<s>
In	O
mathematics	O
,	O
Anderson	B-Algorithm
acceleration	I-Algorithm
,	O
also	O
called	O
Anderson	O
mixing	O
,	O
is	O
a	O
method	O
for	O
the	O
acceleration	O
of	O
the	O
convergence	B-Architecture
rate	I-Architecture
of	O
fixed-point	O
iterations	O
.	O
</s>
<s>
However	O
,	O
the	O
convergence	O
of	O
such	O
a	O
scheme	O
is	O
not	O
guaranteed	O
in	O
general	O
;	O
moreover	O
,	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
usually	O
linear	O
,	O
which	O
can	O
become	O
too	O
slow	O
if	O
the	O
evaluation	O
of	O
the	O
function	O
is	O
computationally	O
expensive	O
.	O
</s>
<s>
Anderson	B-Algorithm
acceleration	I-Algorithm
is	O
a	O
method	O
to	O
accelerate	O
the	O
convergence	O
of	O
the	O
fixed-point	O
sequence	O
.	O
</s>
<s>
For	O
both	O
choices	O
,	O
the	O
optimization	O
problem	O
is	O
in	O
the	O
form	O
of	O
an	O
unconstrained	O
linear	B-Algorithm
least-squares	I-Algorithm
problem	O
,	O
which	O
can	O
be	O
solved	O
by	O
standard	O
methods	O
including	O
QR	O
decomposition	O
and	O
singular	O
value	O
decomposition	O
,	O
possibly	O
including	O
regularization	O
techniques	O
to	O
deal	O
with	O
rank	O
deficiencies	O
and	O
conditioning	B-Algorithm
issues	O
in	O
the	O
optimization	O
problem	O
.	O
</s>
<s>
Solving	O
the	O
least-squares	O
problem	O
by	O
solving	O
the	O
normal	B-Algorithm
equations	I-Algorithm
is	O
generally	O
not	O
advisable	O
due	O
to	O
potential	O
numerical	B-Algorithm
instabilities	I-Algorithm
and	O
generally	O
high	O
computational	O
cost	O
.	O
</s>
<s>
Similarly	O
,	O
near-stagnation	O
(	O
)	O
results	O
in	O
bad	O
conditioning	B-Algorithm
of	O
the	O
least	O
squares	O
problem	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
choice	O
of	O
the	O
parameter	O
might	O
be	O
relevant	O
in	O
determining	O
the	O
conditioning	B-Algorithm
of	O
the	O
least-squares	O
problem	O
,	O
as	O
discussed	O
below	O
.	O
</s>
<s>
A	O
too	O
large	O
value	O
of	O
may	O
worsen	O
the	O
conditioning	B-Algorithm
of	O
the	O
least	O
squares	O
problem	O
and	O
the	O
cost	O
of	O
its	O
solution	O
.	O
</s>
<s>
One	O
possibility	O
is	O
to	O
choose	O
for	O
each	O
iteration	O
(	O
sometimes	O
referred	O
to	O
as	O
Anderson	B-Algorithm
acceleration	I-Algorithm
without	O
truncation	O
)	O
.	O
</s>
<s>
A	O
more	O
sophisticated	O
technique	O
is	O
based	O
on	O
choosing	O
so	O
as	O
to	O
maintain	O
a	O
small	O
enough	O
conditioning	B-Algorithm
for	O
the	O
least-squares	O
problem	O
.	O
</s>
<s>
Approximating	O
the	O
derivative	O
by	O
means	O
of	O
finite	B-Algorithm
differences	I-Algorithm
is	O
a	O
possible	O
alternative	O
,	O
but	O
it	O
requires	O
multiple	O
evaluations	O
of	O
at	O
each	O
iteration	O
,	O
which	O
again	O
can	O
become	O
very	O
costly	O
.	O
</s>
<s>
Anderson	B-Algorithm
acceleration	I-Algorithm
requires	O
only	O
one	O
evaluation	O
of	O
the	O
function	O
per	O
iteration	O
,	O
and	O
no	O
evaluation	O
of	O
its	O
derivative	O
.	O
</s>
<s>
Several	O
authors	O
have	O
pointed	O
out	O
similarities	O
between	O
the	O
Anderson	B-Algorithm
acceleration	I-Algorithm
scheme	O
and	O
other	O
methods	O
for	O
the	O
solution	O
of	O
non-linear	O
equations	O
.	O
</s>
<s>
Eyert	O
and	O
Fang	O
and	O
Saad	O
interpreted	O
the	O
algorithm	O
within	O
the	O
class	O
of	O
quasi-Newton	B-Algorithm
and	O
multisecant	O
methods	O
,	O
that	O
generalize	O
the	O
well	O
known	O
secant	O
method	O
,	O
for	O
the	O
solution	O
of	O
the	O
non-linear	O
equation	O
;	O
they	O
also	O
showed	O
how	O
the	O
scheme	O
can	O
be	O
seen	O
as	O
a	O
method	O
in	O
the	O
Broyden	O
class	O
;	O
</s>
<s>
Walker	O
and	O
Ni	O
showed	O
that	O
the	O
Anderson	B-Algorithm
acceleration	I-Algorithm
scheme	O
is	O
equivalent	O
to	O
the	O
GMRES	O
method	O
in	O
the	O
case	O
of	O
linear	O
problems	O
(	O
i.e.	O
</s>
<s>
The	O
following	O
is	O
an	O
example	O
implementation	O
in	O
MATLAB	B-Language
language	O
of	O
the	O
Anderson	B-Algorithm
acceleration	I-Algorithm
scheme	O
for	O
finding	O
the	O
fixed-point	O
of	O
the	O
function	O
.	O
</s>
