NAME

       Math::Vector::Real - Real vector arithmetic in Perl

SYNOPSIS

         use Math::Vector::Real;

         my $v = V(1.1, 2.0, 3.1, -4.0, -12.0);
         my $u = V(2.0, 0.0, 0.0,  1.0,   0.3);

         printf "abs(%s) = %d\n", $v, abs($b);
         my $dot = $u * $v;
         my $sub = $u - $v;
         # etc...

DESCRIPTION

       A simple pure perl module to manipulate vectors of any dimension.

       The function "V", always exported by the module, allows one to create new vectors:

         my $v = V(0, 1, 3, -1);

       Vectors are represented as blessed array references. It is allowed to manipulate the
       arrays directly as far as only real numbers are inserted (well, actually, integers are
       also allowed because from a mathematical point of view, integers are a subset of the real
       numbers).

       Example:

         my $v = V(0.0, 1.0);

         # extending the 2D vector to 3D:
         push @$v, 0.0;

         # setting some component value:
         $v->[0] = 23;

       Vectors can be used in mathematical expressions:

         my $u = V(3, 3, 0);
         $p = $u * $v;       # dot product
         $f = 1.4 * $u + $v; # scalar product and vector addition
         $c = $u x $v;       # cross product, only defined for 3D vectors
         # etc.

       The currently supported operations are:

         + * /
         - (both unary and binary)
         x (cross product for 3D vectors)
         += -= *= /= x=
         == !=
         "" (stringfication)
         abs (returns the norm)
         atan2 (returns the angle between two vectors)

       That, AFAIK, are all the operations that can be applied to vectors.

       When an array reference is used in an operation involving a vector, it is automatically
       upgraded to a vector. For instance:

         my $v = V(1, 2);
         $v += [0, 2];

   Extra methods
       Besides the common mathematical operations described above, the following methods are
       available from the package.

       Note that all these methods are non destructive returning new objects with the result.

       $v = Math::Vector::Real->new(@components)
           Equivalent to "V(@components)".

       $zero = Math::Vector::Real->zero($dim)
           Returns the zero vector of the given dimension.

       $v = Math::Vector::Real->cube($dim, $size)
           Returns a vector of the given dimension with all its components set to $size.

       $u = Math::Vector::Real->axis_versor($dim, $ix)
           Returns a unitary vector of the given dimension parallel to the axis with index $ix
           (0-based).

           For instance:

             Math::Vector::Real->axis_versor(5, 3); # V(0, 0, 0, 1, 0)
             Math::Vector::Real->axis_versor(2, 0); # V(1, 0)

       @b = Math::Vector::Real->canonical_base($dim)
           Returns the canonical base for the vector space of the given dimension.

       $u = $v->versor
           Returns the versor for the given vector.

           It is equivalent to:

             $u = $v / abs($v);

       $wrapped = $w->wrap($v)
           Returns the result of wrapping the given vector in the box (hyper-cube) defined by $w.

           Long description:

           Given the vector "W" and the canonical base "U1, U2, ...Un" such that "W = w1*U1 +
           w2*U2 +...+ wn*Un". For every component "wi" we can consider the infinite set of
           affine hyperplanes perpendicular to "Ui" such that they contain the point "j * wi *
           Ui" being "j" an integer number.

           The combination of all the hyperplanes defined by every component define a grid that
           divides the space into an infinite set of affine hypercubes. Every hypercube can be
           identified by its lower corner indexes "j1, j2, ..., jN" or its lower corner point
           "j1*w1*U1 + j2*w2*U2 +...+ jn*wn*Un".

           Given the vector "V", wrapping it by "W" is equivalent to finding where it lays
           relative to the lower corner point of the hypercube inside the grid containing it:

             Wrapped = V - (j1*w1*U1 + j2*w2*U2 +...+ jn*wn*Un)

             such that ji*wi <= vi <  (ji+1)*wi

       $max = $v->max_component
           Returns the maximum of the absolute values of the vector components.

       $min = $v->min_component
           Returns the minimum of the absolute values of the vector components.

       $d2 = $b->norm2
           Returns the norm of the vector squared.

       $d = $v->dist($u)
           Returns the distance between the two vectors.

       $d = $v->dist2($u)
           Returns the distance between the two vectors squared.

       ($bottom, $top) = Math::Vector::Real->box($v0, $v1, $v2, ...)
           Returns the two corners of a hyper-box containing all the given vectors.

       $v->set($u)
           Equivalent to "$v = $u" but without allocating a new object.

           Note that this method is destructive.

       $d = $v->max_component_index
           Return the index of the vector component with the maximum size.

       ($p, $n) = $v->decompose($u)
           Decompose the given vector $u in two vectors: one parallel to $v and another normal.

           In scalar context returns the normal vector.

       @b = Math::Vector::Real->complementary_base(@v)
           Returns a base for the subspace complementary to the one defined by the base @v.

           The vectors on @v must be linearly independent. Otherwise a division by zero error may
           pop up or probably due to rounding errors, just a wrong result may be generated.

       @b = $v->normal_base
           Returns a set of vectors forming an ortonormal base for the hyperplane normal to $v.

           In scalar context returns just some unitary vector normal to $v.

           Note that this two expressions are equivalent:

             @b = $v->normal_base;
             @b = Math::Vector::Real->complementary_base($v);

       ($i, $j, $k) = $v->rotation_base_3d
           Given a 3D vector, returns a list of 3 vectors forming an orthonormal base where $i
           has the same direction as the given vector $v and "$k = $i x $j".

       @r = $v->rotate_3d($angle, @s)
           Returns the vectors @u rotated around the vector $v an angle $angle in radians in
           anticlockwise direction.

           See <http://en.wikipedia.org/wiki/Rotation_operator_(vector_space)>.

       @s = $center->select_in_ball($radius, $v1, $v2, $v3, ...)
           Selects from the list of given vectors those that lay inside the n-ball determined by
           the given radius and center ($radius and $center respectively).

   Zero vector handling
       Passing the zero vector to some methods (i.e. "versor", "decompose", "normal_base", etc.)
       is not acceptable. In those cases, the module will croak with an "Illegal division by
       zero" error.

       "atan2" is an exceptional case that will return 0 when any of its arguments is the zero
       vector (for consistency with the "atan2" builtin operating over real numbers).

       In any case note that, in practice, rounding errors frequently cause the check for the
       zero vector to fail resulting in numerical instabilities.

       The correct way to handle this problem is to introduce in your code checks of this kind:

         if ($v->norm2 < $epsilon2) {
           croak "$v is too small";
         }

       Or even better, reorder the operations to minimize the chance of instabilities if the
       algorithm allows it.

SEE ALSO

       Math::Vector::Real::Random extends this module with random vector generation methods.

       Math::GSL::Vector, PDL.

       There are other vector manipulation packages in CPAN (Math::Vec, Math::VectorReal,
       Math::Vector), but they can only handle 3 dimensional vectors.

COPYRIGHT AND LICENSE

       Copyright (C) 2009-2012 by Salvador Fandin~o (sfandino@yahoo.com)

       This library is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself, either Perl version 5.10.0 or, at your option, any later version of
       Perl 5 you may have available.