1. This is a problem on Gaussian quadrature and related.
(a) Explain the principle of the Gaussian quadrature.
(b) Show that the Gaussian quadrature
can be exact for all polynomials of degree 3 with 2 points x0 = 1p3 and x1 = 1p3. Find c0 and c1 as
well. [Hint: Follow class notes closely.]
(c) Determine the values of ci and xi, i = 0;1 so that the quadrature formula
∫ 11x2f(x)dx = c0f(x0) + c1f(x1)
will be exact for all polynomials of degree 3. [Hint: Follow the procedures solving part (b) closely.]
2. Mass spectrometry analysis gives a series of peak height readings for various ion masses. For each peak,
the height hi is contributed to by various constituents (measured by the index j). The jth component
with per unit concentration pj makes the contribution cij to the ith peak, so that the following relation
holds.
hi =n∑j=1cijpj;
where n is the number of components present. Carnahan (1964) gives the cij values shown in the table
below:
Peak number unknown CH4 C2H4 C2H6 C3H6 C3H8
1 0.165 0.202 0.317 0.234 0.182
2 27.7 0.862 0.062 0.073 0.131
3 22.35 13.05 4.42 6.001
4 11.28 0 1.11
5 9.85 1.684
If a sample had measured peak heights h1 = 5:20, h2 = 61:7, h3 = 149:2, h4 = 79:4, and h5 = 89:3.
Calculate the values of pj for each component. The total of all the pj values was 51.53. Use Jacobi’s
and Gauss-Seidal iteration approaches, till adjacent iterations is within 10 6.
Requirements: Rearrange the measurements to take advantages of the principle and strength of each
iteration method