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Take the polynomial $x^2+(-4*10^3)x+2$.
In the floating-point system with $b=10$, $m=4$, $e=4$, if I wanted to find the roots using the quadratic formula what would be the values of the roots?
I got 3.999 as one of my roots and 1.000 as the other but my 1.000 root does not make sense in terms of relative error (I get a huge number).
Could anyone help me with this by going through the steps?
If you apply the classical method for the solution of quadratic equations, the roots are given by $$x_{\pm}=\frac{4\times 10^3\pm \sqrt{16\times 10 ^6-8}}{2}=2\times 10^3\pm \sqrt{4\times 10 ^6-2}$$ $$x_{\pm}=2000 \Big(1\pm\sqrt{1-\frac{1}{2\times 10^6}}\Big)=2000 \Big(1\pm\sqrt{1-{5\times 10^{-7}}}\Big)$$ Now, using Taylor $$\sqrt{1-\epsilon}=1-\frac{\epsilon }{2}-\frac{\epsilon ^2}{8}-\frac{\epsilon ^3}{16}+O\left(\epsilon ^4\right)$$ Make $\epsilon=5 \times 10^{-7}$ and get, for any order of the expansion, the value of $\sqrt{1-\epsilon}$; this will give the values of $x_{\pm}$ to the desired accuracy. For example, if you use the third order, you will get the values with more than $20$ correct decimal places.
There are various approaches. One can say that for the large root, the $2$ is insignificant, so the large root is about $4\times 10^3$. For the small root, $x^2$ is insignificant, so the small root is about $\frac{2}{400\times 10^3}$.
For greater accuracy, find the large root $R$ using the Quadratic Formula. The product of the roots is $2$, so the small root is $\frac{2}{R}$. No loss of significant digits from subtracting two large nearly equal quantities.
Or else (equivalently) for the small root we can use the Citardauq Formula $\frac{2c}{-b\pm \sqrt{b^2-4ac}}$. The small root is equal to $\frac{4}{4\times 10^3 +\sqrt{(4\times 10^3)^2-8}}$.
