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  "body": "\n## Definition\n\nLet $H \\colon \\R^{n+1} \\to \\R^n$ be a [[continuously differentiable|0HHHDCXZV4]] map, and let $u_0 \\in \\R^{n+1}$ satisfy $H(u_0) = 0$.\nA point $u$ with $H(u) = 0$ is *regular* if the [[Jacobian matrix|047NDTEPCV]] $H'(u)$, of size $n \\times (n+1)$, has rank $n$.\nNear a regular point, the zero set $H^{-1}(0)$ is a smooth curve, by the [[implicit function theorem|0BXBADR94C]].\nA *numerical continuation method* for $H$ from $u_0$ is an algorithm that produces points $u_1, u_2, \\dots$ such that:\n\n1. each $u_k$ approximately satisfies $H(u_k) = 0$, within a prescribed tolerance;\n2. each $u_{k+1}$ is obtained from $u_k$ (and possibly earlier points) by a step of controlled length along the curve;\n3. the points follow the connected component of $H^{-1}(0)$ through $u_0$ in one direction.\n\n## Remarks\n\nThe usual scheme is predictor–corrector.\nThe predictor moves from $u_k$ to $v = u_k + h t_k$ for a step length $h \u003e 0$.\nHere the unit tangent $t_k$ is a unit vector in the kernel of $H'(u_k)$, oriented in the direction of travel.\nThe corrector applies [[Newton's method|07EXVNF9VP]] to the $n + 1$ equations $H(u) = 0$ and $t_k \\cdot (u - v) = 0$; this is *pseudo-arclength continuation*.\nExample: for $H(x, \\lambda) = x^2 + \\lambda^2 - 1$ on $\\R^2$, every zero is regular, and pseudo-arclength continuation from $(1, 0)$ traces the whole unit circle.\nNon-example: *natural parameter continuation* fixes $\\lambda$ at successive values and solves for $x$; on the same circle it fails at $(0, \\pm 1)$, where $\\partial H / \\partial x = 0$, because the curve turns back in $\\lambda$.\nSuch points are called turning points or folds.\n\n## Explanations\n\n### L1 human\n\nContinuation follows a curve of solutions in small steps.\nEach step guesses ahead along the tangent and then corrects the guess back onto the curve.\nMeasuring progress by arc length, rather than by the parameter, lets the method go around bends where the parameter turns back.\n",
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