Virtual Simplex Method Pivot-In-Place Linear Programming

The patent introduces an intermediary representation called the E matrix that acts as a mediator between the original linear programming problem and the computational process. This E matrix stores the problem data in a format that allows exact arithmetic operations, preventing the accumulation of numerical errors while maintaining computational efficiency. The intermediary structure enables the system to achieve both speed and accuracy by decoupling the computational process from direct numerical manipulation.

The patent creates a virtual copy of the linear programming problem through the E matrix representation. Instead of directly manipulating the original problem data which is susceptible to numerical errors, the system works with this virtual copy that preserves the exact structure and relationships of the original problem. This copying approach allows the computational process to proceed without introducing numerical inaccuracies.

The patent implements a dynamic system where the E matrix can be updated incrementally as constraints change. Instead of requiring complete reinitialization when constraints are modified, the system dynamically adjusts the E matrix representation and continues the optimization process. This dynamic approach maintains adaptability while preserving the simplicity of the initial setup, as the same data structure handles both static and dynamic scenarios.

The E matrix representation serves multiple functions: it stores the initial problem data, enables the optimization process, and facilitates handling of changing constraints. This universal data structure eliminates the need for separate handling mechanisms for different scenarios, allowing the system to maintain ease of setup while gaining adaptability to dynamic changes.

The E matrix serves as an intermediary that prevents direct numerical manipulation and the associated error accumulation. By introducing this intermediate representation layer, the system maintains straightforward implementation while eliminating the harmful effect of accumulated computer errors. The intermediary structure allows exact arithmetic to be performed on the problem data without introducing numerical inaccuracies.

The patent converts the potential harm of direct numerical computation into a benefit by using the E matrix representation. The structure that would normally introduce errors is transformed into a mechanism that prevents error accumulation. The same computational operations that would normally cause problems are redirected through the E matrix, turning a harmful process into a beneficial one that ensures solution correctness.