# Singular Perturbation - Examples of Singular Perturbative Problems - Algebraic Equations

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.

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