<s>
Truncation	B-Algorithm
errors	I-Algorithm
in	O
numerical	B-Algorithm
integration	I-Algorithm
are	O
of	O
two	O
kinds	O
:	O
</s>
<s>
global	B-Algorithm
truncation	I-Algorithm
errors	I-Algorithm
–	O
the	O
cumulative	O
error	O
caused	O
by	O
many	O
iterations	O
.	O
</s>
<s>
The	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
the	O
error	O
that	O
our	O
increment	O
function	O
,	O
,	O
causes	O
during	O
a	O
single	O
iteration	O
,	O
assuming	O
perfect	O
knowledge	O
of	O
the	O
true	O
solution	O
at	O
the	O
previous	O
iteration	O
.	O
</s>
<s>
More	O
formally	O
,	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
,	O
,	O
at	O
step	O
is	O
computed	O
from	O
the	O
difference	O
between	O
the	O
left	O
-	O
and	O
the	O
right-hand	O
side	O
of	O
the	O
equation	O
for	O
the	O
increment	O
:	O
</s>
<s>
The	O
numerical	O
method	O
is	O
consistent	O
if	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
(	O
this	O
means	O
that	O
for	O
every	O
there	O
exists	O
an	O
such	O
that	O
for	O
all	O
;	O
see	O
little-o	O
notation	O
)	O
.	O
</s>
<s>
Furthermore	O
,	O
we	O
say	O
that	O
the	O
numerical	O
method	O
has	O
order	O
if	O
for	O
any	O
sufficiently	O
smooth	O
solution	O
of	O
the	O
initial	O
value	O
problem	O
,	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
(	O
meaning	O
that	O
there	O
exist	O
constants	O
and	O
such	O
that	O
for	O
all	O
)	O
.	O
</s>
<s>
The	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
the	O
accumulation	O
of	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
over	O
all	O
of	O
the	O
iterations	O
,	O
assuming	O
perfect	O
knowledge	O
of	O
the	O
true	O
solution	O
at	O
the	O
initial	O
time	O
step	O
.	O
</s>
<s>
More	O
formally	O
,	O
the	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
,	O
,	O
at	O
time	O
is	O
defined	O
by	O
:	O
</s>
<s>
The	O
numerical	O
method	O
is	O
convergent	O
if	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
goes	O
to	O
zero	O
as	O
the	O
step	O
size	O
goes	O
to	O
zero	O
;	O
in	O
other	O
words	O
,	O
the	O
numerical	O
solution	O
converges	O
to	O
the	O
exact	O
solution	O
:	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
possible	O
to	O
calculate	O
an	O
upper	O
bound	O
on	O
the	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
,	O
if	O
we	O
already	O
know	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
The	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
satisfies	O
the	O
recurrence	O
relation	O
:	O
</s>
<s>
The	O
next	O
iterate	O
of	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
depends	O
on	O
the	O
previous	O
s	O
iterates	O
.	O
</s>
<s>
Thus	O
,	O
in	O
the	O
definition	O
for	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
,	O
it	O
is	O
now	O
assumed	O
that	O
the	O
previous	O
s	O
iterates	O
all	O
correspond	O
to	O
the	O
exact	O
solution	O
:	O
</s>
<s>
The	O
definition	O
of	O
the	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
also	O
unchanged	O
.	O
</s>
<s>
The	O
relation	O
between	O
local	O
and	O
global	B-Algorithm
truncation	I-Algorithm
errors	I-Algorithm
is	O
slightly	O
different	O
from	O
in	O
the	O
simpler	O
setting	O
of	O
one-step	O
methods	O
.	O
</s>
<s>
For	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
an	O
additional	O
concept	O
called	O
zero-stability	O
is	O
needed	O
to	O
explain	O
the	O
relation	O
between	O
local	O
and	O
global	B-Algorithm
truncation	I-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
Linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
that	O
satisfy	O
the	O
condition	O
of	O
zero-stability	O
have	O
the	O
same	O
relation	O
between	O
local	O
and	O
global	O
errors	O
as	O
one-step	O
methods	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
zero-stable	O
and	O
consistent	O
,	O
then	O
it	O
converges	O
.	O
</s>
<s>
And	O
if	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
zero-stable	O
and	O
has	O
local	O
error	O
,	O
then	O
its	O
global	O
error	O
satisfies	O
.	O
</s>
