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  "body": "\n## Definition\n\nLet $X$ and $Y$ be normed spaces with [[norms|0V2WP9B1MF]] $\\norm{\\cdot}$, and let $f \\colon X \\to Y$ be a problem.\nLet $x \\in X$ with $x \\ne 0$, and let $\\hat y \\in Y$ be a computed approximation of $y = f(x)$.\n\n1. The *forward error* of $\\hat y$ is $\\norm{\\hat y - y}$, and its relative forward error is $\\norm{\\hat y - y}/\\norm{y}$ when $y \\ne 0$.\n2. The *relative backward error* of $\\hat y$ is\n$$\\eta(\\hat y) = \\inf \\set{ \\frac{\\norm{\\Delta x}}{\\norm{x}} : f(x + \\Delta x) = \\hat y },$$\nwith $\\eta(\\hat y) = \\infty$ when no such $\\Delta x$ exists.\n3. *Backward error analysis* of an algorithm is a bound on $\\eta(\\hat y)$ for its computed outputs $\\hat y$, usually by a modest multiple of the unit roundoff.\n\n## Remarks\n\nExample: in floating-point arithmetic the computed sum is $\\hat s = \\mathrm{fl}(a + b) = (a + b)(1 + \\delta)$ with $\\abs{\\delta} \\le u$.\nIt equals the exact sum of $a(1 + \\delta)$ and $b(1 + \\delta)$.\nSo for the problem $(a, b) \\mapsto a + b$ with the max norm, the relative backward error is at most $u$.\nNon-example: a bound on $\\norm{\\hat y - y}$ alone, with no statement about a perturbed input, is a forward error analysis.\nIf $f$ is differentiable with relative [[condition number|AB3S6MHJ8X]] $\\kappa$ at $x$, then to first order the relative forward error is at most $\\kappa \\, \\eta(\\hat y)$.\nComponentwise backward errors, which bound each $\\abs{\\Delta x_i}/\\abs{x_i}$ separately, are also common.\n",
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