<s>
Numerical	B-General_Concept
analysis	I-General_Concept
is	O
the	O
study	O
of	O
algorithms	O
that	O
use	O
numerical	B-General_Concept
approximation	I-General_Concept
(	O
as	O
opposed	O
to	O
symbolic	B-Algorithm
manipulations	I-Algorithm
)	O
for	O
the	O
problems	O
of	O
mathematical	O
analysis	O
(	O
as	O
distinguished	O
from	O
discrete	O
mathematics	O
)	O
.	O
</s>
<s>
It	O
is	O
the	O
study	O
of	O
numerical	B-Algorithm
methods	I-Algorithm
that	O
attempt	O
at	O
finding	O
approximate	O
solutions	O
of	O
problems	O
rather	O
than	O
the	O
exact	O
ones	O
.	O
</s>
<s>
Numerical	B-General_Concept
analysis	I-General_Concept
finds	O
application	O
in	O
all	O
fields	O
of	O
engineering	O
and	O
the	O
physical	O
sciences	O
,	O
and	O
in	O
the	O
21st	O
century	O
also	O
the	O
life	O
and	O
social	O
sciences	O
,	O
medicine	O
,	O
business	O
and	O
even	O
the	O
arts	O
.	O
</s>
<s>
Current	O
growth	O
in	O
computing	O
power	O
has	O
enabled	O
the	O
use	O
of	O
more	O
complex	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
providing	O
detailed	O
and	O
realistic	O
mathematical	O
models	O
in	O
science	O
and	O
engineering	O
.	O
</s>
<s>
Examples	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
include	O
:	O
ordinary	O
differential	O
equations	O
as	O
found	O
in	O
celestial	O
mechanics	O
(	O
predicting	O
the	O
motions	O
of	O
planets	O
,	O
stars	O
and	O
galaxies	O
)	O
,	O
numerical	O
linear	O
algebra	O
in	O
data	O
analysis	O
,	O
and	O
stochastic	O
differential	O
equations	O
and	O
Markov	O
chains	O
for	O
simulating	O
living	O
cells	O
in	O
medicine	O
and	O
biology	O
.	O
</s>
<s>
Before	O
modern	O
computers	O
,	O
numerical	B-Algorithm
methods	I-Algorithm
often	O
relied	O
on	O
hand	O
interpolation	B-Algorithm
formulas	O
,	O
using	O
data	O
from	O
large	O
printed	O
tables	O
.	O
</s>
<s>
A	O
tablet	O
from	O
the	O
Yale	O
Babylonian	O
Collection	O
(	O
YBC	O
7289	O
)	O
,	O
gives	O
a	O
sexagesimal	O
numerical	B-General_Concept
approximation	I-General_Concept
of	O
the	O
square	O
root	O
of	O
2	O
,	O
the	O
length	O
of	O
the	O
diagonal	O
in	O
a	O
unit	O
square	O
.	O
</s>
<s>
Numerical	B-General_Concept
analysis	I-General_Concept
continues	O
this	O
long	O
tradition	O
:	O
rather	O
than	O
giving	O
exact	O
symbolic	O
answers	O
translated	O
into	O
digits	O
and	O
applicable	O
only	O
to	O
real-world	O
measurements	O
,	O
approximate	O
solutions	O
within	O
specified	O
error	O
bounds	O
are	O
used	O
.	O
</s>
<s>
The	O
overall	O
goal	O
of	O
the	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
is	O
the	O
design	O
and	O
analysis	O
of	O
techniques	O
to	O
give	O
approximate	O
but	O
accurate	O
solutions	O
to	O
hard	O
problems	O
,	O
the	O
variety	O
of	O
which	O
is	O
suggested	O
by	O
the	O
following	O
:	O
</s>
<s>
Advanced	O
numerical	B-Algorithm
methods	I-Algorithm
are	O
essential	O
in	O
making	O
numerical	B-General_Concept
weather	I-General_Concept
prediction	I-General_Concept
feasible	O
.	O
</s>
<s>
Computing	O
the	O
trajectory	O
of	O
a	O
spacecraft	O
requires	O
the	O
accurate	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
a	O
system	O
of	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
Such	O
simulations	O
essentially	O
consist	O
of	O
solving	O
partial	O
differential	O
equations	O
numerically	B-General_Concept
.	O
</s>
<s>
Hedge	O
funds	O
(	O
private	O
investment	O
funds	O
)	O
use	O
