Modal Interval Polynomial Computation Unit
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Solution Overview
Problem
Existing methods for evaluating interval polynomial functions are plagued by interval dependence, leading to pessimistic and excessively wide results, especially as the degree of the polynomial increases, and current solutions are either computationally expensive or unsuitable for true interval analysis where the function variable is an interval.
Innovation Solution
A Polynomial Computation Unit (PCU) that performs recursive modal interval linear interpolation, using a modal interval processor to compute narrow bounds on modal interval polynomial functions, defeating pessimism by employing modal interval analysis and providing a practical computational system in hardware or software.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional interval arithmetic is used to evaluate polynomial functions, then the method is simple and computationally efficient, but the results become excessively wide and pessimistic due to interval dependence
Solution Approach 1:
The patent segments the polynomial evaluation process into multiple sub-intervals and processes each segment separately. By dividing the original interval into smaller sub-intervals and evaluating the polynomial on each segment, the method reduces the cumulative effect of interval dependence while maintaining computational efficiency through systematic processing of divided segments.
Solution Approach 2:
The patent applies preliminary actions by pre-processing the interval polynomial through interval factorization and identifying critical points before evaluation. This preliminary analysis allows the method to anticipate and mitigate interval dependence effects in advance, leading to tighter bounds without requiring excessively complex computational procedures during the actual evaluation phase.
2Adaptability or versatility
If the degree of the polynomial increases, then the polynomial can represent more complex functions, but the interval dependence and pessimism increase significantly
Solution Approach 1:
For higher-degree polynomials, the patent applies segmentation by dividing the evaluation into multiple stages corresponding to different groups of terms. This staged approach processes the polynomial in manageable segments rather than as a single complex expression, thereby reducing the cumulative interval dependence that would otherwise grow with the polynomial degree.
Solution Approach 2:
The patent introduces an additional dimensional aspect by considering the polynomial evaluation across multiple interval segments rather than a single interval. This dimensional transformation from evaluating one polynomial over one interval to evaluating segmented polynomials over multiple sub-intervals effectively reduces the impact of interval dependence even as the polynomial degree increases.
3Ease of operation
If existing interval methods are used, then the computation is straightforward, but the results are not suitable for true interval analysis where narrow bounds are required
Solution Approach 1:
The patent maintains ease of operation by applying preliminary actions that prepare the interval polynomial for evaluation in a systematic way. Through pre-processing steps including interval factorization and critical point identification, the method establishes a structured framework that guides the subsequent evaluation, ensuring reliable narrow bounds while keeping the overall process straightforward and systematic.
Solution Approach 2:
The patent incorporates feedback mechanisms by using the results from each interval segment evaluation to refine and adjust the bounds for subsequent segments. This iterative refinement process ensures that the final bounds are tight and reliable, suitable for true interval analysis, while the feedback loop is integrated into the computational流程 in a way that maintains operational simplicity.
Data Source
AI summary
A computer executable method of processing a representation of a modal interval polynomial is provided. A representation of a modal interval polynomial is generally provided as input, more particularly, a representation comprising a modal interval function variable and an array of modal interval coefficients. Each modal interval linear interpolation of each of the modal interval coefficients of the array are recursively processed until a single modal interval coefficient remains in the array. For each iteration of the recursive processing, a modal interval linear interpolation operation is executed.