The transport of images method: computing all zeros of harmonic mappings by continuation
- Resource Type
- Authors
- Olivier Sète; Jan Zur
- Source
- Subject
- Mathematics - Complex Variables
Applied Mathematics
General Mathematics
Track (disk drive)
Root (chord)
Harmonic (mathematics)
Numerical Analysis (math.NA)
Residual
Computational Mathematics
Continuation
General purpose
FOS: Mathematics
Applied mathematics
Mathematics - Numerical Analysis
Complex Variables (math.CV)
MATLAB
Complex plane
computer
65H20, 31A05, 30C55
computer.programming_language
Mathematics
- Language
We present a continuation method to compute all zeros of a harmonic mapping $\,f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solutions of $f(z) = \eta $ with $\lvert \eta \rvert{}$ sufficiently large and then track all solutions as $\eta $ tends to $0$ to finally obtain all zeros of $f$. Using theoretical results on harmonic mappings we analyze where and how the number of solutions of $f(z) = \eta $ changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular. In our numerical examples the method always terminates with the correct number of zeros, is very fast compared to general purpose root finders and is highly accurate in terms of the residual. An easy-to-use MATLAB implementation is freely available online.