<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
is	O
a	O
form	O
of	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
for	O
approximating	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
a	O
function	O
.	O
</s>
<s>
Many	O
algorithms	O
have	O
been	O
developed	O
for	O
computing	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
rules	O
.	O
</s>
<s>
In	O
2014	O
,	O
Ignace	O
Bogaert	O
presented	O
explicit	O
asymptotic	O
formulas	O
for	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
weights	O
and	O
nodes	O
,	O
which	O
are	O
accurate	O
to	O
within	O
double-precision	O
machine	B-Algorithm
epsilon	I-Algorithm
for	O
any	O
choice	O
of	O
n≥21	O
.	O
</s>
<s>
Carl	O
Friedrich	O
Gauss	O
was	O
the	O
first	O
to	O
derive	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
rule	O
,	O
doing	O
so	O
by	O
a	O
calculation	O
with	O
continued	O
fractions	O
in	O
1814	O
.	O
</s>
<s>
For	O
integrating	O
f	O
over	O
with	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
,	O
the	O
associated	O
orthogonal	O
polynomials	O
are	O
Legendre	O
polynomials	O
,	O
denoted	O
by	O
.	O
</s>
<s>
Several	O
researchers	O
have	O
developed	O
algorithms	O
for	O
computing	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
nodes	O
and	O
weights	O
based	O
on	O
the	O
Newton	O
–	O
Raphson	O
method	O
for	O
finding	O
roots	O
of	O
functions	O
.	O
</s>
<s>
Some	O
methods	O
utilize	O
formulas	O
to	O
approximate	O
the	O
weights	O
and	O
then	O
use	O
a	O
few	O
iterations	O
of	O
Newton-Raphson	O
to	O
lower	O
the	O
error	O
of	O
the	O
approximation	O
to	O
below	O
machine	B-Algorithm
precision	I-Algorithm
.	O
</s>
<s>
In	O
1969	O
,	O
Golub	O
and	O
Welsch	O
published	O
their	O
method	O
for	O
computing	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
rules	O
given	O
the	O
three	O
term	O
recurrence	O
relation	O
that	O
the	O
underlying	O
orthogonal	O
polynomials	O
satisfy	O
.	O
</s>
<s>
They	O
reduce	O
the	O
problem	O
of	O
computing	O
the	O
nodes	O
of	O
a	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
rule	O
to	O
the	O
problem	O
of	O
finding	O
the	O
eigenvalues	O
of	O
a	O
particular	O
symmetric	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
By	O
taking	O
advantage	O
of	O
the	O
symmetric	O
tridiagonal	B-Algorithm
structure	O
,	O
the	O
eigenvalues	O
can	O
be	O
computed	O
in	O
time	O
,	O
as	O
opposed	O
to	O
the	O
time	O
expected	O
for	O
a	O
generic	O
eigenvalue	O
problem	O
.	O
</s>
<s>
In	O
a	O
2014	O
paper	O
,	O
Ignace	O
Bogaert	O
derives	O
asymptotic	O
formulas	O
for	O
the	O
nodes	O
that	O
are	O
exact	O
up	O
to	O
machine	B-Algorithm
precision	I-Algorithm
for	O
and	O
for	O
the	O
weights	O
that	O
are	O
exact	O
up	O
to	O
machine	B-Algorithm
precision	I-Algorithm
for	O
.	O
</s>
<s>
Johansson	O
and	O
Mezzarobba	O
describe	O
a	O
strategy	O
to	O
compute	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
rules	O
in	O
arbitrary-precision	B-Algorithm
arithmetic	I-Algorithm
,	O
allowing	O
both	O
small	O
and	O
large	O
.	O
</s>
<s>
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
is	O
optimal	O
in	O
a	O
very	O
narrow	O
sense	O
for	O
computing	O
integrals	B-Algorithm
of	O
a	O
function	O
f	O
over	O
,	O
since	O
no	O
other	O
quadrature	O
rule	O
integrates	O
all	O
degree	O
polynomials	O
exactly	O
when	O
using	O
n	O
sample	O
points	O
.	O
</s>
<s>
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
is	O
based	O
on	O
approximating	O
f	O
by	O
a	O
polynomial	O
interpolant	O
at	O
Chebyshev	B-Algorithm
nodes	I-Algorithm
and	O
integrates	O
polynomials	O
of	O
degree	O
up	O
to	O
n	O
exactly	O
when	O
given	O
n	O
samples	O
.	O
</s>
<s>
Even	O
though	O
it	O
does	O
not	O
integrate	O
polynomials	O
or	O
other	O
functions	O
that	O
are	O
analytic	O
in	O
a	O
large	O
neighborhood	O
of	O
as	O
well	O
as	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
,	O
Clenshaw	O
–	O
Curtis	O
converges	O
at	O
approximately	O
the	O
same	O
rate	O
as	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
for	O
most	O
(	O
non-analytic	O
)	O
integrands	O
.	O
</s>
<s>
Also	O
,	O
Clenshaw	O
–	O
Curtis	O
shares	O
the	O
properties	O
that	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
enjoys	O
of	O
convergence	O
for	O
all	O
continuous	O
integrands	O
f	O
and	O
robustness	O
to	O
numerical	O
rounding	O
errors	O
.	O
</s>
<s>
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
quadrature	I-Algorithm
is	O
based	O
on	O
approximating	O
f	O
by	O
a	O
polynomial	O
interpolant	O
at	O
equally-spaced	O
points	O
in	O
,	O
and	O
like	O
Clenshaw	O
–	O
Curtis	O
also	O
integrates	O
polynomials	O
of	O
degree	O
up	O
to	O
n	O
exactly	O
when	O
given	O
n	O
samples	O
.	O
</s>
