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fuzzy test for row equivalency #1427

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b83c694
fuzzy test for row equivalency
Alex-Jordan Jun 2, 2026
3ffd905
normalized gram schmit method
Alex-Jordan Jun 20, 2026
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137 changes: 137 additions & 0 deletions lib/Value/Matrix.pm
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Original file line number Diff line number Diff line change
Expand Up @@ -631,6 +631,143 @@ sub isRREF {
return 1;
}

=head3 C<isREQ>

Fuzzy, probabilistic check for if two matrices are row equivalent.

Usage:

$A = Matrix([3, 1], [1, 0.3333]);
$B = Matrix([1, 1/3], [0, 0]);
$A->isREQ($B); # true

=cut

sub isREQ {
my ($l, $r) = @_;
my @ldim = $l->dimensions;
my @rdim = $r->dimensions;
Value::Error('Cannot compare row equivalency of matrices of different degree') unless scalar @ldim == scalar @rdim;
for my $i (0 .. $#ldim) {
Value::Error('Cannot compare row equivalency because matrices differ in the '
. $l->NameForNumber($i + 1)
. ' dimension')
unless $ldim[$i] == $rdim[$i];
}
return 1 if $l->isZero && $r->isZero;
return 0 if $l->isZero && !$r->isZero || !$l->isZero && $r->isZero;

if (scalar @ldim == 1) {
# simply need to determine if rows are parallel
my ($lk, $rk);
for my $i (1 .. $ldim[0]) {
if ($l->element($i) != 0) {
$lk = $l->element($i);
$rk = $r->element($i);
last;
}
}
return $lk * $r == $rk * $l;
} elsif (scalar @ldim == 2) {
my @rgs = $r->NGS;
my @lgs = $l->NGS;

return 0 if scalar @lgs != scalar @rgs;

# project each row from $rrows onto @lgs;
# if the complement is nonzero, the row spaces disagree
my @rrows = map { $r->new($_) } $r->value;
for my $v (@rrows) {
my $proj = $r->new((0) x $ldim[1]);
for my $w (@lgs) {
$proj += ($v * $w) * $w;
}
return 0 unless $v == $proj;
}
# and vice versa
my @lrows = map { $l->new($_) } $l->value;
for my $v (@lrows) {
my $proj = $r->new((0) x $rdim[1]);
for my $w (@rgs) {
$proj += ($v * $w) * $w;
}
return 0 unless $v == $proj;
}

# if we got this far, the row spaces are (fuzzy) equal
# so the matrices are row equivalent
return 1;
} else {
for my $i (1 .. $ldim[0]) {
return 0 unless $l->new($l->{data}[ $i - 1 ])->isREQ($r->new($r->{data}[ $i - 1 ]));
}
return 1;
}
}

=head3 C<NGS>

Normalized Gram Schmidt basis for a given list of vectors. This uses "fuzzy" equivalency to avoid machine rounding
issues. If called as a method on a Matrix, the rows of that Matrix are used as the list of vectors. If called as a
method with the option C<< cols => 1 >>, then the columns are used as the list of vectors.

In array context, returns an array of 1D Matrix objects. In scalar context, returns a Matrix with those rows.

If C<< cols => 1 >> was used, return the Matrix objects in column form.


Usage:

$A = Matrix([1, 2, 2], [2, 2, 1]);

@X = $A->NGS # @X is (Matrix(1, 2, 2)/3, Matrix(10, 2, -7)/sqrt(153))
@X = Value::Matrix->NGS([1, 2, 2], [2, 2, 1]) # @X is the same as above
$X = $A->NGS # $X is Matrix([1/3, 2/3, 2/3], [10/sqrt(153), 2/sqrt(153), -7/sqrt(153)])
$X = Value::Matrix->NGS([1, 2, 2], [2, 2, 1]) # $X is the same as above

@X = $A->NGS(cols => 1) # @X is (Matrix([1/sqrt(5)], [2/sqrt(5)]), Matrix([2/sqrt(5)], [-1/sqrt(5)]))
$X = $A->NGS(cols => 1) # $X is Matrix([1/sqrt(5), 2/sqrt(5)], [2/sqrt(5), -1/sqrt(5)])

=cut

sub NGS {
my ($self, @args) = @_;
my %options;
my @rows;
if (ref($self) eq 'Value::Matrix') {
%options = @args;
@rows = $options{cols} ? $self->transpose->value : $self->value;
} else {
@rows = @args;
}
Value::Error('You must provide vectors to apply Gram Schmidt to') if !@rows;
@rows = map { Value::Matrix->new($_) } @rows;
@rows = grep { !$_->isZero } @rows;
Value::Error('You must provide at least one nonzero row for Gram Schmidt') unless @rows;
my $n = ($rows[0]->dimensions)[0];
for my $r (@rows) {
Value::Error('Rows provided for Gram Schmidt should not be nested arrays') unless $r->degree == 1;
Value::Error('All rows provided for Gram Schmidt should have the same length') unless ($r->dimensions)[0] == $n;
}
@rows = map { $_ / ($_ * $_)**0.5 } @rows;
my @gs = ($rows[0]);
for my $i (1 .. $#rows) {
my $proj = Value::Matrix->new((0) x $n);
for my $v (@gs) {
$proj += ($rows[$i] * $v) * $v;
}
my $gs = $rows[$i] - $proj;
push @gs, $gs / ($gs * $gs)**0.5 unless $rows[$i] == $proj;
}
return
$options{cols}
? wantarray
? map { $_->transpose } @gs
: Value::Matrix->new(@gs)->transpose
: wantarray ? @gs
: Value::Matrix->new(@gs);
}

