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- CSCI 5606, Assignment 8

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Problem 1. Find a formula of the form

Z 1

−1

f(x)dx ≈ c

X

2

i=0

f(xi)

that is exact for all quadratic polynomials.

Problem 2. Show how the Gaussian quadrature rule

Z 1

−1

f(x)dx ≈

5

9

f(−

s

3

5

) + 8

9

f(0) + 5

9

f(

s

3

5

)

can be used for R a

b

f(x)dx. Apply this result to evaluate

(a) Z π/2

0

xdx

(b) Z 4

0

(sin t)/tdt.

Problem 3. Apply the Romberg algorithm to find R(2, 2) for these integrals:

(a)

Z 3

1

1/xdx

(b)

Z π/2

0

(x/π)

2

dx.

1