tools	O
from	O
all	O
fields	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
to	O
attempt	O
to	O
calculate	O
the	O
value	O
of	O
stocks	O
and	O
derivatives	B-Algorithm
more	O
precisely	O
than	O
other	O
market	O
participants	O
.	O
</s>
<s>
The	O
rest	O
of	O
this	O
section	O
outlines	O
several	O
important	O
themes	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
The	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
predates	O
the	O
invention	O
of	O
modern	O
computers	O
by	O
many	O
centuries	O
.	O
</s>
<s>
Linear	O
interpolation	B-Algorithm
was	O
already	O
in	O
use	O
more	O
than	O
2000	O
years	O
ago	O
.	O
</s>
<s>
Many	O
great	O
mathematicians	O
of	O
the	O
past	O
were	O
preoccupied	O
by	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
as	O
is	O
obvious	O
from	O
the	O
names	O
of	O
important	O
algorithms	O
like	O
Newton	O
's	O
method	O
,	O
Lagrange	O
interpolation	B-Algorithm
polynomial	O
,	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
or	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
origins	O
of	O
modern	O
numerical	B-General_Concept
analysis	I-General_Concept
are	O
often	O
linked	O
to	O
a	O
1947	O
paper	O
by	O
John	O
von	O
Neumann	O
and	O
Herman	O
Goldstine	O
,	O
</s>
<s>
but	O
others	O
consider	O
modern	O
numerical	B-General_Concept
analysis	I-General_Concept
to	O
go	O
back	O
to	O
work	O
by	O
E	O
.	O
T	O
.	O
Whittaker	O
in	O
1912	O
.	O
</s>
<s>
To	O
facilitate	O
computations	O
by	O
hand	O
,	O
large	O
books	O
were	O
produced	O
with	O
formulas	O
and	O
tables	O
of	O
data	O
such	O
as	O
interpolation	B-Algorithm
points	O
and	O
function	O
coefficients	O
.	O
</s>
<s>
Using	O
these	O
tables	O
,	O
often	O
calculated	O
out	O
to	O
16	O
decimal	B-Application
places	O
or	O
more	O
for	O
some	O
functions	O
,	O
one	O
could	O
look	O
up	O
values	O
to	O
plug	O
into	O
the	O
formulas	O
given	O
and	O
achieve	O
very	O
good	O
numerical	O
estimates	O
of	O
some	O
functions	O
.	O
</s>
<s>
But	O
the	O
invention	O
of	O
the	O
computer	O
also	O
influenced	O
the	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
since	O
now	O
longer	O
and	O
more	O
complicated	O
calculations	O
could	O
be	O
done	O
.	O
</s>
<s>
The	O
Leslie	O
Fox	O
Prize	O
for	O
Numerical	B-General_Concept
Analysis	I-General_Concept
was	O
initiated	O
in	O
1985	O
by	O
the	O
Institute	O
of	O
Mathematics	O
and	O
its	O
Applications	O
.	O
</s>
<s>
For	O
the	O
iterative	B-Algorithm
method	I-Algorithm
,	O
apply	O
the	O
bisection	O
method	O
to	O
f(x )	O
=	O
3x3	O
24	O
.	O
</s>
<s>
A	O
discretization	B-Algorithm
would	O
be	O
to	O
say	O
that	O
the	O
speed	O
of	O
the	O
car	O
was	O
constant	O
from	O
0:00	O
to	O
0:40	O
,	O
then	O
from	O
0:40	O
to	O
1:20	O
and	O
finally	O
from	O
1:20	O
to	O
2:00	O
.	O
</s>
<s>
Ill-conditioned	B-Algorithm
problem	O
:	O
Take	O
the	O
function	O
.	O
</s>
<s>
Evaluating	O
f(x )	O
near	O
x	O
=	O
1	O
is	O
an	O
ill-conditioned	B-Algorithm
problem	O
.	O
</s>
<s>
Well-conditioned	B-Algorithm
problem	O
:	O
By	O
contrast	O
,	O
evaluating	O
the	O
same	O
function	O
near	O
x	O
=	O
10	O
is	O
a	O
well-conditioned	B-Algorithm
problem	O
.	O
</s>
<s>
Direct	B-Algorithm
methods	I-Algorithm
compute	O
the	O
solution	O
to	O
a	O
problem	O
in	O
a	O
finite	O
number	O
of	O
steps	O
.	O
</s>
<s>
These	O
methods	O
would	O
give	O
the	O
precise	O
answer	O
if	O
they	O
were	O
performed	O
in	O
