2.6 Numerical solution of equations

If f is continuous and f(a) and f(b) have opposite signs, at least one root lies in (a,b). A bracket locates a root; it does not necessarily prove uniqueness.

Check continuity and sign values, then narrow the interval by bisection or another allowed method. State the interval and its width when reporting accuracy.

f(1)<0 and f(2)>0 implies a root in (1,2) for continuous f; repeated bisection gives a guaranteed bracket.

A sign change can hide an odd number of roots, and a zero endpoint must be treated separately rather than called an interior root.

Iteration replaces x by x_{n+1}=g(x_n). A fixed point α satisfies g(α)=α, corresponding to a root of the rearranged equation.

Choose a starting value in the stated interval, compute enough figures during iteration, and stop using a tolerance on successive values or the residual. Different rearrangements can converge differently.

For x=cos x, starting x₀=0 gives a sequence approaching about 0.739; starting values should remain in a region where g behaves stably.

A few stable-looking digits do not prove convergence, and iteration can diverge or enter a cycle even when the equation has a root.

For x_{n+1}=g(x_n), convergence near a fixed point is encouraged when |g′(x)|<1 there; the derivative also indicates how quickly errors shrink locally.

Derive g from the original equation without changing the root set, select an admissible initial value, and verify the final approximation in the original function.

Rearranging x³+x−1=0 as x=(1−x)^{1/3} may behave differently from another rearrangement; compare the local slope before iterating.

An iteration formula is not automatically valid just because it is algebraically equivalent; convergence is a separate question.