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Math-Bacovia-0.04
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/ Math::Bacovia

NAME

Math::Bacovia - Symbolic math library with support for alternative representations.

VERSION

Version 0.04

SYNOPSIS

Math::Bacovia is an experimental symbolic math library, with support for alternative representations, simplification of expressions and numerical evaluation.

    use 5.014;
    use Math::Bacovia qw(:all);

    my $x = Symbol('x');
    my $y = Symbol('y');

    say $x+$y;               #=> Sum(Symbol("x"), Symbol("y"))
    say $x-$y;               #=> Difference(Symbol("x"), Symbol("y"))
    say $x*$y;               #=> Product(Symbol("x"), Symbol("y"))
    say $x/$y;               #=> Fraction(Symbol("x"), Symbol("y"))

    say $x**$y;              #=> Power(Symbol("x"), Symbol("y"))

    say Log($x);             #=> Log(Symbol("x"))
    say Log($x)+Log($y);     #=> Log(Product(Symbol("x"), Symbol("y")))

    say Exp($x);             #=> Exp(Symbol("x"))
    say Exp($x)*Exp($y);     #=> Exp(Sum(Symbol("x"), Symbol("y")))

    say "\n=> Sum:";
    my $sum = Fraction(0, 1);

    for my $n (1..10) {
        $sum += Fraction(1, $n);
    }
    say $sum;                     #=> Fraction(10628640, 3628800)
    say $sum->numeric;            #=> 7381/2520

    say "\n=> Product:";
    my $prod = Product();

    for my $n (1..3) {
        $prod *= Exp(Fraction(1, $n));
    }

    say $prod->pretty;            #=> (exp(1) * exp(1/2) * exp(1/3))
    say $prod->simple->pretty;    #=> exp(11/6)
    say $prod->numeric;           #=> 6.25470095193632871640207...

    say "\n=> Alternative representations:";
    say join ', ', Power(3, 5)->alternatives(full => 1);   #=> Power(3, 5), Exp(Product(Log(3), 5)), 243

TYPES

The types supported by this library are described bellow:

Symbol(name, value=undef)

Represents a symbolic value. Optionally, it can have a numerical value (or any other value).

Number(value)

Represents a numerical value.

Fraction(numerator, denominator)

Represents a symbolic fraction.

Difference(minuend, subtrahend)

Represents a symbolic difference.

Power(base, power)

Represents a symbolic exponentiation in a symbolic base.

Log(x)

Represents the natural logarithm of a symbolic value.

Exp(x)

Represents the natural exponentiation of a symbolic value.

Sum(a, b, c, ...)

Represents a summation of an arbitrary (finite) number of symbolic values.

Product(a, b, c, ...)

Represents a product of an arbitrary (finite) number of symbolic values.

EXPORT

Importing a name from Math::Bacovia can be done using the following syntax:

    use Math::Bacovia qw(Fraction);     # imports the Fraction() function

i / e / pi / tau

Math::Bacovia also provides a small list of useful constants which can be imported:

    i      the imaginary unit                sqrt(-1)
    e      the e mathematical constant       Exp(1)
    pi     the PI mathematical constant      Log(-1) * -sqrt(-1)
    tau    the TAU mathmematical constant    Log(-1) * -sqrt(-1) * 2

For importing a constant, the same syntax can be used as in importing a function:

    use Math::Bacovia qw(i tau);

By specifying the :all special keyword, Math::Bacovia will export all the provided functions and constants.

    use Math::Bacovia qw(:all);

Nothing is exported by default.

METHODS

This section describes the special methods provided by Math::Bacovia.

inv

    1 / $x
    $x->inv

Multiplicative inverse of x, defined as:

    Fraction(1, $x)

neg

    -$x
    $x->neg

Additive inverse of x, defined as:

    Difference(0, $x)

add

    $x + $y
    $x->add($y)

Summation of x and y, defined as:

    Sum($x, $y)

sub

    $x - $y
    $x->sub($y)

Difference of x from y, defined as:

    Difference($x, $y)

mul

    $x * $y
    $x->mul($y)

Product of x and y, defined as:

    Product($x, $y)

div

    $x / $y
    $x->div($y)

Fraction x over y, defined as:

    Fraction($x, $y)

pow

    $x ** $y
    $x->pow($y)

x to the power y, defined as:

    Power($x, $y)

sqrt

    sqrt($x)
    $x->sqrt

Square root of x, defined as:

    $x ** Fraction(1, 2)

log

    log($x)
    $x->log

Natural logarithm of x, defined as:

    Log($x)

exp

    exp($x)
    $x->exp

Natural exponentiation of x, defined as:

    Exp($x)

int

    int($x)
    $x->int

Evaluates x numerically and truncates the result to an integer, returing a Math::AnyNum object.

It may return NaN when this conversion is not possible.

