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Memory Cards — Chapter 2
Each card contains a compact summary of what must be remembered from the chapter.
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Multiple roots
Multiple roots slow Newton's method. Multiplicity correction can restore fast convergence when the multiplicity is known.
Bisection
Bisection is globally reliable under a sign change. It is linearly convergent and preserves a bracket at every step.
Fixed-point iteration
The iteration \(x_{n+1}=g(x_n)\) converges locally when \(|g'(x_\ast)|<1\).
Newton method
Newton is locally quadratic for a simple root if the start is close enough and \(f'\) is not zero near the root.
Root-finding problem
Solve \(f(x)=0\). A complete answer needs assumptions, starting data, stopping rule, residual, and error interpretation.
Hybrid methods
Practical solvers often combine bracketing safety with Newton or secant speed.
Secant method
The secant method avoids derivatives and is superlinear, but it does not preserve a bracket.