3.6 Numerical solution of equations

For a continuous f, opposite signs at a and b guarantee at least one root between them. A root-location interval is evidence of existence, not automatically uniqueness.

Evaluate f at the endpoints accurately, keep the interval ordered, and report its width or decimal bounds as the requested accuracy.

If f(1)=−0.4 and f(2)=0.3, continuity gives a root in (1,2); bisection can reduce this to a bracket of width below 0.01.

A sign change can contain several roots, and a root exactly at an endpoint should be stated separately.

With x_{n+1}=g(x_n), a fixed point α satisfies g(α)=α. Starting from x₀ creates a sequence intended to approach α.

Choose a valid starting value, retain guard digits, and stop when successive values or the residual meet the tolerance. Verify the final value in f(x)=0.

For x=cosx, x₀=1 gives iterates that approach approximately 0.739; reporting the last iterate without a tolerance is incomplete.

Iteration can oscillate or diverge even when a root exists; apparent agreement of early digits is not a proof.

For x_{n+1}=g(x_n), a fixed point is locally attractive when |g′(α)|<1; a smaller magnitude generally means faster local error reduction.

Derive g from the original equation, check the derivative on the intended interval and compare stopping errors with the function residual.

Two algebraically equivalent rearrangements of x³+x−1=0 can have different g′ near the root; choose the one with smaller local magnitude.

Algebraic equivalence of equations does not imply numerical equivalence of iterations.