<s>
Quasi-Newton	B-Algorithm
methods	I-Algorithm
are	O
methods	O
used	O
to	O
either	O
find	O
zeroes	O
or	O
local	O
maxima	O
and	O
minima	O
of	O
functions	O
,	O
as	O
an	O
alternative	O
to	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
"	O
full	O
"	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
requires	O
the	O
Jacobian	O
in	O
order	O
to	O
search	O
for	O
zeros	O
,	O
or	O
the	O
Hessian	O
for	O
finding	O
extrema	O
.	O
</s>
<s>
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
to	O
find	O
zeroes	O
of	O
a	O
function	O
of	O
multiple	O
variables	O
is	O
given	O
by	O
,	O
where	O
is	O
the	O
left	O
inverse	O
of	O
the	O
Jacobian	O
matrix	O
of	O
evaluated	O
for	O
.	O
</s>
<s>
Strictly	O
speaking	O
,	O
any	O
method	O
that	O
replaces	O
the	O
exact	O
Jacobian	O
with	O
an	O
approximation	O
is	O
a	O
quasi-Newton	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
methods	O
given	O
below	O
for	O
optimization	O
refer	O
to	O
an	O
important	O
subclass	O
of	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
,	O
secant	O
methods	O
.	O
</s>
<s>
Other	O
methods	O
that	O
can	O
be	O
used	O
are	O
the	O
column-updating	O
method	O
,	O
the	O
inverse	O
column-updating	O
method	O
,	O
the	O
quasi-Newton	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
method	I-Algorithm
and	O
the	O
quasi-Newton	B-Algorithm
inverse	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
More	O
recently	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
have	O
been	O
applied	O
to	O
find	O
the	O
solution	O
of	O
multiple	O
coupled	O
systems	O
of	O
equations	O
(	O
e.g.	O
</s>
<s>
Therefore	O
,	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
can	O
be	O
readily	O
applied	O
to	O
find	O
extrema	O
of	O
a	O
function	O
.	O
</s>
<s>
Most	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
used	O
in	O
optimization	O
exploit	O
this	O
property	O
.	O
</s>
<s>
In	O
optimization	O
,	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
(	O
a	O
special	O
case	O
of	O
variable-metric	B-Algorithm
methods	I-Algorithm
)	O
are	O
algorithms	O
for	O
finding	O
local	O
maxima	O
and	O
minima	O
of	O
functions	O
.	O
</s>
<s>
Quasi-Newton	B-Algorithm
methods	I-Algorithm
are	O
based	O
on	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
to	O
find	O
the	O
stationary	O
point	O
of	O
a	O
function	O
,	O
where	O
the	O
gradient	O
is	O
0	O
.	O
</s>
<s>
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
assumes	O
that	O
the	O
function	O
can	O
be	O
locally	O
approximated	O
as	O
a	O
quadratic	O
in	O
the	O
region	O
around	O
the	O
optimum	O
,	O
and	O
uses	O
the	O
first	O
and	O
second	O
derivatives	B-Algorithm
to	O
find	O
the	O
stationary	O
point	O
.	O
</s>
<s>
In	O
higher	O
dimensions	O
,	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
uses	O
the	O
gradient	O
and	O
the	O
Hessian	O
matrix	O
of	O
second	O
derivatives	B-Algorithm
of	O
the	O
function	O
to	O
be	O
minimized	O
.	O
</s>
<s>
In	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
the	O
Hessian	O
matrix	O
does	O
not	O
need	O
to	O
be	O
computed	O
.	O
</s>
<s>
Quasi-Newton	B-Algorithm
methods	I-Algorithm
are	O
a	O
generalization	O
of	O
the	O
secant	O
method	O
to	O
find	O
the	O
root	O
of	O
the	O
first	B-Algorithm
derivative	I-Algorithm
for	O
multidimensional	O
problems	O
.	O
</s>
<s>
In	O
multiple	O
dimensions	O
the	O
secant	O
equation	O
is	O
under-determined	O
,	O
and	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
differ	O
in	O
how	O
they	O
constrain	O
the	O
solution	O
,	O
typically	O
by	O
adding	O
a	O
simple	O
low-rank	O
update	O
to	O
the	O
current	O
estimate	O
of	O
the	O
Hessian	O
.	O
</s>
<s>
The	O
first	O
quasi-Newton	B-Algorithm
algorithm	I-Algorithm
was	O
proposed	O
by	O
William	O
C	O
.	O
Davidon	O
,	O
a	O
physicist	O
working	O
at	O
Argonne	O
National	O
Laboratory	O
.	O
</s>
<s>
He	O
developed	O
the	O
first	O
quasi-Newton	B-Algorithm
algorithm	I-Algorithm
in	O
1959	O
:	O
the	O
DFP	B-Algorithm
updating	I-Algorithm
formula	I-Algorithm
,	O
which	O
was	O
later	O
popularized	O
by	O
Fletcher	O
and	O
Powell	O
in	O
1963	O
,	O
but	O
is	O
rarely	O
used	O
today	O
.	O
</s>
<s>
The	O
most	O
common	O
quasi-Newton	B-Algorithm
algorithms	I-Algorithm
are	O
currently	O
the	O
SR1	B-Algorithm
formula	I-Algorithm
(	O
for	O
"	O
symmetric	B-Algorithm
rank-one	I-Algorithm
"	O
)	O
,	O
the	O
BHHH	B-Algorithm
method	O
,	O
the	O
widespread	O
BFGS	B-Algorithm
