<s>
A	O
finite	B-Algorithm
difference	I-Algorithm
is	O
a	O
mathematical	O
expression	O
of	O
the	O
form	O
.	O
</s>
<s>
If	O
a	O
finite	B-Algorithm
difference	I-Algorithm
is	O
divided	O
by	O
,	O
one	O
gets	O
a	O
difference	B-Algorithm
quotient	I-Algorithm
.	O
</s>
<s>
The	O
approximation	O
of	O
derivatives	B-Algorithm
by	O
finite	B-Algorithm
differences	I-Algorithm
plays	O
a	O
central	O
role	O
in	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
for	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
differential	O
equations	O
,	O
especially	O
boundary	O
value	O
problems	O
.	O
</s>
<s>
A	O
difference	O
equation	O
is	O
a	O
functional	O
equation	O
that	O
involves	O
the	O
finite	B-Algorithm
difference	I-Algorithm
operator	O
in	O
the	O
same	O
way	O
as	O
a	O
differential	O
equation	O
involves	O
derivatives	B-Algorithm
.	O
</s>
<s>
Certain	O
recurrence	O
relations	O
can	O
be	O
written	O
as	O
difference	O
equations	O
by	O
replacing	O
iteration	O
notation	O
with	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
finite	B-Algorithm
differences	I-Algorithm
are	O
widely	O
used	O
for	O
approximating	O
derivatives	B-Algorithm
,	O
and	O
the	O
term	O
"	O
finite	B-Algorithm
difference	I-Algorithm
"	O
is	O
often	O
used	O
as	O
an	O
abbreviation	O
of	O
"	O
finite	B-Algorithm
difference	I-Algorithm
approximation	O
of	O
derivatives	B-Algorithm
"	O
.	O
</s>
<s>
Finite	B-Algorithm
difference	I-Algorithm
approximations	O
are	O
finite	B-Algorithm
difference	I-Algorithm
quotients	O
in	O
the	O
terminology	O
employed	O
above	O
.	O
</s>
<s>
Finite	B-Algorithm
differences	I-Algorithm
were	O
introduced	O
by	O
Brook	O
Taylor	O
in	O
1715	O
and	O
have	O
also	O
been	O
studied	O
as	O
abstract	O
self-standing	O
mathematical	O
objects	O
in	O
works	O
by	O
George	O
Boole	O
(	O
1860	O
)	O
,	O
L	O
.	O
M	O
.	O
Milne-Thomson	O
(	O
1933	O
)	O
,	O
and	O
(	O
1939	O
)	O
.	O
</s>
<s>
Finite	B-Algorithm
differences	I-Algorithm
trace	O
their	O
origins	O
back	O
to	O
one	O
of	O
Jost	O
Bürgi	O
's	O
algorithms	O
(	O
)	O
and	O
work	O
by	O
others	O
including	O
Isaac	O
Newton	O
.	O
</s>
<s>
The	O
formal	O
calculus	O
of	O
finite	B-Algorithm
differences	I-Algorithm
can	O
be	O
viewed	O
as	O
an	O
alternative	O
to	O
the	O
calculus	O
of	O
infinitesimals	O
.	O
</s>
<s>
Three	O
basic	O
types	O
are	O
commonly	O
considered	O
:	O
forward	O
,	O
backward	O
,	O
and	O
central	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
A	O
backward	B-Algorithm
difference	I-Algorithm
uses	O
the	O
function	O
values	O
at	O
and	O
,	O
instead	O
of	O
the	O
values	O
at	O
and	O
:	O
</s>
<s>
Finite	B-Algorithm
difference	I-Algorithm
is	O
often	O
used	O
as	O
an	O
approximation	O
of	O
the	O
derivative	B-Algorithm
,	O
typically	O
in	O
numerical	B-Algorithm
differentiation	I-Algorithm
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
at	O
a	O
point	O
is	O
defined	O
by	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
Hence	O
,	O
the	O
forward	B-Algorithm
difference	I-Algorithm
divided	O
by	O
approximates	O
the	O
derivative	B-Algorithm
when	O
is	O
small	O
.	O
</s>
<s>
The	O
same	O
formula	O
holds	O
for	O
the	O
backward	B-Algorithm
difference	I-Algorithm
:	O
</s>
<s>
The	O
main	O
problem	O
with	O
the	O
central	B-Algorithm
difference	I-Algorithm
method	O
,	O
however	O
,	O
is	O
that	O
oscillating	O
functions	O
can	O
yield	O
zero	O
derivative	B-Algorithm
.	O
</s>
<s>
If	O
for	O
odd	O
,	O
and	O
