Aitken Extrapolation for Matrix Diagonal Estimation
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Solution Overview
Problem
Computational bottlenecks in estimating electronic structures of materials and matrices, particularly due to the high computational costs associated with computing diagonal entries and trace estimates, which are essential in computational chemistry, material engineering, and machine learning, often require a large number of probing vectors, leading to inefficiencies.
Innovation Solution
The implementation of Aitken extrapolation methods to accelerate the estimation of matrix diagonal entries and trace estimates by combining initial probing vector results, reducing the number of central processing unit (CPU) cycles and memory resources needed, thereby improving the efficiency of matrix operations and material analysis.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional probing methods are used to compute diagonal entries of matrices, then measurement precision is improved, but productivity deteriorates due to high computational costs and large number of CPU cycles required
Solution Approach 1:
The patent applies Aitken extrapolation to accelerate the convergence of diagonal entry estimates obtained through probing. By performing preliminary computations with fewer probing vectors and then applying the Aitken acceleration formula, the method achieves accurate diagonal estimates with significantly reduced computational effort, resolving the contradiction between precision and productivity
Solution Approach 2:
The patent transforms the convergence rate parameter by applying Aitken extrapolation to the sequence of diagonal estimates. This parameter transformation accelerates the convergence from linear to quadratic, allowing accurate results to be obtained with fewer probing vectors and reduced CPU cycles
2Measurement precision
If a large number of probing vectors are used to achieve certain accuracy in diagonal estimation, then measurement precision is improved, but loss of time increases due to extended computational processing
Solution Approach 1:
The patent performs preliminary diagonal estimates using a small number of probing vectors, then applies Aitken extrapolation to accelerate convergence to the final accurate result. This two-stage approach achieves high precision trace estimates with minimal processing time, eliminating the need to use large numbers of probing vectors directly
3Manufacturing precision
If conventional matrix diagonal computation methods are used, then manufacturing precision of material property analysis is improved, but device complexity increases due to extensive computational resources required
Solution Approach 1:
The patent performs preliminary analysis with fewer probing vectors to obtain initial diagonal estimates, then applies Aitken extrapolation to achieve accurate material property predictions. This reduces computational resource requirements and simplifies the computational device complexity while maintaining high manufacturing precision in material analysis
Solution Approach 2:
The patent extracts only the essential diagonal entries of the matrix that are needed for material property analysis, rather than computing the entire matrix. By applying Aitken extrapolation to these extracted diagonal elements, the method achieves accurate material predictions with reduced computational complexity
Data Source
AI summary
Using a hardware processor, load a matrix. Compute, using the hardware processor, diagonal entry approximations for the matrix by using one or more probing vectors. Apply, using the hardware processor, an Aitken extrapolation to the diagonal entry approximations. Obtain, using the hardware processor, a final diagonal estimation based on the Aitken extrapolation. Optionally, the matrix comprises a matrix of material properties of a first material, and further actions include, based on the final diagonal estimation for the first material, determining that the first material is suitable for a certain application, based on the determination that the first material is suitable, specifying the first material; and, responsive to the specifying, using the hardware processor to control a machine tool to fabricate a part of the first material.