<s>
In	O
computational	O
mathematics	O
,	O
an	O
iterative	B-Algorithm
method	I-Algorithm
is	O
a	O
mathematical	O
procedure	O
that	O
uses	O
an	O
initial	O
value	O
to	O
generate	O
a	O
sequence	O
of	O
improving	O
approximate	O
solutions	O
for	O
a	O
class	O
of	O
problems	O
,	O
in	O
which	O
the	O
n-th	O
approximation	O
is	O
derived	O
from	O
the	O
previous	O
ones	O
.	O
</s>
<s>
A	O
specific	O
implementation	O
of	O
an	O
iterative	B-Algorithm
method	I-Algorithm
,	O
including	O
the	O
termination	O
criteria	O
,	O
is	O
an	O
algorithm	O
of	O
the	O
iterative	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
An	O
iterative	B-Algorithm
method	I-Algorithm
is	O
called	O
convergent	O
if	O
the	O
corresponding	O
sequence	O
converges	O
for	O
given	O
initial	O
approximations	O
.	O
</s>
<s>
A	O
mathematically	O
rigorous	O
convergence	O
analysis	O
of	O
an	O
iterative	B-Algorithm
method	I-Algorithm
is	O
usually	O
performed	O
;	O
however	O
,	O
heuristic-based	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
also	O
common	O
.	O
</s>
<s>
In	O
contrast	O
,	O
direct	B-Algorithm
methods	I-Algorithm
attempt	O
to	O
solve	O
the	O
problem	O
by	O
a	O
finite	O
sequence	O
of	O
operations	O
.	O
</s>
<s>
In	O
the	O
absence	O
of	O
rounding	B-Algorithm
errors	I-Algorithm
,	O
direct	B-Algorithm
methods	I-Algorithm
would	O
deliver	O
an	O
exact	O
solution	O
(	O
for	O
example	O
,	O
solving	O
a	O
linear	O
system	O
of	O
equations	O
by	O
Gaussian	B-Algorithm
elimination	I-Algorithm
)	O
.	O
</s>
<s>
Iterative	B-Algorithm
methods	I-Algorithm
are	O
often	O
the	O
only	O
choice	O
for	O
nonlinear	O
equations	O
.	O
</s>
<s>
However	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
often	O
useful	O
even	O
for	O
linear	O
problems	O
involving	O
many	O
variables	O
(	O
sometimes	O
on	O
the	O
order	O
of	O
millions	O
)	O
,	O
where	O
direct	B-Algorithm
methods	I-Algorithm
would	O
be	O
prohibitively	O
expensive	O
(	O
and	O
in	O
some	O
cases	O
impossible	O
)	O
even	O
with	O
the	O
best	O
available	O
computing	O
power	O
.	O
</s>
<s>
Here	O
xn	O
is	O
the	O
nth	O
approximation	O
or	O
iteration	O
of	O
x	O
and	O
xn+1	O
is	O
the	O
next	O
or	O
n	O
+	O
1	O
iteration	O
of	O
x	O
.	O
Alternately	O
,	O
superscripts	O
in	O
parentheses	O
are	O
often	O
used	O
in	O
numerical	B-General_Concept
methods	I-General_Concept
,	O
so	O
as	O
not	O
to	O
interfere	O
with	O
subscripts	O
with	O
other	O
meanings	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
system	O
of	O
linear	O
equations	O
,	O
the	O
two	O
main	O
classes	O
of	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
the	O
stationary	O
iterative	B-Algorithm
methods	I-Algorithm
,	O
and	O
the	O
more	O
general	O
Krylov	B-Algorithm
subspace	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Stationary	O
iterative	B-Algorithm
methods	I-Algorithm
solve	O
a	O
linear	O
system	O
with	O
an	O
operator	O
approximating	O
the	O
original	O
one	O
;	O
and	O
based	O
on	O
a	O
measurement	O
of	O
the	O
error	O
in	O
the	O
result	O
(	O
the	B-Algorithm
residual	I-Algorithm
)	O
,	O
form	O
a	O
"	O
correction	O
equation	O
"	O
for	O
which	O
this	O
process	O
is	O
repeated	O
.	O
</s>
<s>
An	O
important	O
theorem	O
states	O
that	O
for	O
a	O
given	O
iterative	B-Algorithm
method	I-Algorithm
and	O
its	O
iteration	O
matrix	O
it	O
is	O
convergent	O
if	O
and	O
only	O
if	O
its	O
spectral	O
radius	O
is	O
smaller	O
than	O
unity	O
,	O
that	O
is	O
,	O
</s>
<s>
where	O
is	O
only	O
the	O
diagonal	O
part	O
of	O
,	O
and	O
is	O
the	O
strict	O
lower	B-Algorithm
triangular	I-Algorithm
part	O
of	O
.	O
</s>
<s>
Respectively	O
,	O
is	O
the	O
strict	O
upper	B-Algorithm
