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lib/Math/Algebra/Symbols/Sum.pm  view on Meta::CPAN

     {$s->{t} = $s->{t}->clone->divideInt($d)   ->z;
      $s->{s} = $s->{s}->clone->timesInt ($d*$d)->z;
     }
   }

# Eliminate duplicate squared factors
  for my $s(@s)
   {my $F = factorize($s->{s}->c);
    my $p = 1;
    for my $f(keys(%$F))
     {$p *= $f**(int($F->{$f}/2)) if $F->{$f} > 1;
     }
    $s->{t} = $s->{t}->clone->timesInt ($p)   ->z;
    $s->{s} = $s->{s}->clone->divideInt($p*$p)->z;

    if ($s->{s}->isOne)
     {push @t, $s->{t}->removeSqrt;
     }
    else
     {push @t, $s->{t}->clone->Sqrt($s->{$s})->z;
     }
   }

# Result
  sigma(@t);
 }


=head3 isEqualSqrt

Check whether one sum is equal to another after multiplying out sqrts.

=cut


sub isEqualSqrt($)
 {my ($C) = @_;

#_______________________________________________________________________
# Each sqrt
#_______________________________________________________________________

  for(1..99)
   {$C = $C->normalizeSqrts;
    my @s = grep { defined($_->Sqrt)} $C->t;
    my @n = grep {!defined($_->Sqrt)} $C->t;
    last unless scalar(@s) > 0;

#_______________________________________________________________________
# Partition by square roots.
#_______________________________________________________________________

    my %S = ();
    for my $t(@s)
     {my $s = $t->Sqrt;
      my $S = $s->signature;
      push @{$S{$S}}, $t;
     }

#_______________________________________________________________________
# Square each partitions, as required by the formulae below.
#_______________________________________________________________________

    my @t;
    push @t, sigma(@n)->power($two) if scalar(@n);  # Non sqrt partition
    for my $s(keys(%S))
     {push @t, sigma(@{$S{$s}})->power($two);       # Sqrt partition
     }

#_______________________________________________________________________
# I can multiply out upto 4 square roots using the formulae below.
# There are formula to multiply out more than 4 sqrts, but they are big.
# These formulae are obtained by squaring out and rearranging:
# sqrt(a)+sqrt(b)+sqrt(c)+sqrt(d) == 0 until no sqrts remain, and
# then matching terms to produce optimal execution.
# This remarkable result was obtained with the help of this package:
# demonstrating its utility in optimizing complex calculations written
# in Perl: which in of itself cannot optimize broadly.
#_______________________________________________________________________

    my $ns = scalar(@t);
# 2016/01/26 12:29:28 No need to die
#   $ns < 5 or die "There are $ns square roots present.  I can handle less than 5";

    my ($a, $b, $c, $d, $e) = @t;

    if    ($ns == 1)
     {$C = $a;
     }

=pod
       𝗮+𝗯        = 0
 => a+b+2𝗮𝗯 = 0
 => 2𝗮𝗯        = -a-b
 => 0       =  aa+bb-2ab
            = (a-b)**2 or (b-a)**2
=cut

    elsif ($ns == 2)
     {$C = $a-$b;
     }
    elsif ($ns == 3)
     {$C = -$a**2+2*$a*$b-$b**2+2*$c*$a+2*$c*$b-$c**2;
     }
    elsif ($ns == 4)
     {my $a2  = $a  * $a;
      my $a3  = $a2 * $a;
      my $a4  = $a3 * $a;
      my $b2  = $b  * $b;
      my $b3  = $b2 * $b;
      my $b4  = $b3 * $b;
      my $c2  = $c  * $c;
      my $c3  = $c2 * $c;
      my $c4  = $c3 * $c;
      my $d2  = $d  * $d;
      my $d3  = $d2 * $d;
      my $d4  = $d3 * $d;
      my $bpd = $b  + $d;
      my $bpc = $b  + $c;
      my $cpd = $c  + $d;
      $C =
-  ($a4 + $b4 + $c4 + $d4)
+ 4*(
   +$a3*($b+$cpd)+$b3*($a+$cpd)+$c3*($a+$bpd)+$d3*($a+$bpc)
   -$a2*($b *($cpd)+ $c*$d)
   -$a *($b2*($cpd)+$d2*($bpc))
    )

- 6*($a2*$b2+($a2+$b2)*($c2+$d2)+$c2*$d2)

- 4*$c*($b2*$d+$b*$d2)
- 4*$c2*($a*($bpd)+$b*$d)
+40*$c*$a*$b*$d