Defined as:

    $x->numeric->int

TRIGONOMETRIC METHODS

sin / sinh / asin / asinh

    sin($x)
    $x->sin
    $x->sinh
    $x->asin
    $x->asinh

Sine, hyperbolic sine, inverse sine and inverse hyperbolic sine.

cos / cosh / acos / acosh

    cos($x)
    $x->cos
    $x->cosh
    $x->acos
    $x->acosh

Cosine, hyperbolic cosine, inverse cosine and inverse hyperbolic cosine.

tan / tanh / atan / atanh

    $x->tan
    $x->tanh
    $x->atan
    $x->atanh

Tangent, hyperbolic tangent, inverse tangent and inverse hyperbolic tangent.

cot / coth / acot / acoth

    $x->cot
    $x->coth
    $x->acot
    $x->acoth

Cotangent, hyperbolic cotangent, inverse cotangent and inverse hyperbolic cotangent.

sec / sech / asec / asech

    $x->sec
    $x->sech
    $x->asec
    $x->asech

Secant, hyperbolic secant, inverse secant and inverse hyperbolic secant.

csc / csch / acsc / acsch

    $x->csc
    $x->csch
    $x->acsc
    $x->acsch

Cosecant, hyperbolic cosecant, inverse cosecant and inverse hyperbolic cosecant.

atan2

    atan2($x, $y)
    $x->atan2($y)

The arc tangent of x and y, defined as:

    atan2(x, y) = -i * log((y + x*i) / sqrt(x^2 + y^2))

SPECIAL METHODS

Each type provided by Math::Bacovia includes the same set of methods, defined bellow:

alternatives

    $x->alternatives(%opt)

This method uses common mathematical identities to create symbolically equivalent expressions from the self-expression.

Example:

    say for Exp(Log(Fraction(1,3)) * 2)->alternatives;

Output:

    Exp(Product(2, Log(Fraction(1, 3))))
    Power(Fraction(1, 3), 2)
    Exp(Product(2, Log(1/3)))
    Power(1/3, 2)

The options supported by this method are:

    log  => 1,    # will try to generate logarithmic alternatives
    full => 1,    # will try to generate more alternatives (it may be slow)

Example:

    say for Power(3, 5)->alternatives(full => 1);

Output:

    Power(3, 5)
    Exp(Product(Log(3), 5))
    243

WARNING: The number of alternative representations grows exponentially! For non-trivial expressions, this process may take a very long time and use lots of memory. In combination with the full option (set to a true value), the returned list may contain hundreds or even thousands of alternative representations.

simple

    $x->simple(%opt)

Returns a simplification of the self-expression, taking the same options as the alternatives() method.

    say Exp(Log(Log(Exp(Exp(Log(Symbol('x')))))))->simple;

Output:

    Symbol("x")

The simplification of an expression is done by generating the list of alternative representations for it, then selecting the shortest expression from this list.

This approach works pretty well even on relatively complex expressions, such as:

    say Log(Symbol('x'))->cosh->simple->pretty;     #=> ((1 + x^2)/(2 * x))

However, the returned simplified expression is not guaranteed to always be "optimal". In some cases, calling the simple() method on an already simplified expression, the simplified expression may get simplified even further:

    $x->simple(full => 1)->simple(full => 1);

WARNING: On long expressions, this process may take a very long time and use lots of memory.

expand

    $x->expand(%opt)

Returns an expanded version of the self-expression, taking the same options as the alternatives() method.

    say Power(Fraction(5, 7), Fraction(1, 3))->expand(full => 1);

Output:

    Exp(Product(Log(Fraction(5, 7)), Fraction(1, 3)))

The expansion of an expression is done by generating the list of alternative representations for it, then selecting the longest expression from this list.

WARNING: On long expressions, this process may take a very long time and use lots of memory.

numeric

    $x->numeric

Evaluates the self-expression numerically and returns the result as a Math::AnyNum object.

Example:

    my $x = Symbol('x', 13);
    my $expr = ($x**2 - $x + 41);

    say $expr->numeric;   #=> 197

pretty

    $x->pretty

Returns a human-readable stringification of the self-expression.

Example:

    say Power(3, Log(Fraction(1, 2)))->pretty;    #=> 3^log(1/2)

SEE ALSO

For a detailed documentation of each class, please see:

AUTHOR

Daniel Șuteu, <trizen at protonmail.com>

BUGS

Please report any bugs or feature requests to https://github.com/trizen/Math-Bacovia. I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

SUPPORT

You can find documentation for this module with the perldoc command.

    perldoc Math::Bacovia

You can also look for information at:

LICENSE AND COPYRIGHT

Copyright 2017-2018 Daniel Șuteu.

This program is free software; you can redistribute it and/or modify it under the terms of the the Artistic License (2.0). You may obtain a copy of the full license at:

http://www.perlfoundation.org/artistic_license_2_0

Any use, modification, and distribution of the Standard or Modified Versions is governed by this Artistic License. By using, modifying or distributing the Package, you accept this license. Do not use, modify, or distribute the Package, if you do not accept this license.

If your Modified Version has been derived from a Modified Version made by someone other than you, you are nevertheless required to ensure that your Modified Version complies with the requirements of this license.

This license does not grant you the right to use any trademark, service mark, tradename, or logo of the Copyright Holder.

This license includes the non-exclusive, worldwide, free-of-charge patent license to make, have made, use, offer to sell, sell, import and otherwise transfer the Package with respect to any patent claims licensable by the Copyright Holder that are necessarily infringed by the Package. If you institute patent litigation (including a cross-claim or counterclaim) against any party alleging that the Package constitutes direct or contributory patent infringement, then this Artistic License to you shall terminate on the date that such litigation is filed.

Disclaimer of Warranty: THE PACKAGE IS PROVIDED BY THE COPYRIGHT HOLDER AND CONTRIBUTORS "AS IS' AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES. THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, OR NON-INFRINGEMENT ARE DISCLAIMED TO THE EXTENT PERMITTED BY YOUR LOCAL LAW. UNLESS REQUIRED BY LAW, NO COPYRIGHT HOLDER OR CONTRIBUTOR WILL BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL DAMAGES ARISING IN ANY WAY OUT OF THE USE OF THE PACKAGE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.