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Symbolic calculations with operator overload

by ambrus (Abbot)
on Oct 09, 2003 at 06:21 UTC ( #297829=note: print w/replies, xml ) Need Help??

in reply to Using overload. Any complete (simple) examples?

Here's some code I wrote last summer (and changerd a bit now).
#!/usr/bin/perl -w # # We will do operations with multi-variable real polinomials and 3d ve +ctors # of these. These are slow algorithms, but might be good examples for # object orientation ond operator overloading in Perl. It is also not # commented well. Many functions are defined but not used, altough I +have # tested them. It is not difficult to add some calculations that use # these functions. #
# Note that the Vector class knows nothing about the Polinom class, st +ill # it can work with vectors of polinomials as well as vectors of number +s # transparently. This is the real advantage of operator overloading, +apart # from easy usage. # # # head ------------------------------------------- use strict; # Polinom ---------------------------------------- { package Polinom; use overload '+', "add", '-', "sub", '*', "mul", '**', "power", '""', "str", '0+', "num", 'bool', "bool", ; # The internal structure of Polinom objects is like this: # 6*x^2+3*x*y+8*x-4 ==> {"x*x", 6, "x*y", 3, "x", 8, "", -4} # The keys of the hash are variables separated with "*", sorted alpha- # betically, so "y*x" cannot be a key in such a hash. The values are # the coeffcients, those are simply a Perl number. # xtract: this internal function does the magic of converting a number # to a polinomial when used in a polinomial operation sub xtract { my $a= shift; if ( ref($a) ) { return %$a } else { return $a ? ("", $a) : (); } } # constant (useless, because a number can always be used instd a polin +om) sub const { return bless {"", $_[1]}, $_[0]; } # variable constructor, returns more Polinoms if given more strings sub var { local $_; my $t= shift; for (@_) { m|^[a-zA-Z_][a-zA-Z0-9_\-.\[\]]*$| or die "wrong variable name for $t->var"; } my @v= (); for (@_) { push @v, bless {$_, 1}, $t } if (wantarray) { return @v; } else { @_==1 or die 'cannot return list to scalar context'; return $v[0]; } } # add method sub add { my $t= ref $_[0]; my %a= %{shift()}; my %b= xtract(shift()); for my $k (keys %b) { exists $a{$k} or $a{$k}= 0; $a{$k}+= $b{$k}; $a{$k} or delete $a{$k}; }; return bless {%a}, $t; }; # subtract method sub sub { my $t= ref $_[0]; my $r= $_[2]; my %a= xtract($_[$r?1:0]); my %b= xtract($_[$r?0:1]); for my $k (keys %b) { exists $a{$k} or $a{$k}= 0; $a{$k}-= $b{$k}; $a{$k} or delete $a{$k}; }; return bless {%a}, $t; }; # monom-multiply (internal) sub mulmonom ($$) { # <---- ELIMINAL @c, STRING HASZNALJ HLYTTE my @a= split /\*/, shift; my @b= split /\*/, shift; my @c= (); while ( @a && @b ) { push @c, ( $a[0] le $b[0] ? shift @a : shift @b ); }; @a ? push (@c, @a) : push (@c, @b); return join "*", @c; } # multiply method # handles mul with number as a separate case sub mul { my $t= ref $_[0]; my %a= %{shift()}; my $b= shift; if (!ref $b) { $b or return bless {}, $t; for my $v (@a{keys %a}) { $v*= $b; }; return