Approximate Function Calculation Device Using Adaptive Taylor Expansion
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Solution Overview
Problem
Conventional methods require a large number of sections to approximate a function over an entire domain with a desired error, especially when using fixed-point numbers, leading to inefficient calculations.
Innovation Solution
The method involves using a Taylor expansion unit to approximate a function by iteratively calculating error bounds and adjusting section boundaries, reducing the number of sections needed for accurate approximation by determining optimal expansion and end positions within each section.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional polynomial approximation methods are used with fixed-point numbers, then approximation can be performed, but a large number of sections are required to achieve desired error tolerance
Solution Approach 1:
The patent changes the parameter of polynomial degree from fixed to variable. By allowing the polynomial degree to vary across different sections based on the function's local characteristics, the approximation accuracy is improved without requiring a large number of sections. This parameter change enables adaptive approximation that matches the function's complexity in each region.
Solution Approach 2:
The patent introduces dynamic adaptation by determining polynomial degrees adaptively for each section rather than using a uniform degree. The system dynamically adjusts the approximation strategy based on local function behavior, allowing fewer sections to achieve the desired accuracy by concentrating computational resources where needed.
2Measurement precision
If the section width is reduced to improve approximation accuracy, then error decreases, but the number of sections increases
Solution Approach 1:
The patent changes the parameter of polynomial degree to resolve the contradiction between accuracy and efficiency. By using higher degree polynomials in sections where the function exhibits complex behavior and lower degree polynomials elsewhere, the method achieves desired accuracy with fewer sections, thereby improving calculation efficiency without sacrificing precision.
Solution Approach 2:
The patent applies local quality by assigning different polynomial degrees to different sections based on their specific characteristics. Rather than uniformly reducing section width throughout the entire domain, the method locally adapts the approximation strategy, using higher degree polynomials only where necessary to capture local function behavior, thus avoiding the need for excessive sections.
Data Source
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AI summary
An approximate function calculation apparatus includes: a Taylor expansion unit which obtains a function gb(x); a first error upper bound calculating unit which obtains an upper bound of an error of an approximation of a function f(x) by the function gb(x) in a section [x0, b]; an expansion position calculating unit which obtains an expansion position b' that represents a maximum b at which the upper bound of an error of an approximation of the function f(x) by the function gb(x) in the section [x0, b] is equal to or smaller than ε; a second error upper bound calculating unit which obtains an upper bound of an error of an approximation of the function f(x) by a function gb'(x) in a section [b', x2]; a right end position calculating unit which calculates a right end position x2' that represents a maximum x2 at which the upper bound of an error of an approximation of the function f(x) by the function gb' in the section [b', x2] is equal to or smaller than ε; an output unit which outputs information related to a section [x0, x2'] and the function gb'(x); and a control unit which repeats processing of the respective units with a section [x2', x1] as the section D.