Solving Equations Numerically
Most equations cannot be solved by algebra — there is no formula for the roots of a general polynomial of degree 5 or higher, let alone equations mixing polynomials, exponentials, and trig. Root-finding algorithms locate the solutions numerically, to any desired accuracy, by iterating.
Bisection is the simplest and most robust. It relies on the Intermediate Value Theorem (Unit 5): if a continuous has opposite signs at and , a root lies between them. Bisection repeatedly halves the interval, keeping the half where the sign change persists. Each step halves the error, so it is guaranteed to converge (given an initial sign change) but only linearly — steady, if slow.
Newton's method is far faster. From a current guess , it follows the tangent line to the curve down to where it crosses zero, giving the next guess: This is the linear approximation of Unit 5 turned into an iteration. Near a root, Newton's method converges quadratically — the number of correct digits roughly doubles each step, astonishingly fast.
But Newton's speed comes with fragility, the crucial trade-off:
- It needs the derivative , and fails if (a horizontal tangent shoots the next guess to infinity).
- It can diverge or cycle from a poor starting guess.
- It only converges reliably when started close to the root.
Bisection is slow but safe; Newton's is fast but needs a good start and a well-behaved function. In practice, hybrid methods combine them — bisect to get close, then switch to Newton for speed — capturing the best of both.
Common pitfall: assuming Newton's method always converges to a root. It converges quadratically only when started near the root and where is well-behaved; from a poor starting guess it can diverge, cycle, or blow up (especially if , dividing by zero). Bisection, by contrast, is guaranteed to converge given a sign change — the safety Newton's method trades away for speed.