infinite	B-Algorithm
precision	I-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Examples	O
include	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
the	O
QR	O
factorization	O
method	O
for	O
solving	O
systems	O
of	O
linear	O
equations	O
,	O
and	O
the	O
simplex	B-Algorithm
method	I-Algorithm
of	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
In	O
practice	O
,	O
finite	B-Algorithm
precision	I-Algorithm
is	O
used	O
and	O
the	O
result	O
is	O
an	O
approximation	O
of	O
the	O
true	O
solution	O
(	O
assuming	O
stability	B-Algorithm
)	O
.	O
</s>
<s>
In	O
contrast	O
to	O
direct	B-Algorithm
methods	I-Algorithm
,	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
not	O
expected	O
to	O
terminate	O
in	O
a	O
finite	O
number	O
of	O
steps	O
.	O
</s>
<s>
Starting	O
from	O
an	O
initial	O
guess	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
form	O
successive	O
approximations	O
that	O
converge	B-Algorithm
to	O
the	O
exact	O
solution	O
only	O
in	O
the	O
limit	O
.	O
</s>
<s>
A	O
convergence	O
test	O
,	O
often	O
involving	O
the	B-Algorithm
residual	I-Algorithm
,	O
is	O
specified	O
in	O
order	O
to	O
decide	O
when	O
a	O
sufficiently	O
accurate	O
solution	O
has	O
(	O
hopefully	O
)	O
been	O
found	O
.	O
</s>
<s>
Even	O
using	O
infinite	B-Algorithm
precision	I-Algorithm
arithmetic	I-Algorithm
these	O
methods	O
would	O
not	O
reach	O
the	O
solution	O
within	O
a	O
finite	O
number	O
of	O
steps	O
(	O
in	O
general	O
)	O
.	O
</s>
<s>
Examples	O
include	O
Newton	O
's	O
method	O
,	O
the	O
bisection	O
method	O
,	O
and	O
Jacobi	B-Algorithm
iteration	I-Algorithm
.	O
</s>
<s>
In	O
computational	O
matrix	O
algebra	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
generally	O
needed	O
for	O
large	O
problems	O
.	O
</s>
<s>
Iterative	B-Algorithm
methods	I-Algorithm
are	O
more	O
common	O
than	O
direct	B-Algorithm
methods	I-Algorithm
in	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
GMRES	O
and	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
these	O
methods	O
the	O
number	O
of	O
steps	O
needed	O
to	O
obtain	O
the	O
exact	O
solution	O
is	O
so	O
large	O
that	O
an	O
approximation	O
is	O
accepted	O
in	O
the	O
same	O
manner	O
as	O
for	O
an	O
iterative	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
continuous	O
problems	O
must	O
sometimes	O
be	O
replaced	O
by	O
a	O
discrete	O
problem	O
whose	O
solution	O
is	O
known	O
to	O
approximate	O
that	O
of	O
the	O
continuous	O
problem	O
;	O
this	O
process	O
is	O
called	O
'	O
discretization	B-Algorithm
 '	O
.	O
</s>
<s>
The	O
study	O
of	O
errors	O
forms	O
an	O
important	O
part	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Round-off	B-Algorithm
errors	I-Algorithm
arise	O
because	O
it	O
is	O
impossible	O
to	O
represent	O
all	O
real	O
numbers	O
exactly	O
on	O
a	O
machine	O
with	O
finite	O
memory	O
(	O
which	O
is	O
what	O
all	O
practical	O
digital	O
computers	O
are	O
)	O
.	O
</s>
<s>
Truncation	B-Algorithm
errors	I-Algorithm
are	O
committed	O
when	O
an	O
iterative	B-Algorithm
method	I-Algorithm
is	O
terminated	O
or	O
a	O
mathematical	O
procedure	O
is	O
approximated	O
and	O
the	O
approximate	O
solution	O
differs	O
from	O
the	O
exact	O
solution	O
.	O
</s>
<s>
Similarly	O
,	O
discretization	B-Algorithm
induces	O
a	O
discretization	B-Algorithm
error	I-Algorithm
because	O
the	O
solution	O
of	O
the	O
discrete	O
problem	O
