<s>
In	O
mathematics	O
,	O
the	O
Kronecker	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
discrete	I-Algorithm
Laplacians	I-Algorithm
,	O
named	O
after	O
Leopold	O
Kronecker	O
,	O
is	O
a	O
discrete	O
version	O
of	O
the	O
separation	O
of	O
variables	O
for	O
the	O
continuous	O
Laplacian	O
in	O
a	O
rectangular	O
cuboid	O
domain	O
.	O
</s>
<s>
In	O
a	O
general	O
situation	O
of	O
the	O
separation	O
of	O
variables	O
in	O
the	O
discrete	O
case	O
,	O
the	O
multidimensional	O
discrete	B-Algorithm
Laplacian	I-Algorithm
is	O
a	O
Kronecker	O
sum	O
of	O
1D	O
discrete	O
Laplacians	O
.	O
</s>
<s>
Both	O
and	O
must	O
correspond	O
to	O
the	O
case	O
of	O
the	O
homogeneous	O
Dirichlet	O
boundary	O
condition	O
at	O
end	O
points	O
of	O
the	O
x	O
-	O
and	O
y-intervals	O
,	O
in	O
order	O
to	O
generate	O
the	O
2D	O
discrete	B-Algorithm
Laplacian	I-Algorithm
L	O
corresponding	O
to	O
the	O
homogeneous	O
Dirichlet	O
boundary	O
condition	O
everywhere	O
on	O
the	O
boundary	O
of	O
the	O
rectangular	O
domain	O
.	O
</s>
<s>
Here	O
is	O
a	O
sample	O
OCTAVE/MATLAB	O
code	O
to	O
compute	O
L	O
on	O
the	O
regular	O
1015	O
2D	O
grid	O
:	O
</s>
<s>
The	O
eigenvalues	O
and	O
eigenvectors	O
of	O
the	O
standard	O
central	O
difference	O
approximation	O
of	O
the	O
second	O
derivative	O
on	O
an	O
interval	O
for	O
traditional	O
combinations	O
of	O
boundary	O
conditions	O
at	O
the	O
interval	O
end	O
points	O
are	O
well	B-Algorithm
known	I-Algorithm
.	O
</s>
<s>
Each	O
1D	O
discrete	B-Algorithm
Laplacian	I-Algorithm
must	O
correspond	O
to	O
the	O
case	O
of	O
the	O
homogeneous	O
Dirichlet	O
boundary	O
condition	O
,	O
in	O
order	O
to	O
generate	O
the	O
3D	O
discrete	B-Algorithm
Laplacian	I-Algorithm
L	O
corresponding	O
to	O
the	O
homogeneous	O
Dirichlet	O
boundary	O
condition	O
everywhere	O
on	O
the	O
boundary	O
.	O
</s>
<s>
An	O
OCTAVE/MATLAB	O
code	O
is	O
available	O
under	O
a	O
BSD	B-Operating_System
License	I-Operating_System
,	O
which	O
computes	O
the	O
sparse	O
matrix	O
of	O
the	O
1	O
,	O
2D	O
,	O
and	O
3D	O
negative	O
Laplacians	O
on	O
a	O
rectangular	O
grid	O
for	O
combinations	O
of	O
Dirichlet	O
,	O
Neumann	O
,	O
and	O
Periodic	O
boundary	O
conditions	O
using	O
Kronecker	O
sums	O
of	O
discrete	O
1D	O
Laplacians	O
.	O
</s>