method	I-Algorithm
(	O
suggested	O
independently	O
by	O
Broyden	O
,	O
Fletcher	O
,	O
Goldfarb	O
,	O
and	O
Shanno	O
,	O
in	O
1970	O
)	O
,	O
and	O
its	O
low-memory	O
extension	O
L-BFGS	B-Algorithm
.	O
</s>
<s>
The	O
Broyden	O
's	O
class	O
is	O
a	O
linear	O
combination	O
of	O
the	O
DFP	B-Algorithm
and	O
BFGS	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
SR1	B-Algorithm
formula	I-Algorithm
does	O
not	O
guarantee	O
the	O
update	O
matrix	O
to	O
maintain	O
positive-definiteness	B-Algorithm
and	O
can	O
be	O
used	O
for	O
indefinite	O
problems	O
.	O
</s>
<s>
One	O
of	O
the	O
chief	O
advantages	O
of	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
over	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
that	O
the	O
Hessian	O
matrix	O
(	O
or	O
,	O
in	O
the	O
case	O
of	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
,	O
its	O
approximation	O
)	O
does	O
not	O
need	O
to	O
be	O
inverted	O
.	O
</s>
<s>
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
and	O
its	O
derivatives	B-Algorithm
such	O
as	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
,	O
require	O
the	O
Hessian	O
to	O
be	O
inverted	O
,	O
which	O
is	O
typically	O
implemented	O
by	O
solving	O
a	O
system	O
of	O
linear	O
equations	O
and	O
is	O
often	O
quite	O
costly	O
.	O
</s>
<s>
In	O
contrast	O
,	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
usually	O
generate	O
an	O
estimate	O
of	O
directly	O
.	O
</s>
<s>
As	O
in	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
one	O
uses	O
a	O
second-order	O
approximation	O
to	O
find	O
the	O
minimum	O
of	O
a	O
function	O
.	O
</s>
<s>
The	O
various	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
differ	O
in	O
their	O
choice	O
of	O
the	O
solution	O
to	O
the	O
secant	O
equation	O
(	O
in	O
one	O
dimension	O
,	O
all	O
the	O
variants	O
are	O
equivalent	O
)	O
.	O
</s>
<s>
Most	O
methods	O
(	O
but	O
with	O
exceptions	O
,	O
such	O
as	O
Broyden	O
's	O
method	O
)	O
seek	O
a	O
symmetric	O
solution	O
(	O
)	O
;	O
furthermore	O
,	O
the	O
variants	O
listed	O
below	O
can	O
be	O
motivated	O
by	O
finding	O
an	O
update	O
that	O
is	O
as	O
close	O
as	O
possible	O
to	O
in	O
some	O
norm	O
;	O
that	O
is	O
,	O
,	O
where	O
is	O
some	O
positive-definite	B-Algorithm
matrix	I-Algorithm
that	O
defines	O
the	O
norm	O
.	O
</s>
<s>
A	O
key	O
property	O
of	O
the	O
BFGS	B-Algorithm
and	O
DFP	B-Algorithm
updates	O
is	O
that	O
if	O
is	O
positive-definite	O
,	O
and	O
is	O
chosen	O
to	O
satisfy	O
the	O
Wolfe	O
conditions	O
,	O
then	O
is	O
also	O
positive-definite	O
.	O
</s>
<s>
When	O
is	O
a	O
convex	O
quadratic	O
function	O
with	O
positive-definite	O
Hessian	O
,	O
one	O
would	O
expect	O
the	O
matrices	O
generated	O
by	O
a	O
quasi-Newton	B-Algorithm
method	I-Algorithm
to	O
converge	O
to	O
the	O
inverse	O
Hessian	O
.	O
</s>
<s>
This	O
is	O
indeed	O
the	O
case	O
for	O
the	O
class	O
of	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
based	O
on	O
least-change	O
updates	O
.	O
</s>
<s>
Implementations	O
of	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
are	O
available	O
in	O
many	O
programming	O
languages	O
.	O
</s>
<s>
GNU	B-Language
Octave	I-Language
uses	O
a	O
form	O
of	O
BFGS	B-Algorithm
in	O
its	O
fsolve	O
function	O
,	O
with	O
trust	B-Algorithm
region	I-Algorithm
extensions	O
.	O
</s>
<s>
GNU	B-Application
Scientific	I-Application
Library	I-Application
implements	O
the	O
Broyden-Fletcher-Goldfarb-Shanno	O
(	O
BFGS	B-Algorithm
)	O
algorithm	O
.	O
</s>
<s>
R	B-Language
's	O
optim	O
general-purpose	O
optimizer	O
routine	O
uses	O
the	O
BFGS	B-Algorithm
method	I-Algorithm
by	O
using	O
method	O
=	O
"	O
BFGS	B-Algorithm
"	O
.	O
</s>
<s>
Scipy.optimize	O
has	O
fmin_bfgs	O
.	O
</s>
<s>
In	O
the	O
SciPy	B-Application
extension	O
to	O
Python	B-Language
,	O
the	O
scipy.optimize.minimize	O
function	O
includes	O
,	O
among	O
other	O
methods	O
,	O
a	O
BFGS	B-Algorithm
implementation	O
.	O
</s>
<s>
Mathematica	B-Language
includes	O
quasi-Newton	O
solvers	O
.	O
</s>
<s>
The	O
NAG	O
Library	O
contains	O
several	O
routines	O
for	O
minimizing	O
or	O
maximizing	O
a	O
function	O
which	O
use	O
quasi-Newton	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
In	O
MATLAB	O
's	O
Optimization	B-Application
Toolbox	I-Application
,	O
the	O
fminunc	O
function	O
uses	O
(	O
among	O
other	O
methods	O
)	O
the	O
BFGS	B-Algorithm
quasi-Newton	B-Algorithm
method	I-Algorithm
.	O
</s>
