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Numerical computation involves the use of mathematical techniques to solve problems that cannot be solved exactly using analytical methods. These techniques rely on approximations, iterative methods, and statistical analysis to produce accurate results. Numerical computation has a wide range of applications, including:
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\sectionFloating-Point Arithmetic Finite-precision arithmetic leads to rounding errors. Julia provides built-in functions to inspect machine precision: \beginlstlisting[caption=Machine epsilon in Julia] eps(Float64) # 2.220446049250313e-16 eps(Float32) # 1.1920929f-7 \endlstlisting A classic caution: subtracting nearly equal numbers causes catastrophic cancellation. \beginlstlisting x = 1e-10 y = (1 - cos(x)) / x^2 # unstable z = 0.5 * (sin(x/2)/(x/2))^2 # stable println("Unstable: $y, Stable: $z") \endlstlisting