output values are BigFloat objects (normalized), except for bstr() and bsstr().

The string output will always have leading and trailing zeros stripped and drop a plus sign. bstr() will give you always the form with a decimal point, while bsstr() (s for scientific) gives you the scientific notation.

Input                   bstr()          bsstr()
'-0'                    '0'             '0E1'
'  -123 123 123'        '-123123123'    '-123123123E0'
'00.0123'               '0.0123'        '123E-4'
'123.45E-2'             '1.2345'        '12345E-4'
'10E+3'                 '10000'         '1E4'

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.

Actual math is done by using the class defined with with => Class; (which defaults to BigInts) to represent the mantissa and exponent.

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.

#`mantissa()`, `exponent()` and `parts()`

mantissa() and exponent() return the said parts of the BigFloat as BigInts 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 giving you both of them.

A zero is represented and returned as 0E1, not 0E0 (after Knuth).

Currently the mantissa is reduced as much as possible, favouring higher exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0). This might change in the future, so do not depend on it.

#Accuracy vs. Precision

#Rounding

#ffround ( +$scale ): Rounds to the $scale'th place left from the '.', counting from the dot. The first digit is numbered 1.
#ffround ( -$scale ): Rounds to the $scale'th place right from the '.', counting from the dot.
#ffround ( 0 ): Rounds to an integer.
#fround ( +$scale ): Preserves accuracy to $scale digits from the left (aka significant digits) and pads the rest with zeros. If the number is between 1 and -1, the significant digits count from the first non-zero after the '.'
#fround ( -$scale ) and fround ( 0 ): These are effectively no-ops.

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

The default rounding mode is 'even'. By using Math::BigFloat->round_mode($round_mode); you can get and set the default mode for subsequent rounding. The usage of $Math::BigFloat::$round_mode is no longer supported. The second parameter to the round functions then overrides the default temporarily.

The as_number() function returns a BigInt from a Math::BigFloat. It uses 'trunc' as rounding mode to make it equivalent to:

$x = 2.5;
$y = int($x) + 2;

You can override this by passing the desired rounding mode as parameter to as_number():

$x = Math::BigFloat->new(2.5);
$y = $x->as_number('odd');      # $y = 3

#METHODS

Math::BigFloat supports all methods that Math::BigInt supports, except it calculates non-integer results when possible. Please see Math::BigInt for a full description of each method. Below are just the most important differences:

#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()" in Math::BigInt, "bround()" in Math::BigInt or "bfround()" in Math::BigInt 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

#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!

#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).

#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::BigFloat->bpi(100), "\n";

Calculate PI to N digits (including the 3 before the dot). The result is rounded according to the current rounding mode, which defaults to "even".