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Related Concept Videos

Linear Approximation in Time Domain01:21

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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

Jieun Song1, Bumjoo Lee1

  • 1Department of Electronic Engineering, Myongji University, Yongin 17058, Republic of Korea.

Sensors (Basel, Switzerland)
|June 27, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces an optimal approximation algorithm (OPP) that efficiently simplifies complex functions into piecewise polynomials. It minimizes computational cost while maintaining accuracy, crucial for real-time systems.

Keywords:
constrained least squaresfunction approximationpiecewise polynomialregression

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Area of Science:

  • Numerical Analysis
  • Computer Science
  • Applied Mathematics

Background:

  • Simplifying non-linear functions and discrete data is essential for computational efficiency.
  • Existing methods often require manual selection of polynomial order and interval number, impacting runtime.
  • Time-sensitive applications and embedded systems demand approximations with both high accuracy and low computational cost.

Purpose of the Study:

  • To propose an optimal approximation algorithm (OPP) for simplifying functions into piecewise polynomials.
  • To minimize computational cost while ensuring approximation error remains below a specified threshold.
  • To automate the determination of optimal polynomial order and number of intervals.

Main Methods:

  • Utilizing constrained least squares to approximate data with smooth piecewise polynomials.
  • Developing an algorithm to search for the optimal piecewise polynomial (OPP) by considering computational cost and error.
  • Off-line calculation and tabulation of computational costs for all possible piecewise polynomial combinations on a target CPU.
  • Employing a random selection method for optimizing combinations with given sample points.

Main Results:

  • The OPP algorithm successfully determines both the optimal polynomial order and number of intervals.
  • Computational cost is minimized by selecting the OPP with the lowest cost that meets the error tolerance.
  • The algorithm demonstrated effective performance in simplifying representative functions.

Conclusions:

  • The proposed OPP algorithm offers an efficient method for function approximation in resource-constrained environments.
  • It automates the selection of optimal parameters, reducing manual effort and improving performance.
  • This approach is valuable for applications where both accuracy and runtime efficiency are critical.