Efficient Closed-Form Solutions for Separable Polynomial Constraints: The Parametric K-Formula with Applications in Positioning and Control

Separable polynomial constraints arise naturally in numerous engineering applications,
including sensor positioning systems, robotic kinematics, and regularized optimization. While
general-purpose iterative solvers are widely available, their computational overhead and
non-deterministic convergence behavior can be limiting factors in real-time and resourceconstrained environments. This paper presents the Parametric K-Formula (PK-Formula),
a straightforward closed-form method for solving a specific but practically important class
of separable polynomial equations. The method achieves O(n) computational complexity
through parametric decomposition, offering 50–120× speedups over Newton-Raphson methods in benchmark tests while maintaining numerical accuracy comparable to iterative approaches. We provide complete implementation details, compare performance against standard solvers (MATLAB’s fsolve, Python’s scipy.optimize), and demonstrate practical
applications in trilateration systems, regularized optimization, and trajectory control. The
method’s simplicity enables straightforward implementation on embedded systems and microcontrollers, where computational resources are limited. Open-source implementations in
MATLAB, Python, and C are provided in an accompanying repository. This work demonstrates that for the specific problem class of separable polynomial constraints, substantial
practical benefits can be achieved through direct analytical approaches rather than generalpurpose iterative methods.