%0 Unpublished Work %1 henderson2023piecewise %A Henderson, Michael E %A Melville, Robert %D 2023 %I Research Square Platform LLC %K 65h20-global-methods-including-homotopy-approaches %R 10.21203/rs.3.rs-3612152/v1 %T Piecewise Linear Continuation: Derivative-free Manifold Generation %U http://dx.doi.org/10.21203/rs.3.rs-3612152/v1 %X We describe the Piecewise Linear Continuation algorithm Allgower and Georg (2000) for finding solutions of a set of nonlinear equations that depend on k parameters Allgower and Schmidt (1985). Unlike predictor–corrector continuation, the algorithm does not require the derivatives of the system; so is perfectly suited for applications like controlling experimental apparatus or extending the capability of legacy codes. While the algorithm is not new, we describe it here with an emphasis on implementation rather than its mathematical formulation. In addition, we present a new result that relates the face-checking step to one step of Newton’s Method, and show how it can be used to refine the solution. The presentation is motivated with a hardware continuation, where the algorithm is used to control an experimental apparatus and traces the I-V characteristic. A reference implementation in C and C++ Melville (2023) has been provided, and its use to solve two example problems is described here.

@unpublished{henderson2023piecewise, abstract = {We describe the Piecewise Linear Continuation algorithm Allgower and Georg (2000) for finding solutions of a set of nonlinear equations that depend on k parameters Allgower and Schmidt (1985). Unlike predictor–corrector continuation, the algorithm does not require the derivatives of the system; so is perfectly suited for applications like controlling experimental apparatus or extending the capability of legacy codes. While the algorithm is not new, we describe it here with an emphasis on implementation rather than its mathematical formulation. In addition, we present a new result that relates the face-checking step to one step of Newton’s Method, and show how it can be used to refine the solution. The presentation is motivated with a hardware continuation, where the algorithm is used to control an experimental apparatus and traces the I-V characteristic. A reference implementation in C and C++ Melville (2023) has been provided, and its use to solve two example problems is described here.}, added-at = {2024-04-12T06:07:18.000+0200}, author = {Henderson, Michael E and Melville, Robert}, biburl = {https://www.bibsonomy.org/bibtex/2608d4ae50c220ea80897503fe2d41a7c/gdmcbain}, doi = {10.21203/rs.3.rs-3612152/v1}, interhash = {05788329852f6fb5828476408b2995bc}, intrahash = {608d4ae50c220ea80897503fe2d41a7c}, keywords = {65h20-global-methods-including-homotopy-approaches}, month = nov, publisher = {Research Square Platform LLC}, timestamp = {2024-04-12T06:07:18.000+0200}, title = {Piecewise Linear Continuation: Derivative-free Manifold Generation}, url = {http://dx.doi.org/10.21203/rs.3.rs-3612152/v1}, year = 2023 }