Hypercomplex numbers

Introduction

Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022

According to the concept of quaternion complex-number space proposed by William Hamiltonian, the rotation angle of a three-dimensional object can be calculated by using quaternions. Compared with a matrix, a quaternion can be regarded as the combination of a scalar and a vector that is calculated more efficiently and is convenient for interpolation. Assuming that a quaternion is a hypercomplex number consisting of the real part w and the imaginary part (i, j, k), the normalized quaternion can be expressed as: q=q0+q1i+q2j+q3k

Multidual Sensitivity Method in Ray-Tracing Transport Simulations

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Journal Information

Published in Nuclear Science and Engineering, 2021

Matthew R. Balcer, Harry Millwater, Jeffrey A. Favorite

Hypercomplex numbers contain a single real part and multiple nonreal parts. Two types of hypercomplex numbers are multicomplex and multidual numbers. The nonreal parts of a multicomplex number are represented by an imaginary number, (Ref. 1). The nonreal parts of a multidual number are represented by the nonreal component of dual number, where (Ref. 2). Hypercomplex differentiation is an extension of the first-order derivative derived from the complex Taylor series, known as the complex-step method.3 Higher-order derivatives are derived with hypercomplex algebra as outlined in Ref. 1 using multicomplex variables and in Ref. 2 using multidual variables.

Hypercomplex numbers

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Introduction

Multidual Sensitivity Method in Ray-Tracing Transport Simulations