for	O
even	O
,	O
then	O
if	O
it	O
is	O
calculated	O
with	O
the	O
central	B-Algorithm
difference	I-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
Authors	O
for	O
whom	O
finite	B-Algorithm
differences	I-Algorithm
mean	O
finite	B-Algorithm
difference	I-Algorithm
approximations	O
define	O
the	O
forward/backward/central	O
differences	O
as	O
the	O
quotients	O
given	O
in	O
this	O
section	O
(	O
instead	O
of	O
employing	O
the	O
definitions	O
given	O
in	O
the	O
previous	O
section	O
)	O
.	O
</s>
<s>
In	O
an	O
analogous	O
way	O
,	O
one	O
can	O
obtain	O
finite	B-Algorithm
difference	I-Algorithm
approximations	O
to	O
higher	O
order	O
derivatives	B-Algorithm
and	O
differential	O
operators	O
.	O
</s>
<s>
For	O
example	O
,	O
by	O
using	O
the	O
above	O
central	B-Algorithm
difference	I-Algorithm
formula	I-Algorithm
for	O
and	O
and	O
applying	O
a	O
central	B-Algorithm
difference	I-Algorithm
formula	I-Algorithm
for	O
the	O
derivative	B-Algorithm
of	O
at	O
,	O
we	O
obtain	O
the	O
central	B-Algorithm
difference	I-Algorithm
approximation	I-Algorithm
of	O
the	O
second	O
derivative	B-Algorithm
of	O
:	O
</s>
<s>
More	O
generally	O
,	O
the	O
th	O
order	O
forward	O
,	O
backward	O
,	O
and	O
central	B-Algorithm
differences	I-Algorithm
are	O
given	O
by	O
,	O
respectively	O
,	O
</s>
<s>
Note	O
that	O
the	O
central	B-Algorithm
difference	I-Algorithm
will	O
,	O
for	O
odd	O
,	O
have	O
multiplied	O
by	O
non-integers	O
.	O
</s>
<s>
Forward	B-Algorithm
differences	I-Algorithm
applied	O
to	O
a	O
sequence	O
are	O
sometimes	O
called	O
the	O
binomial	O
transform	O
of	O
the	O
sequence	O
,	O
and	O
have	O
a	O
number	O
of	O
interesting	O
combinatorial	O
properties	O
.	O
</s>
<s>
Forward	B-Algorithm
differences	I-Algorithm
may	O
be	O
evaluated	O
using	O
the	O
Nörlund	O
–	O
Rice	O
integral	O
.	O
</s>
<s>
The	O
integral	O
representation	O
for	O
these	O
types	O
of	O
series	O
is	O
interesting	O
,	O
because	O
the	O
integral	O
can	O
often	O
be	O
evaluated	O
using	O
asymptotic	O
expansion	O
or	O
saddle-point	O
techniques	O
;	O
by	O
contrast	O
,	O
the	O
forward	B-Algorithm
difference	I-Algorithm
series	O
can	O
be	O
extremely	O
hard	O
to	O
evaluate	O
numerically	B-General_Concept
,	O
because	O
the	O
binomial	O
coefficients	O
grow	O
rapidly	O
for	O
large	O
.	O
</s>
<s>
The	O
relationship	O
of	O
these	O
higher-order	O
differences	O
with	O
the	O
respective	O
derivatives	B-Algorithm
is	O
straightforward	O
,	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
the	O
first-order	O
difference	O
approximates	O
the	O
first-order	O
derivative	B-Algorithm
up	O
to	O
a	O
term	O
of	O
order	O
.	O
</s>
<s>
This	O
can	O
be	O
proven	O
by	O
expanding	O
the	O
above	O
expression	O
in	O
Taylor	O
series	O
,	O
or	O
by	O
using	O
the	O
calculus	O
of	O
finite	B-Algorithm
differences	I-Algorithm
,	O
explained	O
below	O
.	O
</s>
<s>
If	O
necessary	O
,	O
the	O
finite	B-Algorithm
difference	I-Algorithm
can	O
be	O
centered	O
about	O
any	O
point	O
by	O
mixing	O
forward	O
,	O
backward	O
,	O
and	O
central	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
subtracting	O
out	O
the	O
first	O
term	O
,	O
which	O
lowers	O
the	O
polynomial	O
's	O
degree	O
,	O
and	O
finding	O
the	O
finite	B-Algorithm
difference	I-Algorithm
again	O
:	O
</s>
<s>
Using	O
linear	O
algebra	O
one	O
can	O
construct	O
finite	B-Algorithm
difference	I-Algorithm
approximations	O
which	O
utilize	O
an	O
arbitrary	O
number	O
of	O
