QR Factorization Matrix Update for Reduced Computation Time
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Processing data involving large condition numbers and computationally intensive calculations can be time-consuming and error-prone, necessitating more efficient methods for reusing previously generated calculations.
Innovation Solution
The method involves receiving and updating data sets to reuse intermediate data from previous calculations, specifically using the Householder reflection algorithm for QR factorization, allowing for efficient updating of matrices by adding, removing, or modifying rows and columns, thereby reducing the need for recomputing entire solutions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional matrix factorization methods are used to process data with large condition numbers, then computational accuracy is maintained, but processing time increases significantly
Solution Approach 1:
The patent performs QR factorization on the initial matrix A before adding new rows, storing the intermediate results (Q and R matrices). When new data is added, only the necessary updates are computed rather than performing complete factorization again, significantly reducing processing time while maintaining accuracy
Solution Approach 2:
The patent divides the matrix update process into segments: initial complete QR factorization, then incremental updates when rows are added. This segmentation allows the computationally intensive part to be done once, with lighter update operations subsequent to it
2Measurement precision
If complete matrix factorization is performed each time data is updated, then solution accuracy is ensured, but computational efficiency decreases
Solution Approach 1:
The patent maintains continuous computational efficiency by updating the QR factorization incrementally as new rows are added, rather than discontinuously recomputing the entire factorization. The intermediate Q and R matrices are preserved and updated, creating a continuous efficient processing stream
Solution Approach 2:
The initial QR factorization is performed in advance before data updates occur. This preliminary computation stores intermediate results that can be efficiently updated later, avoiding the need to repeat the expensive complete factorization process
3Productivity
If intermediate calculation results are reused when updating matrices, then processing speed increases, but system complexity increases
Solution Approach 1:
The patent segments the matrix into an initial portion (for which complete QR factorization is performed) and an incremental portion (newly added rows). This segmentation allows reuse of intermediate results from the initial factorization while managing the complexity of updates in a structured way
Data Source
AI summary
Methods and systems for processing data are disclosed. In an aspect, a first set of data can be received. A first set of intermediate data can be determined based on the first set of data. A basis for a factorization can be determined based on the first set of intermediate data. A second set of data can be received. The basis for the factorization can be updated based on the second set of data. The factorization can be performed based on the updated basis for the factorization.