Activity5.1.2

Suppose we have a hypothetical computer that represents numbers using only three decimal digits, as we have been discussing. We will consider the linear system

\begin{equation*} \begin{alignedat}{3} 0.0001x \amp {}+{} \amp y \amp {}={} \amp 1 \\ x \amp {}+{} \amp y \amp {}={} \amp 2\text{.} \\ \end{alignedat} \end{equation*}
  1. Show that this system has the unique solution

    \begin{equation*} \begin{aligned} x \amp {}={} \frac{10000}{9999} = 1.00010001\ldots, \\ y \amp {}={} \frac{9998}{9999} = 0.99989998\ldots\text{.} \end{aligned} \end{equation*}
  2. If we represent this solution inside our computer that only holds 3 decimal digits, what do we find for the solution? This is the best that we can hope to find using our computer.

  3. Let's imagine that we use our computer to find the solution using Gaussian elimination; that is, after every arithmetic operation, we keep only three decimal digits. Our first step is to multiply the first equation by 10000 and subtract it from the second equation. If we represent numbers using only three decimal digits, what does this give for the value of \(y\text{?}\)

  4. By substituting our value for \(y\) into the first equation, what do we find for \(x\text{?}\)

  5. Compare the solution we find on our computer with the actual solution and assess the quality of the approximation.

  6. Let's now modify the linear system by simplying interchanging the equations:

    \begin{equation*} \begin{alignedat}{3} x \amp {}+{} \amp y \amp {}={} \amp 2 \\ 0.0001x \amp {}+{} \amp y \amp {}={} \amp 1\text{.} \\ \end{alignedat} \end{equation*}

    Of course, this doesn't change the actual solution. Let's imagine we use our computer to find the solution using Gaussian elimination. Perform the first step where we multiply the first equation by 0.0001 and subtract from the second equation. What does this give for \(y\) if we represent numbers using only three decimal digits?

  7. Substitute the value you found for \(y\) into the first equation and solve for \(x\text{.}\) Then compare the approximate solution found with our hypothetical computer to the exact solution.

  8. Which approach produces the most accurate approximation?

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