points	O
to	O
the	O
left	O
and	O
a	O
(	O
possibly	O
different	O
)	O
number	O
of	O
points	O
to	O
the	O
right	O
of	O
the	O
evaluation	O
point	O
,	O
for	O
any	O
order	O
derivative	B-Algorithm
.	O
</s>
<s>
This	O
involves	O
solving	O
a	O
linear	O
system	O
such	O
that	O
the	O
Taylor	O
expansion	O
of	O
the	O
sum	O
of	O
those	O
points	O
around	O
the	O
evaluation	O
point	O
best	O
approximates	O
the	O
Taylor	O
expansion	O
of	O
the	O
desired	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
constructs	O
finite	B-Algorithm
difference	I-Algorithm
approximations	O
for	O
non-standard	O
(	O
and	O
even	O
non-integer	O
)	O
stencils	O
given	O
an	O
arbitrary	O
stencil	O
and	O
a	O
desired	O
derivative	B-Algorithm
order	O
.	O
</s>
<s>
An	O
important	O
application	O
of	O
finite	B-Algorithm
differences	I-Algorithm
is	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
especially	O
in	O
numerical	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
,	O
which	O
aim	O
at	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
ordinary	O
and	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
The	O
idea	O
is	O
to	O
replace	O
the	O
derivatives	B-Algorithm
appearing	O
in	O
the	O
differential	O
equation	O
by	O
finite	B-Algorithm
differences	I-Algorithm
that	O
approximate	O
them	O
.	O
</s>
<s>
The	O
resulting	O
methods	O
are	O
called	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Common	O
applications	O
of	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
are	O
in	O
computational	O
science	O
and	O
engineering	O
disciplines	O
,	O
such	O
as	O
thermal	O
engineering	O
,	O
fluid	O
mechanics	O
,	O
etc	O
.	O
</s>
<s>
The	O
Newton	O
series	O
consists	O
of	O
the	O
terms	O
of	O
the	O
Newton	B-Algorithm
forward	I-Algorithm
difference	I-Algorithm
equation	I-Algorithm
,	O
named	O
after	O
Isaac	O
Newton	O
;	O
in	O
essence	O
,	O
it	O
is	O
the	O
Newton	O
interpolation	O
formula	O
,	O
first	O
published	O
in	O
his	O
Principia	O
Mathematica	O
in	O
1687	O
,	O
namely	O
the	O
discrete	O
analog	O
of	O
the	O
continuous	O
Taylor	O
expansion	O
,	O
</s>
<s>
which	O
holds	O
for	O
any	O
polynomial	O
function	O
and	O
for	O
many	O
(	O
but	O
not	O
all	O
)	O
analytic	B-Language
functions	I-Language
.	O
</s>
<s>
This	O
is	O
easily	O
seen	O
,	O
as	O
the	O
sine	O
function	O
vanishes	O
at	O
integer	O
multiples	O
of	O
;	O
the	O
corresponding	O
Newton	O
series	O
is	O
identically	O
zero	O
,	O
as	O
all	O
finite	B-Algorithm
differences	I-Algorithm
are	O
zero	O
in	O
this	O
case	O
.	O
</s>
<s>
To	O
illustrate	O
how	O
one	O
may	O
use	O
Newton	O
's	O
formula	O
in	O
actual	O
practice	O
,	O
consider	O
the	O
first	O
few	O
terms	O
of	O
doubling	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
One	O
can	O
find	O
a	O
polynomial	O
that	O
reproduces	O
these	O
values	O
,	O
by	O
first	O
computing	O
a	O
difference	O
table	O
,	O
and	O
then	O
substituting	O
the	O
differences	O
that	O
correspond	O
to	O
(	O
underlined	O
)	O
into	O
the	O
formula	O
as	O
follows	O
,	O
</s>
<s>
For	O
the	O
case	O
of	O
nonuniform	O
steps	O
in	O
the	O
values	O
of	O
,	O
Newton	O
computes	O
the	O
divided	B-Algorithm
differences	I-Algorithm
,	O
</s>
<s>
The	O
Newton	O
series	O
,	O
together	O
with	O
the	O
Stirling	O
series	O
and	O
the	O
Selberg	O
series	O
,	O
is	O
a	O
special	O
case	O
of	O
the	O
general	O
difference	O
series	O
,	O
all	O
of	O
which	O
are	O
defined	O
in	O
terms	O
of	O
suitably	O
scaled	O
