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2 years ago in Complex Analysis , ScientificComputing By Payal Homraj Bhagat
How can Schwarz-Christoffel transformation equations be solved iteratively in MATLAB?
Implementing the Schwarz-Christoffel transformation for conformal mapping is straightforward until solving for the pre-vertices, which requires solving a nonlinear system. My attempts with MATLAB’s fsolve often diverge without good initial guesses. I need a clear, step-by-step iterative strategy perhaps using a physical analogy or a optimization routine to robustly converge to the correct parameters for a trapezoidal target shape.
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By Aamir Answered 1 year ago
This is the core challenge of practical SC mapping. I would recommend a two-stage iterative process. First, use the Cayley transform to get a rough initial guess for the pre-vertices along the real axis, roughly spaced according to your polygon's side lengths. Then, implement an iterative correction using the method of successive relaxation on the side-length conditions. In MATLAB, formulate the nonlinear equations from the SC integral and use lsqnonlin within a loop, updating only one or two parameters per iteration to maintain stability. I have seen adding a small damping factor of 0.5 to updates prevent oscillation. Monitor the maximum vertex error, not just the function norm.
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