Here we were tasked with initializing a matrix of the specified form and solving it for \( n=10 \) grid steps. The specified matrix is of the form:
\( A = \begin{equation*} \begin{pmatrix} 16 & -9 & \frac{8}{3} & -\frac{1}{4} \\ -4 & 6 & -4 & 1 \\ 1 & -4 & 6 & -4 & 1 \\ & 1 & -4 & 6 & -4 & 1 \\ & & \unicode{x22f1} & \unicode{x22f1} & \unicode{x22f1} & \unicode{x22f1} & \unicode{x22f1} \\ & & & 1 & -4 & 6 & -4 & 1 \\ & & & & 1 & -4 & 6 & -4 & 1 \\ & & & & & \frac{16}{17} & -\frac{60}{17} & \frac{72}{17} & -\frac{28}{17} \\ & & & & & -\frac{12}{17} & \frac{96}{17} & -\frac{156}{17} & -\frac{72}{17} \\ \end{pmatrix} \end{equation*} \)
Our force in this case is the constant force \( f = -480wdg\) defined on the main page. Hence, our \( b \) vector is \( \frac{h^4}{EI} \cdot f \) for each of the 10 entries. The code used to obtain the following result my be obtained through the "Get Code" button on the main page or clicking the links on this page. Using the files problem1.m, initmatrix.m, and simpleforces.m we set up the specified matrix and performed Gauss Elimination via the "\" MATLAB command to find our solution vector:
\(y = \begin{equation} \begin{pmatrix} -0.000180624738462 \\ -0.000674847507692 \\ -0.001416986584615 \\ -0.002349087507692 \\ -0.003420923076923 \\ -0.004589993353846 \\ -0.005821525661538 \\ -0.007088474584615 \\ -0.008371521969230 \\ -0.009659076923076 \\ \end{pmatrix} \end{equation} \)
The correct solution is \( y(x) = (\frac{f}{24EI})x^2(x^2-4Lx+6L^2) \), which we plot against our solution in Problem 2.
<< Previous Next >>Basic initialization of matrix with \(n=10\)
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