Numerical techniques are computational methods for approximating solutions to mathematical problems that lack closed-form expressions or are analytically intractable. They encompass algorithms for root finding, numerical integration and differentiation, solution of linear and nonlinear systems, eigenvalue problems, optimization, and numerical solution of ordinary and partial differential equations. These techniques rely on discretization, iteration, and error analysis, with careful attention to stability, convergence, conditioning, and computational complexity. They are implemented using floating-point arithmetic and often exploit matrix factorizations, interpolation, finite difference, finite element, or spectral methods to achieve controllable accuracy within specified tolerances.
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