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Mathematical aspects of numerical continuation and handling of singularities

Consider a smooth function . We want to compute a solution curve of the equation . Numerical continuation is a technique to compute a consecutive sequence of points which approximate the desired branch. Most continuation algorithms implement a predictor-corrector method. The idea behind this method is to generate a sequence of points , along the curve, satisfying a chosen tolerance criterion: for some and an additional accuracy condition where and is the last Newton correction.

To show how the points are generated, suppose we have found a point on the curve. Also suppose we have a normalized tangent vector at , i.e. .

The computation of the next point consists of 2 steps:

- prediction of a new point
- correction of the predicted point

Subsections

- Prediction
- Correction
  - Pseudo-arclength continuation
  - Moore-Penrose continuation
- Stepsize control
- Singularity handling
  - Test functions
  - Multiple test functions
  - Singularity matrix
  - User location