<s>
In	O
mathematics	O
and	O
computational	O
science	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
(	O
also	O
called	O
the	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
)	O
is	O
a	O
first-order	O
numerical	B-General_Concept
procedure	O
for	O
solving	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
with	O
a	O
given	O
initial	O
value	O
.	O
</s>
<s>
It	O
is	O
the	O
most	O
basic	O
explicit	B-Algorithm
method	I-Algorithm
for	O
numerical	B-Algorithm
integration	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
and	O
is	O
the	O
simplest	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
named	O
after	O
Leonhard	O
Euler	O
,	O
who	O
first	O
proposed	O
it	O
in	O
his	O
book	O
Institutionum	O
calculi	O
integralis	O
(	O
published	O
1768	O
–	O
1870	O
)	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
a	O
first-order	O
method	O
,	O
which	O
means	O
that	O
the	O
local	O
error	O
(	O
error	O
per	O
step	O
)	O
is	O
proportional	O
to	O
the	O
square	O
of	O
the	O
step	O
size	O
,	O
and	O
the	O
global	O
error	O
(	O
error	O
at	O
a	O
given	O
time	O
)	O
is	O
proportional	O
to	O
the	O
step	O
size	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
often	O
serves	O
as	O
the	O
basis	O
to	O
construct	O
more	O
complex	O
methods	O
,	O
e.g.	O
,	O
predictor	B-Algorithm
–	I-Algorithm
corrector	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Now	O
,	O
one	O
step	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
from	O
to	O
is	O
:	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
explicit	B-Algorithm
,	O
i.e.	O
</s>
<s>
the	O
solution	O
is	O
an	O
explicit	B-Algorithm
function	O
of	O
for	O
.	O
</s>
<s>
While	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
integrates	O
a	O
first-order	O
ODE	O
,	O
any	O
ODE	O
of	O
order	O
can	O
be	O
represented	O
as	O
a	O
system	O
of	O
first-order	O
ODEs	O
.	O
</s>
<s>
These	O
first-order	O
systems	O
can	O
be	O
handled	O
by	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
,	O
in	O
fact	O
,	O
by	O
any	O
other	O
scheme	O
for	O
first-order	O
systems	O
.	O
</s>
<s>
we	O
would	O
like	O
to	O
use	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
to	O
approximate	O
.	O
</s>
<s>
Although	O
the	O
approximation	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
was	O
not	O
very	O
precise	O
in	O
this	O
specific	O
case	O
,	O
particularly	O
due	O
to	O
a	O
large	O
value	O
step	O
size	O
,	O
its	O
behaviour	O
is	O
qualitatively	O
correct	O
as	O
the	O
figure	O
shows	O
.	O
</s>
<s>
As	O
suggested	O
in	O
the	O
introduction	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
more	O
accurate	O
if	O
the	O
step	O
size	O
is	O
smaller	O
.	O
</s>
<s>
result	O
of	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
!	O
</s>
<s>
The	O
error	O
recorded	O
in	O
the	O
last	O
column	O
of	O
the	O
table	O
is	O
the	O
difference	O
between	O
the	O
exact	O
solution	O
at	O
and	O
the	O
Euler	B-Algorithm
approximation	I-Algorithm
.	O
</s>
<s>
This	O
is	O
true	O
in	O
general	O
,	O
also	O
for	O
other	O
equations	O
;	O
see	O
the	O
section	O
Global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
for	O
more	O
details	O
.	O
</s>
<s>
Other	O
methods	O
,	O
such	O
as	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
also	O
illustrated	O
in	O
the	O
figures	O
,	O
behave	O
more	O
favourably	O
:	O
the	O
global	O
error	O
of	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
roughly	O
proportional	O
to	O
the	O
square	O
of	O
the	O
step	O
size	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
said	O
to	O
be	O
a	O
first-order	O
method	O
,	O
while	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
second	O
order	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
higher-order	O
methods	O
are	O
employed	O
such	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
or	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