does	O
not	O
coincide	O
with	O
the	O
solution	O
of	O
the	O
continuous	O
problem	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
truncation	B-Algorithm
error	I-Algorithm
is	O
roughly	O
0.01	O
.	O
</s>
<s>
A	O
truncation	B-Algorithm
error	I-Algorithm
is	O
created	O
when	O
a	O
mathematical	O
procedure	O
is	O
approximated	O
.	O
</s>
<s>
To	O
integrate	O
a	O
function	O
exactly	O
,	O
an	O
infinite	O
sum	O
of	O
regions	O
must	O
be	O
found	O
,	O
but	O
numerically	B-General_Concept
only	O
a	O
finite	O
sum	O
of	O
regions	O
can	O
be	O
found	O
,	O
and	O
hence	O
the	O
approximation	O
of	O
the	O
exact	O
solution	O
.	O
</s>
<s>
Similarly	O
,	O
to	O
differentiate	O
a	O
function	O
,	O
the	O
differential	O
element	O
approaches	O
zero	O
,	O
but	O
numerically	B-General_Concept
only	O
a	O
nonzero	O
value	O
of	O
the	O
differential	O
element	O
can	O
be	O
chosen	O
.	O
</s>
<s>
Numerical	B-Algorithm
stability	I-Algorithm
is	O
a	O
notion	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
An	O
algorithm	O
is	O
called	O
'	O
numerically	B-Algorithm
stable	I-Algorithm
 '	O
if	O
an	O
error	O
,	O
whatever	O
its	O
cause	O
,	O
does	O
not	O
grow	O
to	O
be	O
much	O
larger	O
during	O
the	O
calculation	O
.	O
</s>
<s>
This	O
happens	O
if	O
the	O
problem	O
is	O
'	O
well-conditioned	B-Algorithm
'	O
,	O
meaning	O
that	O
the	O
solution	O
changes	O
by	O
only	O
a	O
small	O
amount	O
if	O
the	O
problem	O
data	O
are	O
changed	O
by	O
a	O
small	O
amount	O
.	O
</s>
<s>
To	O
the	O
contrary	O
,	O
if	O
a	O
problem	O
is	O
'	O
ill-conditioned	B-Algorithm
'	O
,	O
then	O
any	O
small	O
error	O
in	O
the	O
data	O
will	O
grow	O
to	O
be	O
a	O
large	O
error	O
.	O
</s>
<s>
Both	O
the	O
original	O
problem	O
and	O
the	O
algorithm	O
used	O
to	O
solve	O
that	O
problem	O
can	O
be	O
'	O
well-conditioned	B-Algorithm
'	O
or	O
'	O
ill-conditioned	B-Algorithm
'	O
,	O
and	O
any	O
combination	O
is	O
possible	O
.	O
</s>
<s>
So	O
an	O
algorithm	O
that	O
solves	O
a	O
well-conditioned	B-Algorithm
problem	O
may	O
be	O
either	O
numerically	B-Algorithm
stable	I-Algorithm
or	O
numerically	B-Algorithm
unstable	I-Algorithm
.	O
</s>
<s>
An	O
art	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
is	O
to	O
find	O
a	O
stable	O
algorithm	O
for	O
solving	O
a	O
well-posed	O
mathematical	O
problem	O
.	O
</s>
<s>
Hence	O
,	O
the	O
Babylonian	O
method	O
is	O
numerically	B-Algorithm
stable	I-Algorithm
,	O
while	O
Method	O
X	O
is	O
numerically	B-Algorithm
unstable	I-Algorithm
.	O
</s>
<s>
Numerical	B-Algorithm
stability	I-Algorithm
is	O
affected	O
by	O
the	O
number	O
of	O
the	O
significant	O
digits	O
the	O
machine	O
keeps	O
.	O
</s>
<s>
The	O
desired	O
value	O
,	O
computed	O
using	O
infinite	B-Algorithm
precision	I-Algorithm
,	O
is	O
174755	O
...	O
</s>
<s>
The	O
example	O
is	O
a	O
modification	O
of	O
one	O
taken	O
from	O
Mathew	O
;	O
Numerical	B-Algorithm
methods	I-Algorithm
using	O
MATLAB	B-Language
,	O
3rd	O
ed	O
.	O
</s>
<s>
The	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
includes	O
many	O
sub-disciplines	O
.	O
</s>
<s>
Interpolation	B-Algorithm
:	O
Observing	O
that	O
the	O
temperature	O
varies	O
from	O
20	O
degrees	O
Celsius	O
at	O
1:00	O
to	O
14	O
degrees	O
at	O
3:00	O
,	O
a	O
linear	O
interpolation	B-Algorithm
of	O
this	O
data	O
would	O
conclude	O
that	O
it	O
was	O
17	O
degrees	O
at	O
2:00	O
and	O
18.5	O
degrees	O
