AMATH 341 / CM 271 / CS 371
Assignment 7
Due : Monday March 29, 2010
Instructor: K. D. Papoulia
1. Consider a general quadrature scheme for the interval [
a,b
] that uses distinct quadrature
points
x
1
,...,x
n
and weights
w
1
,...,w
n
where the weights are initially unknown. Suppose
we require this rule to be exact for polynomials of degree up to
n

1. Then an exact
formula for the weight
w
k
is given by the integral over [
a,b
] of the
k
th polynomial in the
Lagrange interpolating basis for the points
x
1
,...,x
n
. (This polynomial appears, e.g., inside
the parentheses in the summation at the bottom of p. 93 of the Moler text, which is p. 1 of
the ‘Interpolation’ chapter online). Explain why this is so. [Hint: what is the outcome of
the quadrature rule if it is applied to this Lagrange basis polynomial?]
Solution.
The
k
th polynomial in the Lagrange interpolating basis is
l
k
(
x
) =
(
x

x
1
)
···
(
x

x
k

1
)(
x

x
k
+1
)
···
(
x

x
n
)
(
x
k

x
1
)
···
(
x
k

x
k

1
)(
x
k

x
k
+1
)
···
(
x
k

x
n
)
.
This polynomial has the property, apparent by inspection, that
l
k
(
x
m
) = 1 if
m
=
k
else
l
k
(
x
m
) = 0. Also apparent is that its degree is
n

1. Since the quadrature rule is supposed
to be exact for polynomials of
n

1 or less, then it is exact for
l
k
, i.e.,
Z
b
a
l
k
(
x
)
dx
=
n
X
i
=1
w
i
l
k
(
x
i
)
.
However, the righthand side simpliﬁes to
w
k
because of the property of the Lagrange basis
polynomials, so we conclude that
w
k
must be the value of the exact integral of the Lagrange
basis polynomial over [
a,b
].
2. Consider integrating a function over the interval [0
,
∞
) using the Matlab function
quad
. This
function takes as its ﬁrst argument a function handle (the integrand), its second and third
arguments the limits of integration, and its fourth the accuracy requirement.
[Special note: the function handle sent to