Pseudobåglängdsmetoden

Pseudobåglängdsmetoden, translated as the pseudo-arc length method, is a numerical technique used to trace solution paths of nonlinear systems of equations, particularly in problems involving bifurcations. It is an extension of Newton's method, designed to follow curves in the solution space where the Jacobian matrix might become singular. The core idea is to introduce an additional constraint equation that couples the change in the solution vector with the change in a parameter. This constraint, the "pseudo-arc length," ensures that the steps taken are not solely determined by the local geometry of the solution curve, which can lead to problems near turning points or bifurcations.

The method augments the original system of equations $F(x, \lambda) = 0$, where $x$ is the state vector and $\lambda$ is a parameter, with a path-following constraint. A common form of this constraint involves the dot product of the solution increment vector $\Delta x$ and a predictor step, which itself is often derived from the tangent to the solution path. This predictor can be obtained by solving the linearized system $\frac{\partial F}{\partial x} p_x + \frac{\partial F}{\partial \lambda} p_\lambda = 0$ and setting $p_\lambda = 1$. The pseudo-arc length condition then ensures that the step taken, $(\Delta x, \Delta \lambda)$, has a specific length relative to this predictor. This approach allows the numerical solver to move smoothly along the solution curve, even when the system undergoes significant qualitative changes, such as bifurcations where multiple stable solutions coexist.