Download Assignment V

Assignment V
1. Write Im for the composite Trapezoidal rule, Sm for the composite Simpson rule and Mm for
the composite midpoint rule, each with m subintervals. Show that
Mm = 2I2m − Im ,
S2m =
4I2m − Im
,
3
Sm =
2Mm + Im
3
2. A quadrature formula on the interval [−1, 1] uses the quadrature points x0 = −α and x1 = α,
where 0 < α ≤ 1:
Z
1
f (x)dx ≈ w0 f (−α) + w1 f (α)
−1
The formula is required to be exact whenever f is a polynomial of degree 1. Show that w0 =
w1 = 1, independent of the value of α. Show also that there is one particular value of α for
which the formula is also exact for all polynomials of degree 2. Find this α and show that, for
this value, the formula is also exact for all polynomials of degree 3.
3. Write down errors in the approximation of
Z 1
x4 dx
Z
and
0
1
x5 dx
0
by the Trapezoidal and Simpson’s rule. Hence find the value of the constant C for which the
Trapezoidal rule gives the correct result for the calculation of
Z 1
(x5 − Cx4 )dx,
0
and show that the Trapezoidal rule gives a more accurate result than Simpson’s rule when
15/14 < C < 85/74.
4. Determine the values of cj , j = −1, 0, 1, 2, such that the quadrature rule
Q(f ) = c−1 f (−1) + c0 f (0) + c1 f (1) + c2 f (2)
gives correct value for the integral
Z
1
f (x)dx
0
when f is any polynomial of degree 3.
5. Find the quadrature weights a0 , a1 , a2 , a3 and the (remaining) quadrature points x1 , x2 of a
quadrature formula of the form
Z 1
f (x)dx ≈ a0 f (−1) + a1 f (x1 ) + a2 f (x2 ) + a3 f (1)
−1
that is exact for all polynomials of degree 5.
6. Consider the integral I(f ) =
R1
0
ex dx and estimate the minimum number m of subintervals that
is needed for computing I(f ) up to an absolute error ≤ 5×10−4 using the composite Trapezoidal
and Simpson’s rules.
7. The function f has a continuous fourth derivative on the interval [−1, 1]. Construct the Hermite
interpolation polynomial of degree 3 using the interpolation points x0 = −1 and x0 = 1. Deduce
that
Z
1
1
f (x)dx = [f (−1) + f (1)] + [f 0 (−1) − f 0 (1)] + E
3
−1
where
|E| ≤
2
max |f iv (x)|
45 x∈[−1,1]
8. Prove that the weights of the Gaussian quadrature formulae are all positive.
9. Show that there exists no polynomial interpolatory quadrature of order n that integrates polynomial of degree 2n + 2 exactly.
10. Suppose that f has a continuous second derivative in [0, 1]. Show that there is a point ζ in (0, 1)
such that
Z
0
1
1
1
xf (x)dx = f (2/3) + f 00 (ζ)
2
72
11. Construct orthogonal polynomial of degree 0, 1, 2 on the interval (0, 1) with the weight function
w(x) = − ln x
12. Suppose that the polynomials φj , j = 0, 1, · · · form an orthogonal system on the interval (0, 1)
with respect to the weight function w(x) = xα , α > 0. Find, in terms of φj , a system of
orthogonal polynomials for the interval (0, b) and the same weight function.
13. Suppose that the polynomials φj , j = 0, 1, · · · form an orthogonal system on the interval (a, b)
with respect to the weight function w(x). Show that, for some value of the constant Cj ,
φj+1 (x) − Cj xφj (x) is a polynomial of degree j, and hence deduce that
φj+1 (x) − Cj xφj (x) =
j
X
αjk φk (x),
αjk ∈ R
k=0
Use the orthogonality properties to show that αjk = 0 for k < j − 1, and deduce that the
polynomials satisfy a recurrence relation of the form
φj+1 (x) − (Cj x + Dj )φj (x) + Ej φj−1 (x) = 0,
j≥1
14. The Lagurre polynomials are orthogonal in the interval (0, ∞) with respect to the weight function
w(x) = e−x . Find the first four Lagurre polynomials such that the coefficient of the leading term
of φk (x) is (−1)k /k!.
15. The Hermite polynomials are orthogonal in the interval (−∞, ∞) with respect to the weight
2
function w(x) = e−x . Find the first four Hermite polynomials such that the coefficient of the
leading term of φk (x) is 2k .