The authors obtain the rates of convergence (or divergence) of Gaussian quadrature on functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a bounded smooth weight function on
[
−
1
,
1
]
[ - 1,1]
, the error in n-point Gaussian quadrature of
f
(
x
)
=
|
x
−
y
|
−
δ
f(x) = |x - y{|^{ - \delta }}
is
O
(
n
−
2
+
2
δ
)
O({n^{ - 2 + 2\delta }})
if
y
=
±
1
y = \pm 1
and
O
(
n
−
1
+
δ
)
O({n^{ - 1 + \delta }})
if
y
∈
(
−
1
,
1
)
y \in ( - 1,1)
, provided we avoid the singularity. If we ignore the singularity y, the error is
O
(
n
−
1
+
2
δ
(
log
n
)
δ
(
log
log
n
)
δ
(
1
+
ε
)
)
O({n^{ - 1 + 2\delta }}{(\log n)^\delta }{(\log \log n)^{\delta (1 + \varepsilon )}})
for almost all choices of y. These assertions are sharp with respect to order.