<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Kronrod	I-Algorithm
quadrature	I-Algorithm
formula	I-Algorithm
is	O
an	O
adaptive	B-Algorithm
method	I-Algorithm
for	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
variant	O
of	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
,	O
in	O
which	O
the	O
evaluation	O
points	O
are	O
chosen	O
so	O
that	O
an	O
accurate	O
approximation	O
can	O
be	O
computed	O
by	O
re-using	O
the	O
information	O
produced	O
by	O
the	O
computation	O
of	O
a	O
less	O
accurate	O
approximation	O
.	O
</s>
<s>
It	O
is	O
an	O
example	O
of	O
what	O
is	O
called	O
a	O
nested	O
quadrature	O
rule	O
:	O
for	O
the	O
same	O
set	O
of	O
function	O
evaluation	O
points	O
,	O
it	O
has	O
two	O
quadrature	B-Algorithm
rules	I-Algorithm
,	O
one	O
higher	O
order	O
and	O
one	O
lower	O
order	O
(	O
the	O
latter	O
called	O
an	O
embedded	O
rule	O
)	O
.	O
</s>
<s>
where	O
wi	O
,	O
xi	O
are	O
the	O
weights	B-Algorithm
and	O
points	O
at	O
which	O
to	O
evaluate	O
the	O
function	O
f(x )	O
.	O
</s>
<s>
Gauss	O
–	O
Kronrod	O
formulas	O
are	O
extensions	O
of	O
the	O
Gauss	B-Algorithm
quadrature	I-Algorithm
formulas	O
generated	O
by	O
adding	O
points	O
to	O
an	O
-point	O
rule	O
in	O
such	O
a	O
way	O
that	O
the	O
resulting	O
rule	O
is	O
of	O
order	O
(	O
;	O
the	O
corresponding	O
Gauss	O
rule	O
is	O
of	O
order	O
)	O
.	O
</s>
<s>
The	O
difference	O
between	O
a	O
Gauss	B-Algorithm
quadrature	I-Algorithm
rule	O
and	O
its	O
Kronrod	O
extension	O
are	O
often	O
used	O
as	O
an	O
estimate	O
of	O
the	O
approximation	O
error	O
.	O
</s>
<s>
For	O
an	O
arbitrary	O
interval	O
the	O
node	O
positions	O
and	O
weights	B-Algorithm
are	O
scaled	O
to	O
the	O
interval	O
as	O
follows	O
:	O
</s>
