Piecewise Polynomial Approximation with Segment Merging
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Solution Overview
Problem
Existing VLSI design techniques for piecewise polynomial approximation face challenges in balancing accuracy and efficiency, as large numbers of segments require significant memory and complex circuitry, leading to increased area and power consumption.
Innovation Solution
A method and system for approximating mathematical functions by initially dividing the range into segments, performing segment-merging iterations to optimize segment size and reduce memory requirements while maintaining accuracy, using a simple segment locator and reduced polynomial coefficients.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If a large number of segments are used to divide the function range for high accuracy approximation, then the approximation accuracy is improved, but the memory requirements increase
Solution Approach 1:
The function range is divided into multiple segments, with each segment having its own approximation polynomial. This allows the system to achieve high approximation accuracy by selecting the appropriate segment based on the input value, while managing memory usage through efficient segment organization and merging capabilities.
Solution Approach 2:
Adjacent segments can be merged into larger segments when the approximation accuracy requirement is still met, reducing the total number of segments and thereby reducing memory requirements. The system dynamically merges segments based on the specified accuracy condition, optimizing the balance between accuracy and memory usage.
2Quantity of substance
If uneven segment division is used to reduce memory requirements, then the memory usage is reduced, but the circuitry complexity increases and calculation speed decreases
Solution Approach 1:
Different segments can have different levels of refinement based on the local characteristics of the function being approximated. Regions where the function changes rapidly can have more segments, while regions with slower variation can have fewer segments, optimizing both accuracy and resource usage.
Solution Approach 2:
The segment structure is dynamic and adaptable. Segments can be merged or split based on the specified accuracy condition, allowing the system to optimize its structure for different operating conditions and accuracy requirements, rather than being fixed.
3Measurement precision
If more segments are used to maintain approximation accuracy, then the accuracy is maintained, but the area and power consumption increase
Solution Approach 1:
Adjacent segments are merged when the approximation accuracy condition is still satisfied, reducing the total number of segments. This directly reduces the circuit area required to store segment boundaries and coefficients, as well as the power consumption associated with processing fewer segments.
Solution Approach 2:
The system changes the parameter of segment count based on the specified accuracy condition. By adjusting the number of segments dynamically through merging operations, the system optimizes the trade-off between approximation accuracy and physical resource consumption (area and power).
Data Source
AI summary
A method for approximating a mathematical function defined over a range includes initially dividing at least part of the range into a set of segments. For at least a subset of the segments, the mathematical function is approximated within each segment by a respective approximation polynomial. A series of one or more segment-merging iterations is performed, a given iteration including: selecting adjacent segments as candidates for merging; approximating the mathematical function by a candidate approximation polynomial, over at least a merged segment formed by merging the adjacent segments; and, if approximation of the mathematical function meets a specified condition, updating the set of segments by (i) replacing the adjacent segments with the merged segment and (ii) replacing the approximation polynomials of the adjacent segments with the candidate approximation polynomial.