(Solved): gaussian elimination with partial pivoting Consider the linear system {x1+x2=3x1+2x2=5 ...

Consider the linear system $$\{\begin{array}{l}?x_1+x_2=3\\ x_1+2x_2=5\end{array}$$ where $$????1$$ is a very small parameter. In this exercise, we will solve this linear system by regular Gaussian elimination and then by Gaussian elimination with partial pivoting. We will see that Gaussian elimination with partial pivoting is preferred when numerical round-off errors are present. a. What is the solution of the linear system when $$?=0$$ ? We can use this solution as a proxy for the exact solution when $$??=0$$ but $$????1$$. $$2$$ b. Let's solve the system by regular Gaussian elimination, subject to numerical round-off. i. Reduce the system to triangular form by regular Gaussian elimination. ii. Since $$????1$$, expressions like $$a+b/?$$ are often rounded to just $$b/?$$ on a computer $${}^1$$. Apply this approximation to your triangular system. Carry out back substitution as usual. Compare your solution with (a). c. Let's solve the system by Gaussian elimination with partial pivoting, subject to numerical round-off. i. Reduce the system to triangular form by Gaussian elimination with partial pivoting. ii. Since $$????1$$, expressions like $$a+b?$$ are often rounded to just $$a$$ on a computer. Apply this approximation to your triangular system. Carry out back substitution as usual. Compare your solution with (a).