Numerical Solution of the Quadratic Formula

Homework Problem 7 in Chapter 11 of Intermediate Physics for Medicine and Biology examines fitting data to a straight line. In that problem, the four data points to be fit are (100, 4004), (101, 4017), (102, 4039), and (103, 4063). The goal is to fit the data to the line y = ax + b. For this data, one must perform the calculation of a and b to high precision or else you get large errors. The solution manual (available to instructors upon request) says that

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

the solution is

For example,
 

has two solutions
 

so x = 1 or 2.

Now, suppose the coefficient b is larger,

The solution is

so x = 300 or 0.00667.

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 must subtract 8 from 90,000. You get 89992, which to four significant figures becomes 89990, which has a square root of (again to four significant figures) 300.0. The solutions are therefore x = 300 or 0. The large solution (300) is correct, but the small solution (0 instead of 0.00667) is completely wrong. The main reason is that when using the minus sign for ± you must subtract two numbers that are almost the same (in this case, 300 – 299.98667) to get a much smaller number.

You might say “so what! Who uses only four significant figures in their calculations?” Okay, try solving

where I increased b from 300 to 3000. You’ll find that using even six significant figures gives one nonsense solution (try it). As you make b larger and larger, the calculation becomes more and more difficult. The situation can cause unexpected, difficult-to-debug errors.

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:

Simplifying, we get

Voilà! The denominator has the plus sign in front of the square root, so it is not susceptible to numerical error. The numerator is simplicity itself. Try solving x² – 300x + 2 = 0 using math to four significant figures,

 

No error, even with just four sig figs. The problem is fixed!

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.