NAME

Math::GComplex - Generic complex number library.

VERSION

Version 0.13

SYNOPSIS

    use 5.010;
    use Math::GComplex;
    use Math::AnyNum qw(:overload);

    my $x = Math::GComplex->new(3, 4);
    my $y = Math::GComplex->new(7, 5);

    say $x + $y;        #=> (10 9)
    say $x - $y;        #=> (-4 -1)
    say $x * $y;        #=> (1 43)
    say $x / $y;        #=> (41/74 13/74)

DESCRIPTION

Math::GComplex is a lightweight library, providing a generic interface to complex number operations, accepting any type of number as a component of a complex number, including native Perl numbers and numerical objects provided by other mathematical libraries, such as Math::AnyNum.

In most cases, it can be used as a drop-in replacement for Math::Complex.

Due to its simple and elegant design, Math::GComplex is between 2x up to 8x faster than Math::Complex.

EXPORT

The following functions are exportable:

    :trig
        sin sinh asin asinh
        cos cosh acos acosh
        tan tanh atan atanh
        cot coth acot acoth
        sec sech asec asech
        csc csch acsc acsch
        atan2 deg2rad rad2deg

    :special
        gcd invmod powmod
        log logn exp pow pown sqrt cbrt root

    :misc
        cplx polar abs acmp sgn conj norm
        inv real imag reals floor ceil round

Multiple functions can be exported as:

    use Math::GComplex qw(acos acosh);

There is also the possibility of exporting an entire group of functions, by specifying their group name, as:

    use Math::GComplex qw(:trig);

The imaginary unit, i = sqrt(-1), is also exportable, as:

    use Math::GComplex qw(i);

Additionally, by specifying the :all keyword, all the exportable functions, including the i constant, will be exported:

    use Math::GComplex qw(:all);

The :overload keyword enables constant overloading, which makes each number a Math::GComplex object and also exports the i constant:

    use Math::GComplex qw(:overload);
    CORE::say 3 + 4*i;                          #=> (3 4)
    CORE::say log(-1);                          #=> (0 3.14159265358979)

NOTE: :overload is lexical to the current scope only.

The syntax for disabling the :overload behavior in the current scope, is:

    no Math::GComplex;        # :overload will be disabled in the current scope

Nothing is exported by default.

INITIALIZATION

new / cplx / make

    my $z = cplx($real, $imag);
    my $z = Math::GComplex->new($real, $imag);
    my $z = Math::GComplex->make($real, $imag);

Create a new complex number, given its Cartesian coordinate form.

cplxe / emake

    my $z = cplxe($r, $theta);
    my $z = Math::GComplex->emake($r, $theta);

Create a new complex number, given its polar form.

i

    my $i = Math::GComplex::i();

Returns the imaginary unit as a Math::GComplex object, equivalent with cplx(0, 1).

BASIC OPERATIONS

This section describes all the basic operations provided by this module.

add

    my $z = $x + $y;
    my $z = $x->add($y);

Addition of x and y, defined as:

    (a + b*i) + (x + y*i) = (a + x) + (b + y)*i

sub

    my $z = $x - $y;
    my $z = $x->sub($y);

Subtraction of y from x, defined as:

    (a + b*i) - (x + y*i) = (a - x) + (b - y)*i

mul

    my $z = $x * $y;
    my $z = $x->mul($y);

Multiplication of x and y, defined as:

    (a + b*i) * (x + y*i) = (a*x - b*y) + (a*y + b*x)*i

div

    my $z = $x / $y;
    my $z = $x->div($y);

Division of x by y, defined as:

    (a + b*i) / (x + y*i) = (a*x + b*y)/(x^2 + y^2) + (b*x - a*y)/(x^2 + y^2)*i

mod

    my $z = $x % $y;
    my $z = $x->mod($y);

Remainder of x when divided by y, defined as:

    mod(a, b) = a - b * floor(a/b)

neg

    my $z = -$x;
    my $z = $x->neg;

Additive inverse of x, defined as:

    neg(a + b*i) = -a - b*i

conj

    my $z = ~$x;
    my $z = $x->conj;

Complex conjugate of x, defined as:

    conj(a + b*i) = a - b*i

inv

    my $z = $x->inv;

Multiplicative inverse of x, defined as:

    inv(x) = 1/x

norm

    my $z = $x->norm;

Normalized value of x, defined as:

    norm(a + b*i) = a^2 + b^2

abs

    my $z = $x->abs;

Absolute value of x, defined as:

    abs(a + b*i) = sqrt(a^2 + b^2)

sgn

    my $z = $x->sgn;

The sign of x, defined as:

    sgn(x) = x / abs(x)

SPECIAL FUNCTIONS

This section describes the special mathematical functions provided by this module.

log

    my $z = log($x);
    my $z = $x->log;

Natural logarithm of x, defined as:

    log(a + b*i) = log(a^2 + b^2)/2 + atan2(b, a) * i

logn

    my $z = $x->logn($y);

Logarithm of x to base y, defined as:

    logn(a, b) = log(a) / log(b)

exp

    my $z = exp($x);
    my $z = $x->exp;

