Abstract
We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.
Highlights
IntroductionSensitivity analysis is an important area in engineering research; when selecting a sensitivity calculation method, the accuracy and computational cost must be considered
Sensitivity analysis is an important area in engineering research; when selecting a sensitivity calculation method, the accuracy and computational cost must be considered.The commonly used method for sensitivity calculation is the finite difference
We propose a definition of the quaternion step derivative for a complex function with reference to the method of deriving the complex step derivative for a real function
Summary
Sensitivity analysis is an important area in engineering research; when selecting a sensitivity calculation method, the accuracy and computational cost must be considered. Further examined the approach of Martins et al [1,2], and based on the results of studying the derivation of the step derivative along the complex direction for a real function using the Taylor series expansion, a complex step differential approximation and its application to a numerical algorithm were presented (see [12,13,14,15]). But unlike [15,19], this study focuses on the definition and structure of quaternions based on the original meaning proposed by Hamilton and proposes a step derivative by recognizing complex number i as the basis of the quaternion This defines the step derivative of the complex function, while retaining the meaning and characteristics of the quaternion. As a future development plan, the step derivative of the quaternionic function is conceived of by defining the step derivative in the base direction of various Clifford algebras
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