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
Famous quotes containing the words instability of, instability and/or equation:
“Not only does the wind of accidents stir me according to its blowing, but I am also stirred and troubled by the instability of my attitude.”
—Michel de Montaigne (1533–1592)
“The difference between tragedy and comedy is the difference between experience and intuition. In the experience we strive against every condition of our animal life: against death, against the frustration of ambition, against the instability of human love. In the intuition we trust the arduous eccentricities we’re born to, and see the oddness of a creature who has never got acclimatized to being created.”
—Christopher Fry (b. 1907)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)