Natural exponentiation of x, defined as:

    exp(a + b*i) = exp(a) * cos(b) + exp(a) * sin(b) * i

pow

    my $z = $x**$y;
    my $z = $x->pow($y);

Raises x to power y and returns the result, defined as:

    a^b = exp(log(a) * b)

pown

    my $z = $x->pown($n);

Raises x to power n, using the exponentiation by squaring method, and returns the result, where n is a native integer.

powmod

    my $z = $x->powmod($n, $m);

Modular exponentiation x^n mod m, where n in an arbitrary large integer.

gcd

    my $z = $n->gcd($k);

Greatest common divisors of two complex numbers.

invmod

    my $x = $n->invmod($m);

Modular multiplicative inverse of two complex numbers.

The returned value is the solution to x in:

    n*x = 1 (mod m)

Returns undef when a multiplicative inverse mod m does not exist.

root

    my $z = $x->root($y);

Nth root of x, defined as:

    root(a, b) = exp(log(a) / b)

sqrt

    my $z = sqrt($x);
    my $z = $x->sqrt;

Square root of x, defined as:

    sqrt(x) = exp(log(x) / 2)

cbrt

    my $z = $x->cbrt;

Cube root of x, defined as:

    cbrt(x) = exp(log(x) / 3)

TRIGONOMETRIC FUNCTIONS

This section includes all the trigonometric functions provied by Math::GComplex.

sin / sinh / asin / asinh

    my $z = $x->sin;
    my $z = $x->sinh;
    my $z = $x->asin;
    my $z = $x->asinh;

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

Defined as:

      sin(x) = (exp(x * i) - exp(-i * x))/(2 * i)
     sinh(x) = (exp(2 * x) - 1) / (2 * exp(x))
     asin(x) = -i * log(i * x + sqrt(1 - x^2))
    asinh(x) = log(sqrt(x^2 + 1) + x)

cos / cosh / acos / acosh

    my $z = $x->cos;
    my $z = $x->cosh;
    my $z = $x->acos;
    my $z = $x->acosh;

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

Defined as:

      cos(x) = (exp(-i * x) + exp(i * x)) / 2
     cosh(x) = (exp(2 * x) + 1) / (2 * exp(x))
     acos(x) = -2 * i * log(i * sqrt((1 - x)/2) + sqrt((1 + x)/2))
    acosh(x) = log(x + sqrt(x - 1) * sqrt(x + 1))

tan / tanh / atan / atanh

    my $z = $x->tan;
    my $z = $x->tanh;
    my $z = $x->atan;
    my $z = $x->atanh;

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

Defined as:

      tan(x) = (2 * i)/(exp(2 * i * x) + 1) - i
     tanh(x) = (exp(2 * x) - 1) / (exp(2 * x) + 1)
     atan(x) = i * (log(1 - i * x) - log(1 + i * x)) / 2
    atanh(x) = (log(1 + x) - log(1 - x)) / 2

cot / coth / acot / acoth

    my $z = $x->cot;
    my $z = $x->coth;
    my $z = $x->acot;
    my $z = $x->acoth;

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

Defined as:

      cot(x) = (2 * i)/(exp(2 * i * x) - 1) + i
     coth(x) = (exp(2 * x) + 1) / (exp(2 * x) - 1)
     acot(x) = atan(1/x)
    acoth(x) = atanh(1/x)

sec / sech / asec / asech

    my $z = $x->sec;
    my $z = $x->sech;
    my $z = $x->asec;
    my $z = $x->asech;

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

Defined as:

      sec(x) = 2/(exp(-i * x) + exp(i * x))
     sech(x) = (2 * exp(x)) / (exp(2 * x) + 1)
     asec(x) = acos(1/x)
    asech(x) = acosh(1/x)

csc / csch / acsc / acsch

    my $z = $x->csc;
    my $z = $x->csch;
    my $z = $x->acsc;
    my $z = $x->acsch;

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

Defined as:

      csc(x) = -(2 * i)/(exp(-i * x) - exp(i * x))
     csch(x) = (2 * exp(x)) / (exp(2 * x) - 1)
     acsc(x) = asin(1/x)
    acsch(x) = asinh(1/x)

atan2

    my $z = atan2($x, $y);
    my $z = $x->atan2($y);

The arc tangent of x and y, defined as:

    atan2(a, b) = -i * log((b + a*i) / sqrt(a^2 + b^2))

deg2rad

    my $rad = $x->deg2rad;

Returns the value of x converted from degrees to radians.

Defined as:

    deg2rad(x) = x / 180 * atan2(0, -abs(x))

rad2deg

    my $deg = $x->rad2deg;

Returns the value of x converted from radians to degrees.

