Numerical roots, linear systems, quadrature and ODE stability
Numerical analysis distinguishes the mathematical problem from the algorithm and the precision of its implementation. Conditioning describes sensitivity of the exact solution to changes in input. Stability concerns how an algorithm propagates perturbations and round-off. A small residual does not always imply a small solution error in an ill-conditioned system. State the norm, scale and stopping criterion before judging an answer. More decimal places in an output do not create information absent from the data.
Bisection maintains a sign-changing bracket for a continuous real function. After each completed halving the bracket width is divided by two, and a midpoint estimate has an error bounded by half the current width if a root remains bracketed. A sign change without continuity does not prove a root. Newton's iteration uses x_next=x−f(x)/f'(x), requiring a meaningful derivative and attention to an appropriate starting value. Rapid local convergence near a simple root is not a global guarantee; a small derivative or poor initialization can cause failure. A fixed-point map needs suitable local or global contraction conditions for the usual convergence theorem.
Linear systems can be solved through elimination and factorization. Pivoting can control harmful divisions but does not cure an intrinsically ill-conditioned problem. Distinguish an exact singular system, which may have no solution or many, from a nearly singular system whose numerical solution is sensitive. Iterative methods need convergence conditions; convergence of one example is not proof for every matrix. Document the residual and its scale, and avoid declaring a model valid solely because an algorithm returned a vector.
Interpolation, regression and quadrature serve different purposes. Interpolation fits prescribed values, while regression accommodates a model and error structure. Adding many equally spaced polynomial interpolation points can worsen oscillation in some problems. Trapezoidal and Simpson rules approximate definite integrals with error orders that require smoothness and suitable mesh conditions. Composite Simpson's usual one-third rule uses an even number of equal subintervals. Check the supplied function, interval and convention rather than selecting a memorized coefficient without its setting.
For ordinary differential equations, a method's accuracy and absolute stability are separate concerns. Explicit Euler applied to y'=λy gives y_next=(1+hλ)y. For real negative λ, decay of the discrete solution requires |1+hλ|<1, hence0<h<2/|λ|. A method can be stable at a chosen step yet inaccurate for the desired tolerance. Stiff problems can impose severe explicit-step restrictions even when the exact solution changes slowly. Local truncation error, accumulated global error and floating-point error differ. Compare mesh refinements and use an appropriate method rather than weakening mathematical requirements to obtain a convenient result.