Piecewise Linear Approximation Coefficient Selection

The patent divides the function approximation into multiple segments or pieces, where each segment uses a simple linear model. Instead of using a complex Taylor expansion across the entire domain, the function is segmented into multiple intervals, each approximated by a linear equation with pre-calculated coefficients. This segmentation reduces the computational complexity from high-order polynomial calculations to simple linear operations while maintaining overall accuracy.

The patent changes the parameters of the approximation method by switching from Taylor expansion parameters (derivatives, higher-order terms) to piecewise linear parameters (segment boundaries, linear coefficients). By pre-calculating and storing these linear coefficients for each segment, the system transforms a computationally intensive continuous approximation into a efficient discrete lookup and evaluation process, significantly reducing processing requirements.

The patent replaces expensive, complex polynomial calculations with cheap, simple linear operations. Each piecewise segment uses basic arithmetic operations (multiplication and addition) instead of expensive polynomial evaluation. These simple linear models act as disposable approximations that can be quickly evaluated and discarded, providing sufficient accuracy without the overhead of complex polynomial arithmetic.

The patent performs preliminary action by pre-calculating the coefficients for each piecewise linear segment and storing them in lookup tables. This pre-computation phase separates the complex work from the runtime execution, allowing the actual function approximation to proceed with simple coefficient retrieval and linear evaluation. The heavy lifting of determining optimal coefficients is done beforehand, reducing the instruction count during actual computation.