<s>
The	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
is	O
a	O
direct	O
numerical	O
method	O
of	O
approximating	O
eigenvalues	O
,	O
originated	O
in	O
the	O
context	O
of	O
solving	O
physical	O
boundary	O
value	O
problems	O
and	O
named	O
after	O
Lord	O
Rayleigh	O
and	O
Walther	O
Ritz	O
.	O
</s>
<s>
According	O
to	O
,	O
citing	O
Richard	O
Courant	O
,	O
both	O
Lord	O
Rayleigh	O
and	O
Walther	O
Ritz	O
independently	O
conceived	O
the	O
idea	O
of	O
utilizing	O
the	O
equivalence	O
between	O
boundary	O
value	O
problems	O
of	O
partial	O
differential	O
equations	O
on	O
the	O
one	O
hand	O
and	O
problems	O
of	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
on	O
the	O
other	O
hand	O
for	O
numerical	O
calculation	O
of	O
the	O
solutions	O
,	O
by	O
substituting	O
for	O
the	O
variational	B-Algorithm
problems	I-Algorithm
simpler	O
approximating	O
extremum	O
problems	O
in	O
which	O
a	O
finite	O
number	O
of	O
parameters	O
need	O
to	O
be	O
determined	O
;	O
see	O
the	O
article	O
Ritz	O
method	O
for	O
details	O
.	O
</s>
<s>
Ironically	O
for	O
the	O
debate	O
,	O
the	O
modern	O
justification	O
of	O
the	O
algorithm	O
drops	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
in	O
favor	O
of	O
the	O
simpler	O
and	O
more	O
general	O
approach	O
of	O
orthogonal	O
projection	O
as	O
in	O
Galerkin	B-Algorithm
method	I-Algorithm
named	O
after	O
Boris	O
Galerkin	O
,	O
thus	O
leading	O
also	O
to	O
the	O
Ritz-Galerkin	B-Algorithm
method	I-Algorithm
naming	O
.	O
</s>
<s>
In	O
the	O
finite	B-Application
element	I-Application
method	I-Application
context	O
,	O
mathematically	O
the	O
same	O
algorithm	O
is	O
commonly	O
called	O
the	O
Ritz-Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
or	O
Ritz	O
method	O
terminology	O
is	O
typical	O
in	O
mechanical	O
and	O
structural	O
engineering	O
to	O
approximate	O
the	O
eigenmodes	O
and	O
resonant	B-Application
frequencies	I-Application
of	O
a	O
structure	O
.	O
</s>
<s>
for	O
the	O
matrix	O
of	O
size	O
using	O
a	O
projected	O
matrix	O
of	O
a	O
smaller	O
size	O
,	O
generated	O
from	O
a	O
given	O
matrix	O
with	O
orthonormal	B-Algorithm
columns	O
.	O
</s>
<s>
If	O
the	O
subspace	O
with	O
the	O
orthonormal	B-Algorithm
basis	O
given	O
by	O
the	O
columns	O
of	O
the	O
matrix	O
contains	O
vectors	O
that	O
are	O
close	O
to	O
eigenvectors	O
of	O
the	O
matrix	O
,	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
above	O
finds	O
Ritz	O
vectors	O
that	O
well	O
approximate	O
these	O
eigenvectors	O
.	O
</s>
<s>
Thus	O
,	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
turns	O
into	O
computing	O
of	O
the	O
Rayleigh	O
quotient	O
if	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
,	O
its	O
Rayleigh	O
quotient	O
(	O
and	O
thus	O
its	O
every	O
Ritz	O
value	O
)	O
is	O
real	O
and	O
takes	O
values	O
within	O
the	O
closed	O
interval	O
of	O
the	O
smallest	O
and	O
largest	O
eigenvalues	O
of	O
.	O
</s>
<s>
Truncated	O
singular	O
value	O
decomposition	O
(	O
SVD	O
)	O
in	O
numerical	O
linear	O
algebra	O
can	O
also	O
use	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
to	O
find	O
approximations	O
to	O
left	O
and	O
right	O
singular	O
vectors	O
of	O
the	O
matrix	O
of	O
size	O
in	O
given	O
subspaces	O
by	O
turning	O
the	O
singular	O
value	O
problem	O
into	O
an	O
eigenvalue	O
problem	O
.	O
</s>
<s>
Having	O
found	O
one	O
set	O
(	O
left	O
of	O
right	O
)	O
of	O
approximate	O
singular	O
vectors	O
and	O
singular	O
values	O
by	O
applying	O
naively	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
to	O
the	O
Hermitian	B-Algorithm
normal	O
matrix	O
or	O
,	O
whichever	O
one	O
is	O
smaller	O
size	O
,	O
one	O
could	O
determine	O
the	O
other	O
set	O
of	O
left	O
of	O
right	O
singular	O
vectors	O
simply	O
by	O
dividing	O
by	O
the	O
singular	O
values	O
,	O
i.e.	O
,	O
and	O
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
used	O
as	O
a	O
post-processing	O
step	O
where	O
the	O
matrix	O
is	O
an	O
output	O
of	O
an	O
eigenvalue	O
solver	O
,	O
e.g.	O
,	O
such	O
as	O
LOBPCG	B-Application
,	O
approximating	O
numerically	O
selected	O
eigenvectors	O
of	O
the	O
normal	O
matrix	O
.	O
</s>
<s>
For	O
an	O
arbitrary	O
matrix	O
,	O
we	O
obtain	O
approximate	O
singular	O
triplets	O
which	O
are	O
optimal	O
given	O
in	O
the	O
sense	O
of	O
optimality	O
of	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
.	O
</s>
