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formula + \(\int_{-1}^{1} f(x) \, dx \approx \sum_{i=0}^{n-1} c_i + f(x_i)\) is exact for polynomials of the highest possible + degree. For \( n \) nodes, it achieves exactness for + polynomials up to degree \( 2n-1 \). The nodes are roots of + orthogonal polynomials such as the Legendre polynomials, and + the weights ensure exact integration for lower-degree polynomials. + \item[b.] Show that the Gaussian quadrature \(\int_{-1}^{1} f(x) dx = c_0 f(x_0) + c_1 f(x_1)\) can be exact for all polynomials of degree 3 with 2 points \(x_0 = - \frac{1}{\sqrt{3}}\) and \(x_1 = \frac{1}{\sqrt{3}}\). Find \(c_0\) and \(c_1\) as well. + + \underline{Sol}: + \begin{itemize} + \item For \( f(x) = 1 \): + \[ + \int_{-1}^{1} 1 \, dx = 2 \implies c_0 + c_1 = 2 + \] + \item For \( f(x) = x \): + \[ + \int_{-1}^{1} x \, dx = 0 \implies c_0 + \left(-\frac{1}{\sqrt{3}}\right) + c_1 + \left(\frac{1}{\sqrt{3}}\right) = 0 \implies c_0 = c_1 + \] + \end{itemize} + Solving \( c_0 + c_1 = 2 \) and \( c_0 = c_1 \), we get \( + c_0 = c_1 = 1 \). + \begin{itemize} + \item Verification for \( f(x) = x^2 \): + \[ + \int_{-1}^{1} x^2 \, dx = \frac{2}{3} \quad \text{and} + \quad 1 \cdot \left(\frac{1}{\sqrt{3}}\right)^2 + 1 + \cdot \left(-\frac{1}{\sqrt{3}}\right)^2 = \frac{2}{3} + \] + \item Verification for \( f(x) = x^3 \): + \[ + \int_{-1}^{1} x^3 \, dx = 0 \quad \text{and} \quad 1 + \cdot \left(\frac{1}{\sqrt{3}}\right)^3 + 1 \cdot + \left(-\frac{1}{\sqrt{3}}\right)^3 = 0 + \] + \end{itemize} + From this, we get: + \[ + \boxed{c_0 = 1}, \quad \boxed{c_1 = 1} + \] + \item[c.] Determine the values of \(c_i\) and \(x_i, i = 0, 1\) so that the quadrature formula \(\int_{-1}^{1} x^2 f(x) dx = c_0 f(x_0) + c_1 f(x_1)\) will be exact for all polynomials of degree 3. + + \begin{itemize} + \item For \( f(x) = 1 \): + \[ + \int_{-1}^{1} x^2 \, dx = \frac{2}{3} = c_0 + c_1 + \implies c_0 + c_1 = \frac{2}{3} + \] + \item For \( f(x) = x \): + \[ + \int_{-1}^{1} x^3 \, dx = 0 = c_0 x_0 + c_1 x_1 + \] + \item For \( f(x) = x^2 \): + \[ + \int_{-1}^{1} x^4 \, dx = \frac{2}{5} = c_0 x_0^2 + c_1 x_1^2 + \] + \item For \( f(x) = x^3 \): + \[ + \int_{-1}^{1} x^5 \, dx = 0 = c_0 x_0^3 + c_1 x_1^3 + \] + \end{itemize} + Assume symmetry \( x_1 = -x_0 \). Then \( c_0 = c_1 \), and + \( c_0 = c_1 = \frac{1}{3} \). Substituting into \( c_0 x_0^2 + + c_1 x_1^2 = \frac{2}{5} \): + \[ + \frac{2}{3} x_0^2 = \frac{2}{5} \implies x_0 = \sqrt{\frac{3}{5}} + \] + \[ + \boxed{c_0 = \frac{1}{3}}, \quad \boxed{c_1 = \frac{1}{3}}, + \quad \boxed{x_0 = \sqrt{\frac{3}{5}}}, \quad \boxed{x_1 = + -\sqrt{\frac{3}{5}}} + \] \end{enumerate} + + \item[2.] Mass spectrometry analysis gives a series of peak height + readings for various ion masses. For each peak, + the height \(h_i\) is contributed to by various constituents + (measured by the index \(j\)). The \(j^{th}\) component + with per unit concentration \(p_j\) makes the contribution + \(c_{ij}\) to the \(i^{th}\) peak, so that the relation + \(h_i = \sum_{j=1}^{n} c_{ij} p_j\) holds. \(n\) is the number of + components present. Carnahan (1964) gives the \(c_{ij}\) values + shown in the table below: + + \[ + \begin{array}{|c|c|c|c|c|c|c|} + \hline + \textrm{Peak number} & \textrm{Unknown} & CH_4 & C_2 H_4 & + C_2 H_6 & C_3 H_6 & C_3 H_8 \\ + \hline + 1 & & 0.165 & 0.202 & 0.317 & 0.234 & 0.182 \\ + 2 & & 27.7 & 0.862 & 0.062 & 0.073 & 0.131 \\ + 3 & & & 22.35 & 13.05 & 4.42 & 6.001 \\ + 4 & & & & 11.28 & 0 & 1.11 \\ + 5 & & & & & 9.85 & 1.684 \\ + \hline + \end{array} + \] + + If a sample had measured peak heights \(h_i = 5.20, h_2 = 61.7, + h_3 = 149.2, h_4 = 79.4\), and \(h_5 = 89.3\). Calculate the + values of \(p_j\) for each component. The total of all the + \(p_j\) values was 51.53. Use Jacobi's and Gauss-Seidel iteration + approaches, till adjacent iterations is within \(10^{-6}\). + + \textbf{Requirements}: Rearrange the measurements to take + advantages of the principle and strength of each iteration method. + + \underline{Sol}:\\ + + \boxed{ + \begin{aligned} + \text{Unknown} & : \boxed{30.0696} \\ + \text{CH}_4 & : \boxed{2.1701} \\ + \text{C}_2\text{H}_4 & : \boxed{0.0000} \\ + \text{C}_2\text{H}_6 & : \boxed{6.6100} \\ + \text{C}_3\text{H}_6 & : \boxed{8.3206} \\ + \text{C}_3\text{H}_8 & : \boxed{4.3597} \\ + \text{Total} & : \boxed{51.53} + \end{aligned} + } \end{enumerate} \end{document}