bless {%a}, $t; } else { my %b= %$b; my %c= (); my $k; for my $a (keys %a) { for my $b (keys %b) { $k= mulmonom($a,$b); exists $c{$k} or $c{$k}= 0; $c{$k}+= $a{$a}*$b{$b}; $c{$k} or delete $c{$k}; } } return bless {%c}, $t; }; }; # multiply method with debug sub mul_g { my $t= ref $_[0]; my %a= %{shift()}; my $b= shift; if (!ref $b) { $b or return bless {}, $t; for my $v (@a{keys %a}) { $v*= $b; }; return bless {%a}, $t; } else { my %b= %$b; my %c= (); my $k; my $ga= keys(%a); print "[poli-mul ".$ga." ".keys(%b).": "; for my $a (keys %a) { print $ga--." "; for my $b (keys %b) { $k= mulmonom($a,$b); exists $c{$k} or $c{$k}= 0; $c{$k}+= $a{$a}*$b{$b}; $c{$k} or delete $c{$k}; } } print "]\n"; return bless {%c}, $t; }; }; # integer power method sub power { my $t= ref $_[0]; my ($a, $n, $rev)= @_; $rev and die "Wrong op: scalar**Polinom"; $n==int($n) or die "Wrong op: Polinom**fraction"; my $x= 1; my $c= 1; while ($n) { $x&$n and $c*= $a, $n-= $x; $a*= $a; $x+= $x; } $c; }; # with: method for applying polinomials # Usage: $polinomial->with (variable=>value, variable=>value, ...); # Where variable can be string or polinomial consisting of just a vari +able; # values can be numbers or polinomials (but numbers are faster of cour +se). # Substitutions are not done simultanously, so do not try to do # $poli->with ($x->$y, $y->x); # _with_{n,z,p}: internal functions for setting a variable to {number, # zero,polinomial} resp. sub _with_n ($$$) { # <--- TEST local $_; my %a= %{shift()}; my $v= shift; my $w= shift; my %r= (); for my $k (keys %a) { my @c= split /\*/, $k; my $n= ""; my $y= $a{$k}; for (@c) { if ($v eq $_) { $y*= $w } else { $n.= "*$_" } } $n=~ s/^\*//; exists $r{$n} or $r{$n}= 0; $r{$n}+= $y; } return %r; } sub _with_z ($$) { my %a= %{shift()}; my $v= shift; my $vre= qr!(^|\*)$v(\*|$)!; for my $k (keys %a) { $k=~$vre and delete $a{$k}; } return %a; } sub _with_p ($$$$) { local $_; my %a=%{shift()}; my $v= shift; my $w= shift; my $t= shift; my $r= 0; my $x; for my $k (keys %a) { my @c= split /\*/, $k; $x= $a{$k}; for (@c) { if ($v eq $_) { $x*= $w } else { $x*= bless({$_,1},$t) } } $r+= $x; } return %$r; } # now the definition of with: sub with { my $o= shift; my %r= %$o; while (@_>=2) { my $v= shift; my $w= shift; if (ref $v) { my @c= %$v; @c==2 || !$c[0] || $c[0]=~/\*/ || $c[1]!=1 or die ref($o).'->with called with wrong polinom as va +r'; $v= $c[0]; }; if (ref $w) {%r= _with_p \%r, $v, $w, ref $o} elsif ($w) {%r= _with_n \%r, $v, $w} else {%r= _with_z \%r, $v} } @_ and die 'odd-sized hash to '.ref $o.'->with'; return bless {%r}, ref $o; } # overloaded stringify method -- does not print powers of variables. # A parital solution would be to filter the output through # perl -wpe's!([a-z])((\*\1)+)!"$1^".(length($2)/2+1)!ge' # which transforms 3*x*x*x-4*x*x to 3*x^3-4*x^2. sub str { my %a= xtract shift; my $r= ""; my $v; for my $k (sort keys %a) { $v= $a{$k}; $r.= $v<0?"-":$r?"+":""; if ($k) { $r.= abs($v)."*" unless abs($v)==1; } else {$r.= abs($v); } $r.