triangular	I-Algorithm
part	O
of	O
.	O
</s>
<s>
Richardson	B-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Jacobi	B-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Damped	O
Jacobi	B-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Successive	B-Algorithm
over-relaxation	I-Algorithm
method	I-Algorithm
(	O
SOR	O
)	O
:	O
</s>
<s>
Symmetric	B-Algorithm
successive	B-Algorithm
over-relaxation	I-Algorithm
(	O
SSOR	O
)	O
:	O
</s>
<s>
Linear	O
stationary	O
iterative	B-Algorithm
methods	I-Algorithm
are	O
also	O
called	O
relaxation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Krylov	B-Algorithm
subspace	I-Algorithm
methods	I-Algorithm
work	O
by	O
forming	O
a	O
basis	O
of	O
the	O
sequence	O
of	O
successive	O
matrix	O
powers	O
times	O
the	O
initial	O
residual	B-Algorithm
(	O
the	O
Krylov	O
sequence	O
)	O
.	O
</s>
<s>
The	O
approximations	O
to	O
the	O
solution	O
are	O
then	O
formed	O
by	O
minimizing	O
the	B-Algorithm
residual	I-Algorithm
over	O
the	O
subspace	O
formed	O
.	O
</s>
<s>
The	O
prototypical	O
method	O
in	O
this	O
class	O
is	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
(	O
CG	O
)	O
which	O
assumes	O
that	O
the	O
system	O
matrix	O
is	O
symmetric	B-Algorithm
positive-definite	B-Algorithm
.	O
</s>
<s>
For	O
symmetric	B-Algorithm
(	O
and	O
possibly	O
indefinite	O
)	O
one	O
works	O
with	O
the	O
minimal	O
residual	B-Algorithm
method	O
(	O
MINRES	O
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
non-symmetric	O
matrices	O
,	O
methods	O
such	O
as	O
the	O
generalized	O
minimal	O
residual	B-Algorithm
method	O
(	O
GMRES	O
)	O
and	O
the	O
biconjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
(	O
BiCG	B-Algorithm
)	O
have	O
been	O
derived	O
.	O
</s>
<s>
However	O
,	O
in	O
the	O
presence	O
of	O
rounding	B-Algorithm
errors	I-Algorithm
this	O
statement	O
does	O
not	O
hold	O
;	O
moreover	O
,	O
in	O
practice	O
N	O
can	O
be	O
very	O
large	O
,	O
and	O
the	O
iterative	O
process	O
reaches	O
sufficient	O
accuracy	O
already	O
far	O
earlier	O
.	O
</s>
<s>
The	O
approximating	O
operator	O
that	O
appears	O
in	O
stationary	O
iterative	B-Algorithm
methods	I-Algorithm
can	O
also	O
be	O
incorporated	O
in	O
Krylov	B-Algorithm
subspace	I-Algorithm
methods	I-Algorithm
such	O
as	O
GMRES	O
(	O
alternatively	O
,	O
preconditioned	O
Krylov	O
methods	O
can	O
be	O
considered	O
as	O
accelerations	O
of	O
stationary	O
iterative	B-Algorithm
methods	I-Algorithm
)	O
,	O
where	O
they	O
become	O
transformations	O
of	O
the	O
original	O
operator	O
to	O
a	O
presumably	O
better	O
conditioned	O
one	O
.	O
</s>
<s>
Jamshīd	O
al-Kāshī	O
used	O
iterative	B-Algorithm
methods	I-Algorithm
to	O
calculate	O
the	O
sine	O
of	O
1°	O
and	O
in	O
The	O
Treatise	O
of	O
Chord	O
and	O
Sine	O
to	O
high	O
precision	O
.	O
</s>
<s>
An	O
early	O
iterative	B-Algorithm
method	I-Algorithm
for	O
solving	O
a	O
linear	O
system	O
appeared	O
in	O
a	O
letter	O
of	O
Gauss	O
to	O
a	O
student	O
of	O
his	O
.	O
</s>
<s>
He	O
proposed	O
solving	O
a	O
4-by-4	O
system	O
of	O
equations	O
by	O
repeatedly	O
solving	O
the	O
component	O
in	O
which	O
the	B-Algorithm
residual	I-Algorithm
was	O
the	O
largest	O
.	O
</s>
<s>
The	O
theory	O
of	O
stationary	O
iterative	B-Algorithm
methods	I-Algorithm
was	O
solidly	O
established	O
with	O
the	O
work	O
of	O
D.M.	O
</s>
<s>
The	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
was	O
also	O
invented	O
in	O
the	O
1950s	O
,	O
with	O
independent	O
developments	O
by	O
Cornelius	O
Lanczos	O
,	O
Magnus	O
Hestenes	O
and	O
Eduard	O
Stiefel	O
,	O
but	O
its	O
nature	O
and	O
applicability	O
were	O
misunderstood	O
at	O
the	O
time	O
.	O
</s>
