Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System
The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-08-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/10/3/206 |
| _version_ | 1797520220916547584 |
|---|---|
| author | Ji-Eun Kim |
| author_facet | Ji-Eun Kim |
| author_sort | Ji-Eun Kim |
| collection | DOAJ |
| description | The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains. |
| first_indexed | 2024-03-10T07:53:43Z |
| format | Article |
| id | doaj.art-eaea29b9bc36487abada76e14f0c1530 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T07:53:43Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-eaea29b9bc36487abada76e14f0c15302023-11-22T12:02:58ZengMDPI AGAxioms2075-16802021-08-0110320610.3390/axioms10030206Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion SystemJi-Eun Kim0Department of Mathematics, Dongguk University, Gyeongju 38066, KoreaThe step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.https://www.mdpi.com/2075-1680/10/3/206complex functionsquaternionstep derivativesnon-commutativity |
| spellingShingle | Ji-Eun Kim Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System Axioms complex functions quaternion step derivatives non-commutativity |
| title | Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System |
| title_full | Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System |
| title_fullStr | Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System |
| title_full_unstemmed | Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System |
| title_short | Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System |
| title_sort | approximation of directional step derivative of complex valued functions using a generalized quaternion system |
| topic | complex functions quaternion step derivatives non-commutativity |
| url | https://www.mdpi.com/2075-1680/10/3/206 |
| work_keys_str_mv | AT jieunkim approximationofdirectionalstepderivativeofcomplexvaluedfunctionsusingageneralizedquaternionsystem |