<s>
While	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
of	O
dimension	O
can	O
be	O
represented	O
efficiently	O
in	O
units	O
of	O
storage	O
by	O
storing	O
only	O
its	O
non-zero	O
entries	O
,	O
a	O
non-sparse	O
matrix	O
would	O
require	O
units	O
of	O
storage	O
,	O
and	O
using	O
this	O
type	O
of	O
matrices	O
for	O
large	O
problems	O
would	O
therefore	O
be	O
prohibitively	O
expensive	O
in	O
terms	O
of	O
storage	O
and	O
computing	O
time	O
.	O
</s>
<s>
Hierarchical	B-Algorithm
matrices	I-Algorithm
provide	O
an	O
approximation	O
requiring	O
only	O
units	O
of	O
storage	O
,	O
where	O
is	O
a	O
parameter	O
controlling	O
the	O
accuracy	O
of	O
the	O
approximation	O
.	O
</s>
<s>
Hierarchical	B-Algorithm
matrices	I-Algorithm
rely	O
on	O
local	O
low-rank	O
approximations	O
:	O
</s>
<s>
In	O
this	O
sense	O
,	O
hierarchical	B-Algorithm
matrices	I-Algorithm
can	O
be	O
considered	O
the	O
algebraic	O
counterparts	O
of	O
these	O
techniques	O
.	O
</s>
<s>
appearing	O
in	O
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
where	O
and	O
are	O
families	O
of	O
finite	B-Application
element	I-Application
basis	O
functions	O
.	O
</s>
<s>
and	O
spectral	B-Algorithm
method	I-Algorithm
can	O
be	O
approximated	O
by	O
a	O
hierarchical	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
most	O
important	O
innovation	O
of	O
the	O
hierarchical	B-Algorithm
matrix	I-Algorithm
method	O
is	O
the	O
development	O
of	O
efficient	O
algorithms	O
for	O
performing	O
(	O
approximate	O
)	O
matrix	O
arithmetic	O
operations	O
on	O
non-sparse	O
matrices	O
,	O
e.g.	O
,	O
to	O
compute	O
approximate	O
inverses	O
,	O
LU	O
decompositions	O
and	O
solutions	O
to	O
matrix	O
equations	O
.	O
</s>
<s>
for	O
hierarchical	B-Algorithm
matrices	I-Algorithm
and	O
a	O
scalar	O
factor	O
.	O
</s>
<s>
In	O
order	O
to	O
treat	O
very	O
large	O
problems	O
,	O
the	O
structure	O
of	O
hierarchical	B-Algorithm
matrices	I-Algorithm
can	O
be	O
improved	O
:	O
</s>
<s>
replace	O
the	O
general	O
low-rank	O
structure	O
of	O
the	O
blocks	O
by	O
a	O
hierarchical	O
representation	O
closely	O
related	O
to	O
the	O
fast	B-Algorithm
multipole	I-Algorithm
method	I-Algorithm
in	O
order	O
to	O
reduce	O
the	O
storage	O
complexity	O
to	O
.	O
</s>
<s>
is	O
an	O
implementation	O
of	O
the	O
core	O
hierarchical	B-Algorithm
matrix	I-Algorithm
algorithms	O
for	O
commercial	O
applications	O
.	O
</s>
<s>
is	O
an	O
open	O
source	O
implementation	O
of	O
hierarchical	B-Algorithm
matrix	I-Algorithm
algorithms	O
intended	O
for	O
research	O
and	O
teaching	O
.	O
</s>
<s>
is	O
a	O
Julia	O
package	O
implementing	O
hierarchical	B-Algorithm
matrices	I-Algorithm
.	O
</s>
