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  "body": "\n## Summary\n\nNumerical linear algebra is the study of algorithms that solve problems of linear algebra on a computer, with floating-point numbers.\nIts main problems are linear systems, least squares, [[eigenvalues|0BQAA2BK3Q]] and [[singular values|09VMDSRXWJ]].\nIt asks how much work an algorithm needs and how rounding errors affect the answer, using notions such as the [[condition number|AB3S6MHJ8X]] and backward stability.\nMost computations in science and engineering end in one of its problems.\n\n## Explanations\n\n### L1 human\n\nOn paper, linear algebra is exact; on a computer, every number is rounded to a fixed number of digits.\nNumerical linear algebra designs methods that stay accurate despite these small errors and that stay fast for matrices with millions of rows.\nFor example, [[Gaussian elimination|0JVJ2WCXZH]] with row exchanges solves a linear system reliably.\nOn the other hand, eigenvalues are almost never computed as roots of the [[characteristic polynomial|0VSSPAFM2C]], because those roots react strongly to tiny errors in the coefficients.\nMethods such as [[LU decomposition|0RKG5YW66Y]], the [[QR algorithm|B6F8YWHYZ5]] and [[power iteration|H4MQD841BR]] belong to this field.\n",
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