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Enabling quaternion derivatives: the generalized HR calculus.

Dongpo Xu1, Cyrus Jahanchahi2, Clive C Took3

  • 1School of Mathematics and Statistics , Northeast Normal University , Changchun 130024, People's Republic of China.

Royal Society Open Science
|September 12, 2015
PubMed
Summary

This study introduces generalized HR (GHR) derivatives for quaternion calculus, enabling analysis of non-analytic functions. GHR derivatives offer rigorous product and chain rules, advancing quaternion analysis in complex optimization problems.

Keywords:
generalized HR calculusnon-analytic quaternion functionnonlinear quaternion functionsquaternion derivativesquaternion least mean square

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Signal Processing

Background:

  • Quaternion derivatives are limited to analytic functions, restricting their application.
  • Existing HR calculus has limitations with non-analytic functions and lacks rigorous rules for quaternion algebra.
  • Non-analytic, real-valued functions are common in applications like mean square error optimization.

Purpose of the Study:

  • Introduce generalized HR (GHR) derivatives for quaternion calculus.
  • Address limitations of existing quaternion derivative methods for non-analytic functions.
  • Develop rigorous product and chain rules for quaternion analysis.

Main Methods:

  • Employ quaternion rotations within a general orthogonal system to define GHR derivatives.
  • Develop left- and right-hand versions of quaternion derivatives for general functions.
  • Extend principles to other functional calculi in non-commutative settings.

Main Results:

  • GHR derivatives handle both analytic and non-analytic quaternion functions.
  • Established rigorous product and chain rules, mean-value theorem, and Taylor's theorem for quaternion calculus.
  • Demonstrated applicability in statistical learning theory and adaptive signal processing.

Conclusions:

  • GHR calculus provides a robust framework for quaternion analysis, overcoming limitations of previous methods.
  • The proposed method enhances the utility of quaternion calculus in diverse scientific and engineering fields.
  • Quaternion rotations are key to extending calculus principles in non-commutative algebra.