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  "body": "\n## Summary\n\nNumerical analysis is the study of algorithms that compute approximate solutions to problems of continuous mathematics.\nIt designs methods for tasks such as solving equations, computing integrals, finding eigenvalues and solving differential equations.\nIt also proves how large the error of such a method can be, and how fast the method converges.\nTwo typical sources of error are the [[truncation error|4AKCQERM89]] of a finite method and the [[round-off error|CPN6SD3BTV]] of [[floating-point arithmetic|RTKRSC3KPH]].\n\n## Explanations\n\n### L1 human\n\nMany equations have no solution that one can write down with a formula, such as $\\cos x = x$.\nNumerical analysis gives step-by-step recipes that produce a number as close to the true answer as one wishes.\nA computer works with a finite number of digits, so each step can introduce a tiny error.\nPart of the subject is to check that these tiny errors stay small and do not grow out of control.\nFor example, the [[bisection method|7VMYC7MB3Q]] halves an interval around the solution again and again.\nAfter $20$ steps the solution is known to within about one millionth of the starting length.\n",
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