especially	O
if	O
a	O
high	O
accuracy	O
is	O
desired	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
can	O
be	O
derived	O
in	O
a	O
number	O
of	O
ways	O
.	O
</s>
<s>
If	O
this	O
is	O
substituted	O
in	O
the	O
Taylor	O
expansion	O
and	O
the	O
quadratic	O
and	O
higher-order	O
terms	O
are	O
ignored	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
arises	O
.	O
</s>
<s>
The	O
Taylor	O
expansion	O
is	O
used	O
below	O
to	O
analyze	O
the	O
error	O
committed	O
by	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
and	O
it	O
can	O
be	O
extended	O
to	O
produce	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
A	O
closely	O
related	O
derivation	O
is	O
to	O
substitute	O
the	O
forward	O
finite	B-Algorithm
difference	I-Algorithm
formula	O
for	O
the	O
derivative	O
,	O
</s>
<s>
Again	O
,	O
this	O
yields	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
similar	O
computation	O
leads	O
to	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
and	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Combining	O
both	O
equations	O
,	O
one	O
finds	O
again	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
line	O
of	O
thought	O
can	O
be	O
continued	O
to	O
arrive	O
at	O
various	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
the	O
error	O
made	O
in	O
a	O
single	O
step	O
.	O
</s>
<s>
It	O
is	O
the	O
difference	O
between	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
after	O
one	O
step	O
,	O
,	O
and	O
the	O
exact	O
solution	O
at	O
time	O
.	O
</s>
<s>
The	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
(	O
LTE	O
)	O
introduced	O
by	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
given	O
by	O
the	O
difference	O
between	O
these	O
equations	O
:	O
</s>
<s>
This	O
shows	O
that	O
for	O
small	O
,	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
approximately	O
proportional	O
to	O
.	O
</s>
<s>
This	O
makes	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
less	O
accurate	O
(	O
for	O
small	O
)	O
than	O
other	O
higher-order	O
techniques	O
such	O
as	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
and	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
for	O
which	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
proportional	O
to	O
a	O
higher	O
power	O
of	O
the	O
step	O
size	O
.	O
</s>
<s>
A	O
slightly	O
different	O
formulation	O
for	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
can	O
be	O
obtained	O
by	O
using	O
the	O
Lagrange	O
form	O
for	O
the	O
remainder	O
term	O
in	O
Taylor	O
's	O
theorem	O
.	O
</s>
<s>
The	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
the	O
error	O
at	O
a	O
fixed	O
time	O
,	O
after	O
however	O
many	O
steps	O
the	O
method	O
needs	O
to	O
take	O
to	O
reach	O
that	O
time	O
from	O
the	O
initial	O
time	O
.	O
</s>
<s>
The	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
the	O
cumulative	O
effect	O
of	O
the	O
local	B-Algorithm
truncation	I-Algorithm
errors	I-Algorithm
committed	O
in	O
each	O
step	O
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
to	O
be	O
expected	O
that	O
the	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
will	O
be	O
proportional	O
to	O
.	O
</s>
<s>
The	O
precise	O
form	O
of	O
this	O
bound	O
is	O
of	O
little	O
practical	O
importance	O
,	O
as	O
in	O
most	O
cases	O
the	O
bound	O
vastly	O
overestimates	O
the	O
actual	O
error	O
committed	O
by	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
What	O
is	O
important	O
is	O
that	O
it	O
shows	O
that	O
the	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
(	O
approximately	O
)	O
proportional	O
to	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
said	O
to	O
be	O
first	O
order	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
can	O
also	O
be	O
numerically	B-Algorithm
unstable	I-Algorithm
,	O
especially	O
for	O
stiff	B-Algorithm
equations	I-Algorithm
,	O