forward	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
The	O
forward	B-Algorithm
difference	I-Algorithm
can	O
be	O
considered	O
as	O
an	O
operator	O
,	O
called	O
the	O
difference	O
operator	O
,	O
which	O
maps	O
the	O
function	O
to	O
.	O
</s>
<s>
where	O
is	O
the	O
shift	B-Algorithm
operator	I-Algorithm
with	O
step	O
h	O
,	O
defined	O
by	O
,	O
and	O
is	O
the	O
identity	O
operator	O
.	O
</s>
<s>
The	O
finite	B-Algorithm
difference	I-Algorithm
of	O
higher	O
orders	O
can	O
be	O
defined	O
in	O
recursive	O
manner	O
as	O
.	O
</s>
<s>
The	O
difference	O
operator	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
as	O
such	O
it	O
satisfies	O
.	O
</s>
<s>
Similar	O
statements	O
hold	O
for	O
the	O
backward	O
and	O
central	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
where	O
denotes	O
the	O
continuum	O
derivative	B-Algorithm
operator	O
,	O
mapping	O
to	O
its	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
expansion	O
is	O
valid	O
when	O
both	O
sides	O
act	O
on	O
analytic	B-Language
functions	I-Language
,	O
for	O
sufficiently	O
small	O
.	O
</s>
<s>
Even	O
for	O
analytic	B-Language
functions	I-Language
,	O
the	O
series	O
on	O
the	O
right	O
is	O
not	O
guaranteed	O
to	O
converge	O
;	O
it	O
may	O
be	O
an	O
asymptotic	O
series	O
.	O
</s>
<s>
However	O
,	O
it	O
can	O
be	O
used	O
to	O
obtain	O
more	O
accurate	O
approximations	O
for	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
calculus	O
of	O
finite	B-Algorithm
differences	I-Algorithm
is	O
related	O
to	O
the	O
umbral	O
calculus	O
of	O
combinatorics	O
.	O
</s>
<s>
functions	O
thus	O
map	O
systematically	O
to	O
umbral	O
finite-difference	B-Algorithm
analogs	O
involving	O
.	O
</s>
<s>
As	O
in	O
the	O
continuum	B-Algorithm
limit	I-Algorithm
,	O
the	O
eigenfunction	O
of	O
also	O
happens	O
to	O
be	O
an	O
exponential	O
,	O
</s>
<s>
Thus	O
,	O
for	O
instance	O
,	O
the	O
Dirac	O
delta	O
function	O
maps	O
to	O
its	O
umbral	O
correspondent	O
,	O
the	O
cardinal	B-Algorithm
sine	I-Algorithm
function	I-Algorithm
,	O
</s>
<s>
The	O
inverse	O
operator	O
of	O
the	O
forward	B-Algorithm
difference	I-Algorithm
operator	I-Algorithm
,	O
so	O
then	O
the	O
umbral	O
integral	O
,	O
is	O
the	O
indefinite	B-Algorithm
sum	I-Algorithm
or	O
antidifference	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Analogous	O
to	O
rules	O
for	O
finding	O
the	O
derivative	B-Algorithm
,	O
we	O
have	O
:	O
</s>
<s>
A	O
generalized	O
finite	B-Algorithm
difference	I-Algorithm
is	O
usually	O
defined	O
as	O
where	O
is	O
its	O
coefficient	O
vector	O
.	O
</s>
<s>
Another	O
way	O
of	O
generalization	O
is	O
making	O
coefficients	O
depend	O
on	O
point	O
:	O
,	O
thus	O
considering	O
weighted	O
finite	B-Algorithm
difference	I-Algorithm
.	O
</s>
<s>
As	O
a	O
convolution	B-Language
operator	O
:	O
Via	O
the	O
formalism	O
of	O
incidence	O
algebras	O
,	O
difference	O
operators	O
and	O
other	O
Möbius	O
inversion	O
can	O
be	O
represented	O
by	O
convolution	B-Language
with	O
a	O
function	O
on	O
the	O
poset	O
,	O
called	O
the	O
Möbius	O
function	O
;	O
for	O
the	O
difference	O
operator	O
,	O
is	O
the	O
sequence	O
.	O
</s>
<s>
Finite	B-Algorithm
differences	I-Algorithm
can	O
be	O
considered	O
in	O
more	O
than	O
one	O
variable	O
.	O
</s>
<s>
They	O
are	O
analogous	O
to	O
partial	O
derivatives	B-Algorithm
in	O
several	O
variables	O
.	O
</s>
<s>
Some	O
partial	O
derivative	B-Algorithm
approximations	O
are	O
:	O
</s>