= $k; }; $r or $r= "0"; return $r; }; # convert to number method, returns undef if polinom is not constant sub num { my @a= %{shift()}; if (@a>2) { return undef; } if (@a==0) { return 0; } if ($a[0]) { return undef; } return $a[1]; } # convert to boolean method sub bool { return scalar keys(%{shift()}); } } # Vector ------------------------------------------ # Vector objects are vectors of 3 perl scalars. See note at top. { package Vector; use overload '*', "mul", 'x', "cross", '""', "str", ; # the xi+yj+zk constructor sub new { @_==4 or die "a Vector should have 3 coordinates"; my $o= [@_[1..3]]; return bless $o, $_[0]; } # the 0-vector constructor sub zero { my $o= [0,0,0]; return bless $o, $_[0]; } # multiply method, overloads "*" operator # muliplies with scalar or calculates dot product as appropriate sub mul { local $_; my $a= shift; my $b= shift; if ( ref($b) && $b->isa("Vector") ) { my $x= $$a[0]*$$b[0]+$$a[1]*$$b[1]+$$a[2]*$$b[2]; return $x; } else { my @x= ($$a[0]*$b, $$a[1]*$b, $$a[2]*$b); return bless [@x], ref $a; } } # multiply method with debug (to see how much you have to wait) sub mul_g { local $_; my $a= shift; my $b= shift; if ( ref($b) && $b->isa("Vector") ) { print "[vector-inmul "; my $x= 0; $x+= $$a[0]->mul_g($$b[0]); $x+= $$a[1]->mul_g($$b[1]); $x+= $$a[2]->mul_g($$b[2]); print "]\n"; return $x; } else { my @x= ($$a[0]*$b, $$a[1]*$b, $$a[2]*$b); return bless [@x], ref $a; } } # cross product method, overloads "x" sub cross { local $_; my $a= shift; my $b= shift; my @x= ( $$a[1]*$$b[2]-$$a[2]*$$b[1], $$a[2]*$$b[0]-$$a[0]*$$b[2], $$a[0]*$$b[1]-$$a[1]*$$b[0]); return bless [@x], ref $a; } # stringify method sub str { my @o= @{shift()}; return "(".$o[0].",".$o[1].",".$o[2].")"; } } # Test -------------------------------------------- { # empty-subclass test package Poland; use vars qw{@ISA}; @ISA= "Polinom"; } { # empty-subclass test package Hector; use vars qw{@ISA}; @ISA= "Vector"; } no strict "vars"; # This test will calculate Chebyshev-polinomials defined by the recurs +ion # t(0,x)= 1; t(1,x)= x; # t(n+1,x)= 2*x*t(n,x)-t(n-1,x) print "Chebyshev-polinomials\n"; # define the variables x $x= var Polinom qw!x!; ($n, $u, $t)= (1, 1, $x); while ($n<12) { print "t($n,$x)= ", $t, ";\n"; print "2*t($n,$x/2)= ", 2*$t->with ($x, 1/2*$x), ";\n"; print "t($n,-1)= ", $t->with ($x, -1), ";\n"; print "t($n,1/2)= ", $t->with ($x, 1/2), ";\n"; ($n, $u, $t)= ($n+1, $t, 2*$x*$t-$u); }; print "\n"; # This test will verify the Pappos-Pascal-theorem. The final result # should be zero. This test is quite slow. # variables $a= Hector->new(Poland->var(qw(a1 a2 a3))); $b= Hector->new(Poland->var(qw(b1 b2 b3))); $c= Hector->new(Poland->var(qw(c1 c2 c3))); $e= Hector->new(Poland->var(qw(d1 d2 d3))); $f= Hector->new(Poland->var(qw(e1 e2 e3))); $|= 1; print "Verifying the Pappos-Pascal theorem\n"; $v= (($c x $e)x($a x $f)) x (($a x $e)x($c x $f)); print "v= "; # if you see the equation sign, the calculation's been done, # only stringification remains print $v, "\n"; $w= ( (($a x $e)x($b x $f)) x (($b x $e)x($a x $f)) ) x ( (($b x $e)x($c x $f)) x (($c x $e)x($b x $f)) ); print "w= "; print $w, "\n"; $r= $v->mul_g($w); print "v.w="; print $r, "\n". "The final result above should be zero.\n"; __END__

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