Numerical Foundations of Rootfinding Algorithms Based on Newton-Raphson for Non-Holomorphic Complex Systems Using Wirtinger Operators

Abstract

Many nonlinear systems in C^n, such as the power flow equations of electrical networks, are not holomorphic, and are therefore usually solved by Newton-Raphson after splitting every quantity into real and imaginary parts. In contrast to this approach, we develop in this paper the convergence theory of the Newton-Raphson-Wirtinger (NRW) method natively in C^n. The Wirtinger operators df/dz and df/dz-bar are derived as the unique coefficients of the widely-linear first-order approximation of f, which is equivalent to real Frechet differentiability but is posed and computed without realification, and their Lipschitz continuity is shown to be equivalent to real Frechet differentiability with a Lipschitz derivative. On this basis, we prove a Newton-Kantorovich semilocal convergence theorem and a local quadratic convergence theorem, which offer sufficient but not necessary conditions for convergence. These conditions are expressed in terms of constants involving the Wirtinger operators, yielding an a priori convergence certificate computable in complex arithmetic. We give a CBLAS/LAPACKE implementation based on a Schur reduction, with an operation count and an honest comparison against the realified method. Two applications illustrate the results. In a single-generator infinite-bus power flow problem, the Kantorovich constants are computed explicitly, the convergence certificate holds, and the iterates are verified against realified Newton-Raphson. In vortex states of the discretized Gross-Pitaevskii equation, the global phase invariance is removed by a structure-preserving phase pinning, Newton converges quadratically. We also prove an affine-covariant version of the Kantorovich theorem, invariant under all invertible widely-linear transformations of the equations, whose constant can be smaller than the product of the classical ones by orders of magnitude, as the vortex example indicates.