<s>
In	O
mathematics	O
,	O
the	O
weakly	B-Algorithm
chained	I-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrices	I-Algorithm
are	O
a	O
family	O
of	O
nonsingular	O
matrices	O
that	O
include	O
the	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrices	O
.	O
</s>
<s>
We	O
say	O
row	O
of	O
a	O
complex	O
matrix	O
is	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
(	O
SDD	O
)	O
if	O
.	O
</s>
<s>
Weakly	O
diagonally	B-Algorithm
dominant	I-Algorithm
(	O
WDD	O
)	O
is	O
defined	O
with	O
instead	O
.	O
</s>
<s>
Recalling	O
that	O
an	O
irreducible	O
matrix	O
is	O
one	O
whose	O
associated	O
directed	O
graph	O
is	O
strongly	O
connected	O
,	O
a	O
trivial	O
corollary	O
of	O
the	O
above	O
is	O
that	O
an	O
irreducibly	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
(	O
i.e.	O
,	O
an	O
irreducible	O
WDD	O
matrix	O
with	O
at	O
least	O
one	O
SDD	O
row	O
)	O
is	O
nonsingular	O
.	O
</s>
<s>
is	O
a	O
nonsingular	O
WDD	O
M-matrix	B-Algorithm
.	O
</s>
<s>
is	O
a	O
nonsingular	O
WDD	O
L-matrix	B-Algorithm
;	O
</s>
<s>
is	O
a	O
WCDD	O
L-matrix	B-Algorithm
;	O
</s>
<s>
In	O
fact	O
,	O
WCDD	O
L-matrices	B-Algorithm
were	O
studied	O
(	O
by	O
James	O
H	O
.	O
Bramble	O
and	O
B	O
.	O
E	O
.	O
Hubbard	O
)	O
as	O
early	O
as	O
1964	O
in	O
a	O
journal	O
article	O
in	O
which	O
they	O
appear	O
under	O
the	O
alternate	O
name	O
of	O
matrices	O
of	O
positive	O
type	O
.	O
</s>
<s>
Moreover	O
,	O
if	O
is	O
an	O
WCDD	O
L-matrix	B-Algorithm
,	O
we	O
can	O
bound	O
its	O
inverse	O
as	O
follows	O
:	O
</s>
<s>
Tighter	O
bounds	O
for	O
the	O
inverse	O
of	O
a	O
WCDD	O
L-matrix	B-Algorithm
are	O
known	O
.	O
</s>
<s>
Due	O
to	O
their	O
relationship	O
with	O
M-matrices	B-Algorithm
(	O
see	O
above	O
)	O
,	O
WCDD	O
matrices	O
appear	O
often	O
in	O
practical	O
applications	O
.	O
</s>
<s>
WCDD	O
L-matrices	B-Algorithm
arise	O
naturally	O
from	O
monotone	O
approximation	O
schemes	O
for	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
Note	O
that	O
is	O
a	O
WCDD	O
L-matrix	B-Algorithm
.	O
</s>
