The Complex-Step Derivative Approximation on Matrix Lie Groups

Cossette, Charles Champagne; Walsh, Alex; Forbes, James Richard

doi:10.1109/lra.2020.2965882

Cited by 14 publications

(7 citation statements)

References 13 publications

(29 reference statements)

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“…The technique continues to be used (see e.g. [4,17]) and extended: to higher order derivatives [13], matrix functions [2], Lie groups [8] and bicomplex or multicomplex numbers [7,14].…”

Section: Relation To Existing Workmentioning

confidence: 99%

Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

Roelfs

Dudal

Huybrechs

2021

Journal of Computational and Applied Mathematics

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The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof.

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“…The technique continues to be used (see e.g. [4,17]) and extended: to higher order derivatives [13], matrix functions [2], Lie groups [8] and bicomplex or multicomplex numbers [7,14].…”

Section: Relation To Existing Workmentioning

confidence: 99%

Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

Roelfs

Dudal

Huybrechs

2021

Journal of Computational and Applied Mathematics

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show abstract

“…The technique continues to be used (see e.g. [11,12]) and extended: to higher order derivatives [13], matrix functions [14], Lie groups [15] and bicomplex or multicomplex numbers [3,16].…”

Section: Relation To Existing Workmentioning

confidence: 99%

Quaternionic Step Derivative: Machine Precision Differentiation of Holomorphic Functions using Complex Quaternions

Roelfs,

Dudal,

Huybrechs

2020

Preprint

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ComplexStep Derivative (CSD) allows easy and accurate differentiation up to machine precision of real functions by evaluating them a small imaginary step next to the real line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. Additionally, the extra degree of freedom means second derivatives of real functions are straightforwardly computed by taking a small step in both a quaternionic and complex direction, making the technique a powerful extension of CSD. To demonstrate the ease of implementation numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof.

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“…Hence, this technique has been used for sensitivity analysis in a wide variety of environments, including computational fluid dynamics (see [8,9]). Lai and Crassidis [10,11] 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]). Kim et al [16][17][18] investigated the composition and properties of quaternion functions based on the algebraic features of quaternions.…”

Section: Introductionmentioning

confidence: 99%

Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

Kim

2021

Mathematics

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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.

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The Complex-Step Derivative Approximation on Matrix Lie Groups

Cited by 14 publications

References 13 publications

Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

Quaternionic Step Derivative: Machine Precision Differentiation of Holomorphic Functions using Complex Quaternions

Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

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