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Fixed-point iteration
The iteration \(x_{n+1}=g(x_n)\) converges locally when \(|g'(x_\ast)|<1\).
Hybrid methods
Practical solvers often combine bracketing safety with Newton or secant speed.
Root-finding problem
Solve \(f(x)=0\). A complete answer needs assumptions, starting data, stopping rule, residual, and error interpretation.
Secant method
The secant method avoids derivatives and is superlinear, but it does not preserve a bracket.
Bisection
Bisection is globally reliable under a sign change. It is linearly convergent and preserves a bracket at every step.
Multiple roots
Multiple roots slow Newton's method. Multiplicity correction can restore fast convergence when the multiplicity is known.
Newton method
Newton is locally quadratic for a simple root if the start is close enough and \(f'\) is not zero near the root.