---
tag: CP2Z74X1X6
kind: definition
name: Numerical continuation
msc: [65H20, 65P30]
defines: 5V3GJ84CPQ
uses: [0HHHDCXZV4, 047NDTEPCV, 0BXBADR94C, 07EXVNF9VP]
sources:
  - url: https://en.wikipedia.org/wiki/Numerical_continuation
---

## Definition

Let $H \colon \R^{n+1} \to \R^n$ be a [[continuously differentiable|0HHHDCXZV4]] map, and let $u_0 \in \R^{n+1}$ satisfy $H(u_0) = 0$.
A point $u$ with $H(u) = 0$ is *regular* if the [[Jacobian matrix|047NDTEPCV]] $H'(u)$, of size $n \times (n+1)$, has rank $n$.
Near a regular point, the zero set $H^{-1}(0)$ is a smooth curve, by the [[implicit function theorem|0BXBADR94C]].
A *numerical continuation method* for $H$ from $u_0$ is an algorithm that produces points $u_1, u_2, \dots$ such that:

1. each $u_k$ approximately satisfies $H(u_k) = 0$, within a prescribed tolerance;
2. each $u_{k+1}$ is obtained from $u_k$ (and possibly earlier points) by a step of controlled length along the curve;
3. the points follow the connected component of $H^{-1}(0)$ through $u_0$ in one direction.

## Remarks

The usual scheme is predictor–corrector.
The predictor moves from $u_k$ to $v = u_k + h t_k$ for a step length $h > 0$.
Here the unit tangent $t_k$ is a unit vector in the kernel of $H'(u_k)$, oriented in the direction of travel.
The 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*.
Example: 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.
Non-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 $\lambda$.
Such points are called turning points or folds.

## Explanations

### L1 human

Continuation follows a curve of solutions in small steps.
Each step guesses ahead along the tangent and then corrects the guess back onto the curve.
Measuring progress by arc length, rather than by the parameter, lets the method go around bends where the parameter turns back.