<s>
In	O
mathematics	O
,	O
divided	B-Algorithm
differences	I-Algorithm
is	O
an	O
algorithm	O
,	O
historically	O
used	O
for	O
computing	O
tables	O
of	O
logarithms	O
and	O
trigonometric	O
functions	O
.	O
</s>
<s>
Divided	B-Algorithm
differences	I-Algorithm
is	O
a	O
recursive	O
division	O
process	O
.	O
</s>
<s>
where	O
the	O
are	O
assumed	O
to	O
be	O
pairwise	O
distinct	O
,	O
the	O
forward	O
divided	B-Algorithm
differences	I-Algorithm
are	O
defined	O
as	O
:	O
</s>
<s>
To	O
make	O
the	O
recursive	O
process	O
of	O
computation	O
clearer	O
,	O
the	O
divided	B-Algorithm
differences	I-Algorithm
can	O
be	O
put	O
in	O
tabular	O
form	O
,	O
where	O
the	O
columns	O
correspond	O
to	O
the	O
value	O
of	O
j	O
above	O
,	O
and	O
each	O
entry	O
in	O
the	O
table	O
is	O
computed	O
from	O
the	O
difference	O
of	O
the	O
entries	O
to	O
its	O
immediate	O
lower	O
left	O
and	O
to	O
its	O
immediate	O
upper	O
left	O
,	O
divided	O
by	O
a	O
difference	O
of	O
corresponding	O
x-values	O
:	O
</s>
<s>
Note	O
that	O
the	O
divided	B-Algorithm
difference	I-Algorithm
depends	O
on	O
the	O
values	O
and	O
,	O
but	O
the	O
notation	O
hides	O
the	O
dependency	O
on	O
the	O
x-values	O
.	O
</s>
<s>
Several	O
other	O
notations	O
for	O
the	O
divided	B-Algorithm
difference	I-Algorithm
of	O
the	O
function	O
ƒ	O
on	O
the	O
nodes	O
x0	O
,...,	O
xn	O
are	O
also	O
used	O
,	O
for	O
instance	O
:	O
</s>
<s>
Divided	B-Algorithm
differences	I-Algorithm
for	O
and	O
the	O
first	O
few	O
values	O
of	O
:	O
</s>
<s>
Mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
for	I-Algorithm
divided	I-Algorithm
differences	I-Algorithm
:	O
if	O
is	O
n	O
times	O
differentiable	O
,	O
then	O
for	O
a	O
number	O
in	O
the	O
open	O
interval	O
determined	O
by	O
the	O
smallest	O
and	O
largest	O
of	O
the	O
'	O
s	O
.	O
</s>
<s>
The	O
divided	B-Algorithm
difference	I-Algorithm
scheme	O
can	O
be	O
put	O
into	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
Summarised	O
,	O
the	O
matrices	O
of	O
divided	B-Algorithm
difference	I-Algorithm
schemes	O
with	O
respect	O
to	O
the	O
same	O
set	O
of	O
nodes	O
x	O
form	O
a	O
commutative	O
ring	O
.	O
</s>
<s>
Since	O
is	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
its	O
eigenvalues	O
are	O
obviously	O
.	O
</s>
<s>
Let	O
be	O
a	O
Kronecker	O
delta-like	O
function	O
,	O
that	O
is	O
Obviously	O
,	O
thus	O
is	O
an	O
eigenfunction	B-Algorithm
of	O
the	O
pointwise	O
function	O
multiplication	O
.	O
</s>
<s>
contains	O
the	O
divided	B-Algorithm
difference	I-Algorithm
scheme	O
for	O
the	O
identity	O
function	O
with	O
respect	O
to	O
the	O
nodes	O
,	O
thus	O
contains	O
the	O
divided	B-Algorithm
differences	I-Algorithm
for	O
the	O
power	O
function	O
with	O
exponent	O
.	O
</s>
<s>
You	O
can	O
compute	O
the	O
divided	B-Algorithm
difference	I-Algorithm
scheme	O
for	O
by	O
applying	O
the	O
corresponding	O
matrix	O
series	O
to	O
:	O
</s>
<s>
This	O
is	O
a	O
consequence	O
of	O
the	O
Peano	B-Algorithm
kernel	I-Algorithm
theorem	I-Algorithm
;	O
it	O
is	O
called	O
the	O
Peano	O
form	O
of	O
the	O
divided	B-Algorithm
differences	I-Algorithm
and	O
is	O
the	O
Peano	B-Algorithm
kernel	I-Algorithm
for	O
the	O
divided	B-Algorithm
differences	I-Algorithm
,	O
all	O
named	O
after	O
Giuseppe	O
Peano	O
.	O
</s>
<s>
They	O
are	O
easier	O
to	O
calculate	O
than	O
the	O
more	O
general	O
divided	B-Algorithm
differences	I-Algorithm
.	O
</s>