at	O
1:30pm	O
.	O
</s>
<s>
Optimization	O
:	O
Suppose	O
lemonade	O
is	O
sold	O
at	O
a	O
lemonade	B-Application
stand	I-Application
,	O
at	O
$1.00	O
per	O
glass	O
,	O
that	O
197	O
glasses	O
of	O
lemonade	O
can	O
be	O
sold	O
per	O
day	O
,	O
and	O
that	O
for	O
each	O
increase	O
of	O
$0.01	O
,	O
one	O
less	O
glass	O
of	O
lemonade	O
will	O
be	O
sold	O
per	O
day	O
.	O
</s>
<s>
If	O
$1.485	O
could	O
be	O
charged	O
,	O
profit	O
would	O
be	O
maximized	O
,	O
but	O
due	O
to	O
the	O
constraint	B-Application
of	O
having	O
to	O
charge	O
a	O
whole-cent	O
amount	O
,	O
charging	O
$1.48	O
or	O
$1.49	O
per	O
glass	O
will	O
both	O
yield	O
the	O
maximum	O
income	O
of	O
$220.52	O
per	O
day	O
.	O
</s>
<s>
This	O
is	O
called	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
for	O
solving	O
an	O
ordinary	O
differential	O
equation	O
.	O
</s>
<s>
Generally	O
,	O
it	O
is	O
important	O
to	O
estimate	O
and	O
control	O
round-off	B-Algorithm
errors	I-Algorithm
arising	O
from	O
the	O
use	O
of	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Interpolation	B-Algorithm
solves	O
the	O
following	O
problem	O
:	O
given	O
the	O
value	O
of	O
some	O
unknown	O
function	O
at	O
a	O
number	O
of	O
points	O
,	O
what	O
value	O
does	O
that	O
function	O
have	O
at	O
some	O
other	O
point	O
between	O
the	O
given	O
points	O
?	O
</s>
<s>
Extrapolation	O
is	O
very	O
similar	O
to	O
interpolation	B-Algorithm
,	O
except	O
that	O
now	O
the	O
value	O
of	O
the	O
unknown	O
function	O
at	O
a	O
point	O
which	O
is	O
outside	O
the	O
given	O
points	O
must	O
be	O
found	O
.	O
</s>
<s>
Standard	O
direct	B-Algorithm
methods	I-Algorithm
,	O
i.e.	O
,	O
methods	O
that	O
use	O
some	O
matrix	O
decomposition	O
are	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
LU	O
decomposition	O
,	O
Cholesky	O
decomposition	O
for	O
symmetric	B-Algorithm
(	O
or	O
hermitian	B-Algorithm
)	O
and	O
positive-definite	B-Algorithm
matrix	I-Algorithm
,	O
and	O
QR	O
decomposition	O
for	O
non-square	O
matrices	O
.	O
</s>
<s>
Iterative	B-Algorithm
methods	I-Algorithm
such	O
as	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
,	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
,	O
successive	B-Algorithm
over-relaxation	I-Algorithm
and	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
are	O
usually	O
preferred	O
for	O
large	O
systems	O
.	O
</s>
<s>
General	O
iterative	B-Algorithm
methods	I-Algorithm
can	O
be	O
developed	O
using	O
a	O
matrix	B-Algorithm
splitting	I-Algorithm
.	O
</s>
<s>
If	O
the	O
function	O
is	O
differentiable	B-Algorithm
and	O
the	O
derivative	B-Algorithm
is	O
known	O
,	O
then	O
Newton	O
's	O
method	O
is	O
a	O
popular	O
choice	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
spectral	B-General_Concept
image	I-General_Concept
compression	I-General_Concept
algorithm	O
is	O
based	O
on	O
the	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
The	O
corresponding	O
tool	O
in	O
statistics	O
is	O
called	O
principal	B-Application
component	I-Application
analysis	I-Application
.	O
</s>
<s>
The	O
field	O
of	O
optimization	O
is	O
further	O
split	O
in	O
several	O
subfields	O
,	O
depending	O
on	O
the	O
form	O
of	O
the	O
objective	O
function	O
and	O
the	O
constraint	B-Application
.	O
</s>
<s>
For	O
instance	O
,	O
linear	B-Algorithm
programming	I-Algorithm
deals	O
with	O
the	O
case	O
that	O
both	O
the	O
objective	O
