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  "body": "\n## Definition\n\n*Numerical linear algebra* is the part of numerical analysis that studies algorithms in finite-precision arithmetic for these problems.\nIn each, $A$ is a real or complex [[matrix|0VD9ZSWWJG]].\n\n1. solve a linear system $Ax = b$;\n2. find $x$ that minimizes $\\norm{Ax - b}_2$ (least squares);\n3. find the [[eigenvalues|0BQAA2BK3Q]] and eigenvectors of a square matrix $A$;\n4. find the [[singular values|09VMDSRXWJ]] and singular vectors of $A$.\n\nAn algorithm is judged by its cost, counted in arithmetic operations and memory, and by its accuracy under rounding errors.\n\n## Remarks\n\nAn algorithm is *backward stable* when its computed answer is the exact answer for slightly perturbed data.\nFor a linear system this means $(A + \\delta A)\\hat x = b$ with $\\norm{\\delta A}$ of the order of the unit roundoff times $\\norm{A}$.\nFor a backward stable method, the relative error of $\\hat x$ is roughly bounded by the [[condition number|AB3S6MHJ8X]] of $A$ times the unit roundoff.\nExample: Householder [[QR decomposition|0ERWG4NXQ9]] solves least squares problems in a backward stable way with about $2mn^2 - \\tfrac{2}{3}n^3$ operations for an $m \\times n$ matrix.\nNon-example: computing the exact determinant of an integer matrix with rational arithmetic is a task of computer algebra.\nIt is not numerical linear algebra, because it does not work in finite precision and has no rounding error to analyse.\nDirect methods, such as [[LU decomposition|0RKG5YW66Y]], finish in a fixed number of steps; iterative methods, such as [[power iteration|H4MQD841BR]], improve an approximation step by step.\n",
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