Numerical Solution of Equations

• If f is continuous and f(a), f(b) have opposite signs, there is a root between a and b.
• A sketch of two graphs shows how many roots there are and roughly where.
• An iteration x_n+1 = F(x_n) comes from rearranging f(x) = 0 as x = F(x).
• If the iteration converges, its limit α satisfies α = F(α), which is the original equation.
• An iteration may fail to converge; a different rearrangement may then work.
• Confirm a root to a given accuracy by a sign change over the rounding interval.

Trapping a root

Write the equation as f(x) = 0 and evaluate f at both ends of the interval. A sign change shows a root lies between them, provided f is continuous there. Watch for the traps: y = 1/(x − 1) changes sign across its asymptote without a root, and a repeated root (curve touching the axis) gives no sign change at all.

A sketch helps you decide how many roots there are: e^x = 4 − x has exactly one root because the increasing curve y = e^x meets the decreasing line y = 4 − x once.

x_n+1 = F(x_n)

Rearrange f(x) = 0 into the form x = F(x), choose a starting value x₁ and keep substituting. If the values settle down to a limit α, then α = F(α), so α is a root of the original equation. Keep enough figures in each iterate, then give the answer once two successive values agree to the required accuracy. You are not required to know the condition for convergence, but some rearrangements diverge or oscillate.