Numerical Solution of Equations

• If f(x) is continuous and f(a), f(b) have opposite signs, a root of f(x) = 0 lies 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) that converges tends to a root α satisfying α = F(α).
• Iterate until the values agree to the required accuracy, then confirm with a sign change over the rounding interval.
• An iteration may fail to converge even when a root exists.

Trapping a root

Rearrange the equation to f(x) = 0. Evaluate f at two values: a change of sign shows a root between them, provided f is continuous on that interval. Graphically, you can also sketch y = g(x) and y = h(x) and count their intersections for g(x) = h(x).

A sign change can mislead: f(x) = 1/(x − 1) changes sign between 0 and 2 because of an asymptote, not a root. And a repeated root (a curve that touches the axis) gives no sign change at all.

x_{n + 1} = F(x_n)

Rearrange f(x) = 0 into x = F(x). Starting from x₁, compute x₂ = F(x₁), x₃ = F(x₂), … If the values converge to α, then α = F(α), so α is a root of the original equation. To show which equation an iteration solves, put x_{n + 1} = x_n = x and rearrange.

Write down each iterate to more decimal places than the answer needs (usually 2 more), stop when two consecutive values agree to the required accuracy, then (if asked) confirm with a sign change across the rounding interval.