sub _isNumber {
my $n = shift;
return Value::isNumber($n) || Value::classMatch($n, 'Fraction');
Expand Down
82 changes: 82 additions & 0 deletions t/math_objects/matrix.t
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Expand Up @@ -180,6 +180,88 @@ subtest 'Test if Matrix is in (R)REF' => sub {
ok !$B4->isRREF, "$B4 is not in RREF";
};

subtest 'Check if two matrices are (fuzzy) row equivalent' => sub {
my $A1 = Matrix(1, 2, 3);
my $A2 = Matrix([ 1, 2, 3 ], [ 4, 5, 6 ]);
my $A3 = Matrix([ [ 1, 2 ], [ 3, 4 ] ], [ [ 5, 6 ], [ 7, 8 ] ]);
my $B1 = Matrix(2, 4, 6);
my $B2 = Matrix([ 1, 2 ], [ 3, 4 ], [ 5, 6 ]);
my $C1 = Matrix(0, 4, 6);
my $Z1 = Matrix([ 0, 0, 0 ], [ 0, 0, 0 ]);
my $Z2 = Matrix([ 0, 0, 0 ], [ 0, 0, 10**-14 ]);

my $M1 = Matrix([ 3, 1 ], [ 1, 1 / 3 ]);
my $M2 = Matrix([ 3, 1 ], [ 0, 0 ]);
my $M3 = Matrix([ 1, 0.3333 ], [ 0, 0 ]);
my $M4 = Matrix([ 1, 0.33 ], [ 0, 0 ]);
my $M5 = Matrix([ 6, 2 ], [ 9, 3 ]);
my $M6 = Matrix([ 3, 1 ], [ 1, 3 ]);

my $N1 = Matrix([ [ 1, 2 ], [ 2, 4 ] ], [ [ 1, 2 ], [ 3, 4 ] ]);
my $N2 = Matrix([ [ 3, 6 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ]);
my $N3 = Matrix([ [ 3, 6 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ]);

ok $Z1->isREQ($Z2), 'Zero matrices are row equivalent';
ok !$Z1->isREQ($A2), 'A zero matrix is not row equivalent to a nonzero matrix';
ok $A1->isREQ($B1), 'Parallel degree 1 matrices are row equivalent';
ok !$C1->isREQ($B1), 'Non-parallel degree 1 matrices are not row equivalent';
ok $M2->isREQ($M5), 'Row equivalent matrices are row equivalent';
ok $M2->isREQ($M1), 'Row equivalent matrices are row equivalent, even with some machine rounding';
ok $M2->isREQ($M3), 'Row equivalent matrices are row equivalent, even with student rounding';
ok !$M2->isREQ($M4), 'Matrices are not row equivalent if rounding is too much';
ok !$M2->isREQ($M6), 'Non-row equivalent matrices are not row equivalent';
ok $N1->isREQ($N2), 'Row equivalent matrices are row equivalent, even at degree above 2';
ok !$N1->isREQ($N3), 'Non-row equivalent matrices are not row equivalent, even at degree above 2';

like dies {
$A2->isREQ($A3);
}, qr/Cannot compare row equivalency of matrices of different degree/,
'Test that error is thrown for comparing matrices of differing degrees';
like dies {
$A2->isREQ($B2);
}, qr/Cannot compare row equivalency because matrices differ in the first dimension/,
'Test that error is thrown for comparing matrices of differing dimensioon';

};

subtest 'Normalized Gram Schmidt' => sub {
my @A = ([ 1, 2, 2 ], [ 2, 2, 1 ]);
my $A = Matrix(@A);
my @NGSA = $A->NGS;
my @NGSAc = $A->NGS(cols => 1);
my @NGSVMA = Value::Matrix->NGS(@A);
my $NGSA = $A->NGS;
my $NGSAc = $A->NGS(cols => 1);
my $NGSVMA = Value::Matrix->NGS(@A);

is "@NGSA", "[0.333333,0.666667,0.666667] [0.808452,0.16169,-0.565916]",
'Test that NGS finds a correct normalized GS basis';
is "@NGSAc", "[[0.447214],[0.894427]] [[0.894427],[-0.447214]]",
'Test that NGS finds a correct normalized GS basis';
is "@NGSVMA", "[0.333333,0.666667,0.666667] [0.808452,0.16169,-0.565916]",
'Test that NGS finds a correct normalized GS basis';
is "$NGSA", "[[0.333333,0.666667,0.666667],[0.808452,0.16169,-0.565916]]",
'Test that NGS finds a correct normalized GS basis';
is "$NGSAc", "[[0.447214,0.894427],[0.894427,-0.447214]]", 'Test that NGS finds a correct normalized GS basis';
is "$NGSVMA", "[[0.333333,0.666667,0.666667],[0.808452,0.16169,-0.565916]]",
'Test that NGS finds a correct normalized GS basis';

like dies {
Value::Matrix->NGS();
}, qr/You must provide vectors to apply Gram Schmidt to/, 'Test that you must pass something as an argument';
like dies {
Value::Matrix->NGS([ 0, 0 ], [ 0, 10**-16 ]);
}, qr/You must provide at least one nonzero row for Gram Schmidt/,
'Test that you must pass at least one nonzero row';
like dies {
Value::Matrix->NGS([ [ 1, 0 ], [ 1, 0 ] ]);
}, qr/Rows provided for Gram Schmidt should not be nested arrays/,
'Test that rows are rows, not nested matrices';
like dies {
Value::Matrix->NGS([ 1, 0 ], [ 1, 0, 0 ]);
}, qr/All rows provided for Gram Schmidt should have the same length/, 'Test that rows have same length';
};

subtest 'Transpose a Matrix' => sub {
my $A = Matrix([ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ] ]);
my $B = Matrix([ [ 1, 5, 9 ], [ 2, 6, 10 ], [ 3, 7, 11 ], [ 4, 8, 12 ] ]);
Expand Down