<s>
In	O
mathematics	O
and	O
computing	O
,	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
(	O
LMA	O
or	O
just	O
LM	O
)	O
,	O
also	O
known	O
as	O
the	O
damped	O
least-squares	B-Algorithm
(	O
DLS	O
)	O
method	O
,	O
is	O
used	O
to	O
solve	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problems	O
.	O
</s>
<s>
These	O
minimization	O
problems	O
arise	O
especially	O
in	O
least	B-Algorithm
squares	I-Algorithm
curve	B-Algorithm
fitting	I-Algorithm
.	O
</s>
<s>
The	O
LMA	O
interpolates	O
between	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
(	O
GNA	O
)	O
and	O
the	O
method	O
of	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
The	O
LMA	O
is	O
more	O
robust	B-Application
than	O
the	O
GNA	O
,	O
which	O
means	O
that	O
in	O
many	O
cases	O
it	O
finds	O
a	O
solution	O
even	O
if	O
it	O
starts	O
very	O
far	O
off	O
the	O
final	O
minimum	O
.	O
</s>
<s>
LMA	O
can	O
also	O
be	O
viewed	O
as	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
using	O
a	O
trust	B-Algorithm
region	I-Algorithm
approach	O
.	O
</s>
<s>
The	O
LMA	O
is	O
used	O
in	O
many	O
software	O
applications	O
for	O
solving	O
generic	O
curve-fitting	B-Algorithm
problems	O
.	O
</s>
<s>
By	O
using	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
it	O
often	O
converges	O
faster	O
than	O
first-order	O
methods	O
.	O
</s>
<s>
However	O
,	O
like	O
other	O
iterative	B-Algorithm
optimization	O
algorithms	O
,	O
the	O
LMA	O
finds	O
only	O
a	O
local	O
minimum	O
,	O
which	O
is	O
not	O
necessarily	O
the	O
global	O
minimum	O
.	O
</s>
<s>
The	O
primary	O
application	O
of	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
is	O
in	O
the	O
least-squares	B-Algorithm
curve	B-Algorithm
fitting	I-Algorithm
problem	I-Algorithm
:	O
given	O
a	O
set	O
of	O
empirical	O
pairs	O
of	O
independent	O
and	O
dependent	O
variables	O
,	O
find	O
the	O
parameters	O
of	O
the	O
model	O
curve	O
so	O
that	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
deviations	O
is	O
minimized	O
:	O
</s>
<s>
Like	O
other	O
numeric	O
minimization	O
algorithms	O
,	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
iterative	B-Algorithm
procedure	O
.	O
</s>
<s>
In	O
each	O
iteration	B-Algorithm
step	O
,	O
the	O
parameter	O
vector	O
is	O
replaced	O
by	O
a	O
new	O
estimate	O
.	O
</s>
<s>
The	O
above	O
expression	O
obtained	O
for	O
comes	O
under	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
(	O
non-negative	O
)	O
damping	O
factor	O
is	O
adjusted	O
at	O
each	O
iteration	B-Algorithm
.	O
</s>
<s>
If	O
reduction	O
of	O
is	O
rapid	O
,	O
a	O
smaller	O
value	O
can	O
be	O
used	O
,	O
bringing	O
the	O
algorithm	O
closer	O
to	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
,	O
whereas	O
if	O
an	O
iteration	B-Algorithm
gives	O
insufficient	O
reduction	O
in	O
the	O
residual	O
,	O
can	O
be	O
increased	O
,	O
giving	O
a	O
step	O
closer	O
to	O
the	O
gradient-descent	O
direction	O
.	O
</s>
<s>
If	O
either	O
the	O
length	O
of	O
the	O
calculated	O
step	O
or	O
the	O
reduction	O
of	O
sum	O
of	O
squares	O
from	O
the	O
latest	O
parameter	O
vector	O
fall	O
below	O
predefined	O
limits	O
,	O
iteration	B-Algorithm
stops	O
,	O
and	O
the	O
last	O
parameter	O
vector	O
is	O
considered	O
to	O
be	O
the	O
solution	O
.	O
</s>
<s>
Fletcher	O
in	O
his	O
1971	O
paper	O
A	O
modified	O
Marquardt	O
subroutine	O
for	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
simplified	O
the	O
form	O
,	O
replacing	O
the	O
identity	O
matrix	O
with	O
the	O
diagonal	O
matrix	O
consisting	O
of	O
the	O
diagonal	O
elements	O
of	O
:	O
</s>
<s>
A	O
similar	O
damping	O
factor	O
appears	O
in	O
Tikhonov	O
regularization	O
,	O
which	O
is	O
used	O
to	O
solve	O
linear	O
ill-posed	B-Algorithm
problems	I-Algorithm
,	O
as	O
well	O
as	O
in	O
ridge	O
regression	O
,	O
an	O
estimation	O
technique	O
in	O
statistics	O
.	O
</s>
<s>
Theoretical	O
arguments	O
exist	O
showing	O
why	O
some	O
of	O
these	O
choices	O
guarantee	O
local	O
convergence	O
of	O
the	O
algorithm	O
;	O
however	O
,	O
these	O
choices	O
can	O
make	O
the	O
global	O
convergence	O
of	O
the	O
algorithm	O
suffer	O
from	O
the	O
undesirable	O
properties	O
of	O
steepest	B-Algorithm
descent	I-Algorithm
,	O
in	O
particular	O
,	O
very	O
slow	O
convergence	O
close	O
to	O
the	O
optimum	O
.	O
</s>
<s>
The	O
idea	O
behind	O
this	O
strategy	O
is	O
to	O
avoid	O
moving	O
downhill	O
too	O
fast	O
in	O
the	O
beginning	O
of	O
optimization	O
,	O
therefore	O
restricting	O
the	O
steps	O
available	O
in	O
future	O
iterations	B-Algorithm
and	O
therefore	O
slowing	O
down	O
convergence	O
.	O
</s>
<s>
The	O
choice	O
of	O
the	O
finite	B-Algorithm
difference	I-Algorithm
step	O
can	O
affect	O
the	O
stability	O
of	O
the	O
algorithm	O
,	O
and	O
a	O
value	O
of	O
around	O
0.1	O
is	O
usually	O
reasonable	O
in	O
general	O
.	O
</s>
<s>
In	O
this	O
example	O
we	O
try	O
to	O
fit	O
the	O
function	O
using	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
implemented	O
in	O
GNU	B-Language
Octave	I-Language
as	O
the	O
leasqr	O
function	O
.	O
</s>
<s>
is	O
an	O
example	O
of	O
very	O
sensitive	O
initial	O
conditions	O
for	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
