We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type
\[
∫
−
1
1
K
(
x
,
y
)
f
(
x
)
d
x
,
−
1
≤
y
≤
1
,
\int _{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1,
\]
where the kernel
K
(
x
,
y
)
K(x,y)
is weakly singular and the function
f
(
x
)
f(x)
has singularities only at the endpoints
±
1
\pm 1
. In particular, when
K
(
x
,
y
)
=
log
|
x
−
y
|
K(x,y) = \log |x - y|
,
K
(
x
,
y
)
=
|
x
−
y
|
v
K(x,y) = |x - y{|^v}
,
v
>
−
1
v > - 1
, and
f
(
x
)
f(x)
has algebraic singularities of the form
(
1
±
x
)
σ
{(1 \pm x)^\sigma }
,
σ
>
−
1
\sigma > - 1
, we prove that the uniform rate of convergence of the rules is
O
(
m
−
2
−
2
σ
log
2
m
)
O({m^{ - 2 - 2\sigma }}{\log ^2}m)
in the case of the first kernel, and
O
(
m
−
2
−
2
σ
−
2
v
log
m
)
O({m^{ - 2 - 2\sigma - 2v}}\log m)
if
v
≤
0
v \leq 0
, or
O
(
m
−
2
−
2
σ
log
m
)
O({m^{ - 2 - 2\sigma }}\log m)
if
v
>
0
v > 0
, for the second, where m is the number of points in the quadrature rule.