Commutative Extended Complex Numbers and Connected Trigonometry

The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. Inspired by the recent complex step method that is only valid for firstorder sensitivities, the new multicomplex approach is valid to arbitrary order. The mathematical theory behind this approach is revealed, and an efficient procedure for the automatic implementation of the method is described. Several applications are presented to validate and demonstrate the accuracy and efficiency of the algorithm. The results are compared to conventional approaches such as finite differencing, the complex step method, and two separate automatic differentiation tools. The multicomplex method performs favorably in the preliminary comparisons and is therefore expected to be useful for a variety of algorithms that exploit higher order derivatives.

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“…This type of number is now commonly named a multicomplex number. They were studied in detail by Price [1991] and Fleury et al [1993].…”

Section: Definition Of Multicomplex Numbersmentioning

confidence: 99%

Using Multicomplex Variables for Automatic Computation of High-Order Derivatives

Lantoine

Russell

Dargent

2012

ACM Trans. Math. Softw.

127

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“…Many generalizations of traditional plane trigonometry have been developed. They range from the spherical trigonometry of Nenelaus, which goes back to the first century A.D. [3], to trigonometry on SU(3) [6], which is one among a number of relatively recent developments [9,11,13,14,20,25].…”

Section: Trigonometry In Perspectivementioning

confidence: 99%

The combinatorial structure of trigonometry

Antippa

2003

International Journal of Mathematics and Mathematical Sciences

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The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of infinitesimal angles, these identities lead to the power series expansions of the trigonometric functions. When applied to the interior angles of a polygon, they lead to two general constraints satisfied by the corresponding tangents. In the case of multiple equal angles, they reduce to the Bernoulli identities. For the case of two distinct angles, they reduce to the Ptolemy identity. They can also be used to derive the De Moivre-Cotes identity. The above results combined provide an appropriate mathematical combinatorial language and formalism for trigonometry and more generally polygonometry. This latter is the structural language of molecular organization, and is omnipresent in the natural phenomena of molecular physics, chemistry, and biology. Polygonometry is as important in the study of moderately complex structures, as trigonometry has historically been in the study of simple structures

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“…There are four idempotent elements in C 2 , two of these idempotent elements, namely (1 + i 1 i 2 )/2 and (1 − i 1 i 2 )/2 play an important role since every element in C 2 has a unique representation as a linear combination of them. Multicomplex numbers were studied in detail in [2] and [8].…”

Section: Introductionmentioning

confidence: 99%

Multicomplex Taylor Series Expansion for Computing High-Order Derivatives

Millwater¹,

Shirinkam²

2014

Int. J. of Appl. Math.

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Multicomplex Taylor series expansion (MCTSE) is a numerical method for calculating higher-order partial derivatives of a multivariable realvalued and complex-valued analytic function based on Taylor series expansion without subtraction cancelation errors. The implementation has been facilitated using Cauchy-Riemann matrix representation of multicomplex variables. In this paper, we show steps for finding these matrices and, in addition, that the number of appearances of the k th derivatives follows the Pascal's triangle. Also, the situations where the MCTSE is not applicable is determined. Finally, we investigate the application of the method for complex-valued functions.

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Commutative Extended Complex Numbers and Connected Trigonometry

Cited by 29 publications

References 0 publications

Using Multicomplex Variables for Automatic Computation of High-Order Derivatives

Using Multicomplex Variables for Automatic Computation of High-Order Derivatives

The combinatorial structure of trigonometry

Multicomplex Taylor Series Expansion for Computing High-Order Derivatives

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