Defined as:

    rad2deg(x) = x * 180 / atan2(0, -abs(x))

MISCELLANEOUS FUNCTIONS

This section describes the various useful methods provided by this module.

floor

    my $z = $x->floor;

The floor function, defined as:

    floor(a + b*i) = floor(a) + floor(b)*i

ceil

    my $z = $x->ceil;

The ceil function, defined as:

    ceil(a + b*i) = ceil(a) + ceil(b)*i

round

    my $z = $x->round;

The round function, rounding x to the nearest Gaussian integer, defined as:

    round(a + b*i) = round(a) + round(b)*i

This function uses the half-away-from-zero tie-breaking method, defined as:

    round(+0.5) = +1
    round(-0.5) = -1

int

    my $z = int($x);
    my $z = $x->int;

The integer-truncation function, defined as:

    int(a + b*i) = int(a) + int(b)*i

and

    my $z = $x & $y;
    my $z = $x->and($y);

Bitwise AND-logical operation, defined as:

    (a + b*i) & (x + y*i) = (a & x) + (b & y)*i

or

    my $z = $x | $y;
    my $z = $x->or($y);

Bitwise OR-logical operation, defined as:

    (a + b*i) | (x + y*i) = (a | x) + (b | y)*i

xor

    my $z = $x ^ $y;
    my $z = $x->xor($y);

Bitwise XOR-logical operation, defined as:

    (a + b*i) ^ (x + y*i) = (a ^ x) + (b ^ y)*i

lsft

    my $z = $x << $n;
    my $z = $x->lsft($n);

Bitwise left-shift operation, defined as:

    (a + b*i) << n         = (a << n) + (b << n)*i
    (a + b*i) << (x + y*i) = int((a + b*i) * 2**(x + y*i))

rsft

    my $z = $x >> $n;
    my $z = $x->rsft($n);

Bitwise right-shift operation, defined as:

    (a + b*i) >> n         = (a >> n) + (b >> n)*i
    (a + b*i) >> (x + y*i) = int((a + b*i) / 2**(x + y*i))

real

    my $re = $x->real;

Return the real part of x.

imag

    my $im = $x->imag;

Returns the imaginary part of x.

reals

    my ($re, $im) = $x->reals

Returns the real and the imaginary part of x, as real numbers.

* Comparisons

eq

    my $bool = $x == $y;
    my $bool = $x->eq($y);

Equality check: returns a true value when x and y are equal.

ne

    my $bool = $x != $y;
    my $bool = $x->ne($y);

Inequality check: returns a true value when x and y are not equal.

gt

    my $bool = $x > $y;
    my $bool = $x->gt($y);

Returns a true value when x is greater than y.

ge

    my $bool = $x >= $y;
    my $bool = $x->ge($y);

Returns a true value when x is equal or greater than y.

lt

    my $bool = $x < $y;
    my $bool = $x->lt($y);

Returns a true value when x is less than y.

le

    my $bool = $x <= $y;
    my $bool = $x->le($y)

Returns a true value when x is equal or less than y.

cmp

    my $int = $x <=> $y;
    my $int = $x->cmp($y);

Compares x to y and returns a negative value when x is less than y, 0 when x and y are equal, and a positive value when x is greater than y.

Complex numbers are compared as:

    (real($x) <=> real($y)) ||
    (imag($x) <=> imag($y))

acmp

    my $int = $x->acmp($y);

Absolute comparison of x and y, defined as:

    acmp(a, b) = abs(a) <=> abs(b)

* Conversions

polar

    my ($rho, $theta) = $x->polar;

Returns the polar form of x, such that:

    x = rho * exp(theta * i)

boolify

    my $bool = $x->boolify;

Returns a true value when either the real part or the imaginary part of x is non-zero.

numify

    my $num = $x->numify;

Returns the real part of x.

stringify

    my $str = $x->stringify;

Returns a stringification version of x.

Example:

    Math::GComplex->new( 3, -4)->stringify;    # "(3 -4)"
    Math::GComplex->new(-5,  6)->stringify;    # "(-5 6)"

LIMITATIONS

Being a generic interface, it assumes that all the special cases (such as division by zero) are handled by the library of which type the components of a complex number are.

When the components of a complex number are native Perl numbers, the "division by zero" and the "logarithm of zero" cases are implicitly handled by this library.

However the user may still encounter incorrect results due to rounding errors and/or overflow/underflow in some special cases, such as:

    coth(1e6) = (NaN NaN)
    cosh(1e6) = (NaN NaN)

AUTHOR

Daniel Șuteu, <trizen at cpan.org>

BUGS

Please report any bugs or feature requests at https://github.com/trizen/Math-GComplex/issues. 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::GComplex

You can also look for information at:

SEE ALSO

  • Other math libraries

    Math::AnyNum - Arbitrary size precision for integers, rationals, floating-points and complex numbers.

    Math::GMP - High speed arbitrary size integer math.

    Math::GMPz - perl interface to the GMP library's integer (mpz) functions.

    Math::GMPq - perl interface to the GMP library's rational (mpq) functions.

    Math::MPFR - perl interface to the MPFR (floating point) library.

    Math::MPC - perl interface to the MPC (multi precision complex) library.

    Math::Complex - complex numbers and associated mathematical functions.

LICENSE AND COPYRIGHT

Copyright 2018-2019 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.

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