Richardson Extrapolation for Power Method Matrix Approximation
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Solution Overview
Problem
Current power methods for determining the best rank-1 approximation of a matrix require numerous processing iterations and memory accesses, leading to significant consumption of processing power, memory, and electrical power, which can slow down other processes and require complex, expensive hardware.
Innovation Solution
An enhanced power method that reduces the number of processing iterations and memory accesses by determining a point on a line passing through previous singular vector values, allowing for a more efficient computation of the best rank-1 approximation using a modified power method algorithm.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional power method is used to determine best rank-1 approximation of a matrix, then processing accuracy is maintained, but processing time and power consumption increase significantly
Solution Approach 1:
The patent applies preliminary action by performing Richardson extrapolation to estimate the limit value of the power method before completing all traditional iterations. This allows the system to predict the final result based on a few initial iterations, significantly reducing processing time while maintaining accuracy. The extrapolation formula uses the sequence of approximations from early iterations to calculate the limiting value that would be reached after infinite iterations.
Solution Approach 2:
The patent uses cheap short-living objects by replacing the expensive, time-consuming traditional power method with a computationally inexpensive Richardson extrapolation formula. This formula provides a quick approximation that captures the essential result without requiring numerous iterative calculations, effectively using a simple mathematical shortcut instead of a complex computational process.
2Measurement precision
If traditional power method is used to determine best rank-1 approximation of a matrix, then processing accuracy is maintained, but power consumption increases significantly
Solution Approach 1:
The patent applies preliminary action by performing Richardson extrapolation to estimate the limit value of the power method before completing all traditional iterations. This allows the system to predict the final result based on a few initial iterations, significantly reducing processing time and power consumption while maintaining accuracy. The extrapolation formula uses the sequence of approximations from early iterations to calculate the limiting value that would be reached after infinite iterations.
Solution Approach 2:
The patent uses cheap short-living objects by replacing the expensive, power-intensive traditional power method with a computationally inexpensive Richardson extrapolation formula. This formula provides a quick approximation that captures the essential result without requiring numerous iterative calculations, effectively using a simple mathematical shortcut instead of a complex computational process.
3Reliability
If traditional power method is used to determine best rank-1 approximation of a matrix, then processing completeness is achieved, but device complexity increases
Solution Approach 1:
The patent applies mechanics substitution by replacing the mechanical iterative computation system with a mathematical extrapolation formula. Instead of physically executing numerous iterative matrix operations that require complex hardware resources, the system uses Richardson extrapolation—a mathematical technique that computes the limit value directly from a sequence of approximations. This substitution eliminates the need for complex iterative processing hardware while maintaining processing completeness.
4Measurement precision
If traditional power method is used to determine best rank-1 approximation of a matrix, then accurate results are obtained, but available resources for other tasks are reduced
Solution Approach 1:
The patent applies preliminary action by performing Richardson extrapolation to estimate the limit value of the power method before completing all traditional iterations. This allows the system to predict the final result based on a few initial iterations, significantly reducing processing time and power consumption while maintaining accuracy. The extrapolation formula uses the sequence of approximations from early iterations to calculate the limiting value that would be reached after infinite iterations.
Solution Approach 2:
The patent uses cheap short-living objects by replacing the expensive, resource-intensive traditional power method with a computationally inexpensive Richardson extrapolation formula. This formula provides a quick approximation that captures the essential result without requiring numerous iterative calculations, effectively using a simple mathematical shortcut instead of a complex computational process.
Data Source
AI summary
A power method can be enhanced. For example, an electronic communication indicating a job to be performed can be received. A best rank-1 approximation of a matrix associated with the job can be determined using the power method. Each iteration of the power method can include determining a point that lies on a line passing through (i) a first value for a first singular vector from an immediately prior iteration of the power method; and (ii) a second value for the first singular vector from another prior iteration of the power method. Each iteration of the power method can also include determining, by performing the power method using the point, a current value for the first singular vector and a current value for a second singular vector for a current iteration of the power method. The job can then be performed using the best rank-1 approximation of the matrix.