Algebraic Equations
Consider the problem of finding all roots of the polynomial . In the limit, this cubic degenerates into the quadratic with roots at .
Singular perturbation analysis suggests that the cubic has another root . Indeed, with, the roots are -0.955, 1.057, and 9.898. With, the roots are -0.995, 1.005, and 99.990. With, the roots are -0.9995, 1.0005, and 999.999.
In a sense, the problem has two different scales: two of the roots converge to finite numbers as decreases, while the third becomes arbitrarily large.
Read more about this topic: Singular Perturbation, Examples of Singular Perturbative Problems
Famous quotes containing the word algebraic:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)