Instability of The Quadratic Equation
For example, consider the venerable quadratic equation:
- ,
with the two exact solutions:
This formula may not always produce an accurate result. For example, when c is very small, loss of significance can occur in either of the root calculations, depending on the sign of b.
The case, will serve to illustrate the problem:
We have
In real arithmetic, the roots are
In 10-digit floating-point arithmetic,
Notice that the solution of greater magnitude is accurate to ten digits, but the first nonzero digit of the solution of lesser magnitude is wrong.
Because of the subtraction that occurs in the quadratic equation, it does not constitute a stable algorithm to calculate the two roots.
Read more about this topic: Loss Of Significance
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