function	O
and	O
the	O
constraints	O
are	O
linear	O
.	O
</s>
<s>
A	O
famous	O
method	O
in	O
linear	B-Algorithm
programming	I-Algorithm
is	O
the	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Popular	O
methods	O
use	O
one	O
of	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
(	O
like	O
the	O
midpoint	O
rule	O
or	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
)	O
or	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
In	O
higher	O
dimensions	O
,	O
where	O
these	O
methods	O
become	O
prohibitively	O
expensive	O
in	O
terms	O
of	O
computational	O
effort	O
,	O
one	O
may	O
use	O
Monte	B-Algorithm
Carlo	I-Algorithm
or	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
(	O
see	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
)	O
,	O
or	O
,	O
in	O
modestly	O
large	O
dimensions	O
,	O
the	O
method	O
of	O
sparse	B-Algorithm
grids	I-Algorithm
.	O
</s>
<s>
Numerical	B-General_Concept
analysis	I-General_Concept
is	O
also	O
concerned	O
with	O
computing	O
(	O
in	O
an	O
approximate	O
way	O
)	O
the	O
solution	O
of	O
differential	O
equations	O
,	O
both	O
ordinary	O
differential	O
equations	O
and	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
This	O
can	O
be	O
done	O
by	O
a	O
finite	B-Application
element	I-Application
method	I-Application
,	O
a	O
finite	B-Algorithm
difference	I-Algorithm
method	O
,	O
or	O
(	O
particularly	O
in	O
engineering	O
)	O
a	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
theoretical	O
justification	O
of	O
these	O
methods	O
often	O
involves	O
theorems	O
from	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
The	O
Netlib	B-Application
repository	O
contains	O
various	O
collections	O
of	O
software	O
routines	O
for	O
numerical	O
problems	O
,	O
mostly	O
in	O
Fortran	B-Application
and	O
C	B-Language
.	O
Commercial	O
products	O
implementing	O
many	O
different	O
numerical	B-General_Concept
algorithms	I-General_Concept
include	O
the	O
IMSL	B-Library
and	O
NAG	O
libraries	O
;	O
a	O
free-software	B-Application
alternative	O
is	O
the	O
GNU	B-Application
Scientific	I-Application
Library	I-Application
.	O
</s>
<s>
There	O
are	O
several	O
popular	O
numerical	B-General_Concept
computing	I-General_Concept
applications	O
such	O
as	O
MATLAB	B-Language
,	O
TK	B-Application
Solver	I-Application
,	O
S-PLUS	B-Application
,	O
and	O
IDL	B-Language
as	O
well	O
as	O
free	O
and	O
open	O
source	O
alternatives	O
such	O
as	O
FreeMat	B-Language
,	O
Scilab	B-Application
,	O
GNU	B-Language
Octave	I-Language
(	O
similar	O
to	O
Matlab	B-Language
)	O
,	O
and	O
IT++	B-Language
(	O
a	O
C++	O
library	O
)	O
.	O
</s>
<s>
There	O
are	O
also	O
programming	O
languages	O
such	O
as	O
R	B-Language
(	O
similar	O
to	O
S-PLUS	B-Application
)	O
,	O
Julia	B-Application
,	O
and	O
Python	B-Language
with	O
libraries	O
such	O
as	O
NumPy	B-Application
,	O
SciPy	B-Application
and	O
SymPy	B-Application
.	O
</s>
<s>
Many	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
such	O
as	O
Mathematica	B-Language
also	O
benefit	O
from	O
the	O
availability	O
of	O
arbitrary-precision	B-Algorithm
arithmetic	I-Algorithm
which	O
can	O
provide	O
more	O
accurate	O
results	O
.	O
</s>
<s>
Also	O
,	O
any	O
spreadsheet	B-Application
software	I-Application
can	O
be	O
used	O
to	O
solve	O
simple	O
problems	O
relating	O
to	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
