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Problem 358

Use a partial pivoting strategy to solve the system $$ \begin{aligned} &(0.100) 10^{-3} \mathrm{x}_{1}+(0.100) 10^{1} \mathrm{x}_{2}=(0.200) 10^{1} \\\ &(0.100) 10^{1} \mathrm{x}_{1}+(0.100) 10^{1} \mathrm{x}_{2}=(0.300) 10^{1} \end{aligned} $$ Is the answer markedly superior to merely using Gauss elimination?

Expert verified

In this specific problem, using a partial pivoting strategy resulted in the correct solutions \(x_1 \approx 1\) and \(x_2 \approx 2\), while merely using Gauss elimination without row swapping led to an incorrect result. Partial pivoting reduced the chance of round-off errors and provided a markedly superior solution by reordering the rows.

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Chapter 16

Show that the system of linear equations $$ \begin{aligned} &5 x+7 y+6 z+5 u=23 \\ &7 x+10 y+8 z+7 u=32 \\ &6 x+8 y+10 z+9 u=33 \\ &5 x+7 y+9 z+10 u=31 \end{aligned} $$ is ill-conditioned. How can ill-conditioning be measured?

Chapter 16

Construct a flow-chart for deriving the characteristic equation of an \(\mathrm{n} \times \mathrm{n}\) matrix.

Chapter 16

Construct a flow-chart for solving the set of simultaneous linear equations $A X=b$ by finding the inverse of the coefficient matrix.

Chapter 16

The largest eigenvalue of the matrix $$ \mathrm{A}=\begin{array}{|rrr} \mid 0 & 5 & -6 \\ -4 & 12 & -12 \mid \\ \mid-2 & -2 & 10 \end{array} $$ was found, using the power method, to be \(\lambda_{1}=16\). An associated eigenvector was \(\mathrm{v}_{1}=[0.5,1.0,-0.5]\). Find the remaining eigenvalues and eigenvectors of \(\mathrm{A}\)

Chapter 16

Eigensystems arise in the physical sciences when we study vibrations. In such systems the eigenvector corresponding to the smallest eigenvalue will have elements that are all of the same sign. Using this information, estimate the smallest eigenvalue of the following matrix by means of the Rayleigh quotient: $$ \mathrm{A}=\mid \begin{array}{rrr} 1.7 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array} $$

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