Matrix Triangulation Using Product-Sum Operations

The patent applies preliminary action by pre-calculating and storing inverse elements in a lookup table before the main triangulation process. During triangulation, instead of performing recursive operations with multiple reductions, the method directly retrieves pre-computed inverse elements and applies them through product-sum operations. This preliminary preparation eliminates the need for frequent reductions during the actual triangulation, resolving the contradiction between productivity and reduction count.

The patent substitutes the traditional mechanical recursive operation system with a product-sum operation system based on pre-computed inverse elements. Instead of recursively updating matrix entries with repeated reductions, the method uses direct product-sum operations with lookup table references. This substitution replaces the reduction-heavy mechanical process with a more efficient computational approach, reducing the number of reductions while maintaining triangulation functionality.

By pre-computing and storing inverse elements in a lookup table, the patent simplifies the triangulation process while reducing computational complexity. The preliminary action of creating the lookup table transforms complex recursive operations into simple table lookups and product-sum operations, making the overall process easier to execute with lower computational complexity during the actual triangulation phase.

The patent uses copying by creating a lookup table that stores copies of pre-computed inverse elements. Instead of repeatedly computing inverse elements during recursive operations, the method copies relevant inverse elements from the lookup table and applies them directly. This copying approach simplifies the operation process and reduces computational complexity by avoiding redundant calculations.