meaning	O
that	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
grows	O
very	O
large	O
for	O
equations	O
where	O
the	O
exact	O
solution	O
does	O
not	O
.	O
</s>
<s>
However	O
,	O
if	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
applied	O
to	O
this	O
equation	O
with	O
step	O
size	O
,	O
then	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
is	O
qualitatively	O
wrong	O
:	O
It	O
oscillates	O
and	O
grows	O
(	O
see	O
the	O
figure	O
)	O
.	O
</s>
<s>
This	O
is	O
what	O
it	O
means	O
to	O
be	O
unstable	B-Algorithm
.	O
</s>
<s>
If	O
a	O
smaller	O
step	O
size	O
is	O
used	O
,	O
for	O
instance	O
,	O
then	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
does	O
decay	O
to	O
zero	O
.	O
</s>
<s>
In	O
the	O
example	O
,	O
,	O
so	O
if	O
then	O
which	O
is	O
outside	O
the	O
stability	O
region	O
,	O
and	O
thus	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
is	O
unstable	B-Algorithm
.	O
</s>
<s>
This	O
limitation	O
—	O
along	O
with	O
its	O
slow	O
convergence	O
of	O
error	O
with	O
—	O
means	O
that	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
not	O
often	O
used	O
,	O
except	O
as	O
a	O
simple	O
example	O
of	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
In	O
step	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
the	O
rounding	B-Algorithm
error	I-Algorithm
is	O
roughly	O
of	O
the	O
magnitude	O
where	O
is	O
the	O
machine	B-Algorithm
epsilon	I-Algorithm
.	O
</s>
<s>
Assuming	O
that	O
the	O
rounding	B-Algorithm
errors	I-Algorithm
are	O
independent	O
random	O
variables	O
,	O
the	O
expected	O
total	O
rounding	B-Algorithm
error	I-Algorithm
is	O
proportional	O
to	O
.	O
</s>
<s>
Thus	O
,	O
for	O
extremely	O
small	O
values	O
of	O
the	O
step	O
size	O
the	O
truncation	B-Algorithm
error	I-Algorithm
will	O
be	O
small	O
but	O
the	O
effect	O
of	O
rounding	B-Algorithm
error	I-Algorithm
may	O
be	O
big	O
.	O
</s>
<s>
Most	O
of	O
the	O
effect	O
of	O
rounding	B-Algorithm
error	I-Algorithm
can	O
be	O
easily	O
avoided	O
if	O
compensated	B-Algorithm
summation	I-Algorithm
is	O
used	O
in	O
the	O
formula	O
for	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
simple	O
modification	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
which	O
eliminates	O
the	O
stability	O
problems	O
noted	O
above	O
is	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
This	O
differs	O
from	O
the	O
(	O
standard	O
,	O
or	O
forward	O
)	O
Euler	B-Algorithm
method	I-Algorithm
in	O
that	O
the	O
function	O
is	O
evaluated	O
at	O
the	O
end	O
point	O
of	O
the	O
step	O
,	O
instead	O
of	O
the	O
starting	O
point	O
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
,	O
meaning	O
that	O
the	O
formula	O
for	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
has	O
on	O
both	O
sides	O
,	O
so	O
when	O
applying	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
we	O
have	O
to	O
solve	O
an	O
equation	O
.	O
</s>
<s>
Other	O
modifications	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
that	O
help	O
with	O
stability	O
yield	O
the	O
exponential	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
or	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
is	O
illustrated	O
by	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
which	O
is	O
already	O
mentioned	O
in	O
this	O
article	O
:	O
</s>
<s>
This	O
leads	O
to	O
the	O
family	O
of	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
other	O
possibility	O
is	O
to	O
use	O
more	O
past	O
values	O
,	O
as	O
illustrated	O
by	O
the	O
two-step	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
This	O
leads	O
to	O
the	O
family	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
the	O
film	O
Hidden	O
Figures	O
,	O
Katherine	O
Goble	O
resorts	O
to	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
in	O
calculating	O
the	O
re-entry	O
of	O
astronaut	O
John	O
Glenn	O
from	O
Earth	O
orbit	O
.	O
</s>
