Recursive Calculation of Secondary Symmetric Functions
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Solution Overview
Problem
Existing methods for calculating elementary symmetric functions of subsets of a set of numerical elements are inefficient, requiring m^2 * (log m)^2 time due to repeated calls to the calculateESF function, which is computationally expensive.
Innovation Solution
A method and system that recursively calculate secondary symmetric functions by employing a recursive relationship among secondary symmetric functions, previously determined secondary symmetric functions, and elementary symmetric functions, allowing for calculations in an m^2 time-frame.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If repeated calls to calculateESF function are used to calculate elementary symmetric functions of all (m-1)-subsets, then calculation completeness is ensured, but computational time increases to m^2 * (log m)^2
Solution Approach 1:
The patent pre-calculates and stores the elementary symmetric functions of the original set {Z} before processing subsets. By having these pre-computed values available, the algorithm avoids redundant calculations when deriving symmetric functions for (m-1)-subsets, thereby reducing overall computational time from m^2*(log m)^2 to m^2.
Solution Approach 2:
The patent uses a recursive relationship that leverages previously calculated secondary symmetric functions and elementary symmetric functions to compute new secondary symmetric functions. This feedback mechanism allows the algorithm to build upon existing calculations rather than starting from scratch for each subset, significantly improving efficiency.
2Measurement precision
If m calls to calculateESF function are made for each subset, then accuracy of symmetric function calculation is maintained, but computational complexity increases to m^2 * (log m)^2
Solution Approach 1:
The algorithm makes the computational system self-efficient by using the pre-calculated elementary symmetric functions and the recursive relationship to derive secondary symmetric functions automatically. This self-service approach eliminates the need for repeated external calls to calculateESF, reducing computational complexity while maintaining accuracy.
Solution Approach 2:
The patent changes the computational parameters by switching from direct calculation using calculateESF for each subset to a recursive formula-based approach. This parameter change transforms the complexity from m^2*(log m)^2 to m^2 by utilizing mathematical relationships between symmetric functions of different sets.
3Adaptability or versatility
If elementary symmetric functions are calculated for all (m-1)-subsets using standard methods, then completeness of subset analysis is achieved, but processing time increases significantly
Solution Approach 1:
The patent creates a universal algorithm that can calculate symmetric functions for all (m-1)-subsets using a single recursive relationship and pre-computed elementary symmetric functions. This multi-functional approach handles all subset calculations uniformly without requiring separate processing for each subset, achieving both completeness and efficiency.
Data Source
AI summary
A method and system that calculate elementary symmetric functions of subsets of a set {Z} comprising a total of m numerical elements in an m2 time-frame. In one embodiment the method comprises: receiving (310) a set {Z} comprising a total of m numerical elements, wherein z(i) denotes the ith element of set {Z} indexed by number i; obtaining (320) a plurality of elementary symmetric functions for set {Z}, wherein esf(j) denotes the jth elementary symmetric function of set {Z} indexed by number j; and determining (330) a plurality of secondary symmetric functions for set {Z}, wherein for each value of index i, ssf(j,i) denotes the jth elementary symmetric function of a subset {Z\z(i)} comprising set {Z} with element z(i) removed from set {Z}, and wherein for a particular value of index i, the secondary symmetric functions are determined in accordance with a recursive relationship among the secondary symmetric function to be determined, a previously determined secondary symmetric function, one of the obtained elementary symmetric functions, and the element z(i) removed from set {Z}.