Numerical Solution of the Quadratic Formula
This problem illustrates how students can run into numerical problems if they are not careful. With modern calculators that carry many significant figures, this may seem like a moot point. But the idea is still important and can creep subtly into computer computations and cause unexpected, difficult-to-debug errors.
Numerical Recipes |
Are there other examples of numerical errors creeping into calculations? Yes. You can find one discussed in Numerical Recipes that involves the quadratic formula.
We all know the quadratic formula from high school. If you have a quadratic equation of the form
This calculation is susceptible to numerical error. For instance, suppose all numerical calculations are performed to only four significant figures. Then when you reach the step
You might say “so what! Who uses only four significant figures in their calculations?” Okay, try solving
What’s the moral to this story? Is it simply that you must use high precision when doing calculations? No. We can do better. Notice that the solution is fine when using the plus sign in the quadratic equation. We need make no changes. It’s the negative sign that gives the problem,
Let’s try a trick; multiply the expression by a very special form of one:
I should note that the problem is fixed only for negative values of b. If b is positive, you can use an analogous approach to get a slightly different form of the solution (I’ll leave that as an exercise for the reader).
So, the moral of the story is: if you find that your numerical calculation is susceptible to numerical error, fix it! Look for a trick that eliminates the problem. Often you can find one.
Source: http://hobbieroth.blogspot.com/2024/05/numerical-solution-of-quadratic-formula.html
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