A new proof of the generalized Hamiltonian–Real calculus
Keyword(s):
The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.
2010 ◽
Vol 54
(1-2)
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pp. 59
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1992 ◽
1994 ◽
Vol 04
(01)
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pp. 23-51
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2015 ◽
Vol 84
(3)
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pp. 399-412
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1971 ◽
Vol 19
(4)
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pp. 718-728
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