Higher Order Real Derivatives Using Parabolic Analytic Functions

Amid the bidimensional hypercomplex numbers, parabolic numbers are defined as $\{z=x+\imath y:\; x,y\in \mathbb{R}, \imath^2=0, \imath\neq 0\}$. The analytic functions of a parabolic variable have been introduced as an analytic continuation of the real function of a real variable. Also, their algebraic property has already been discussed. This paper will show the $n$-th derivative of the real functions using parabolic numbers to further generalize the automatic differentiation. Also, we shall show some of the applications of it.

Keywords:

Parabolic, Analytic functions, Dual number, Higher order derivative automatic differentiation, Hypercomplex numbers.,

PDF

___

[1] Bischof, C.H., Pusch, G.O., Knoesel, R.: Sensitivity analysis of the mm5 weather model using automatic differentiation. Computers in Physics 10(6), 605–612 (1996)
[2] Bisi, C., De Martino, A., Winkelmann, J.: On a runge theorem over R3. Annali di Matematica 202(4), 1531–1556 (2023)
[3] Bisi, C., Winkelmann, J.: On a quaternionic Picard theorem. Proc. Amer. Math. Soc., Ser. B, 7, 106–117 (2020)
[4] Bisi, C., Winkelmann, J.: The harmonicity of slice regular functions. The Journal of Geometric Analysis 31(8), 7773–7811 (2021)
[5] Casanova, G.: Parabolic analytic functions. Adv. Appl. Clifford Algebras 9(2), 221–224 (1999)
[6] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: N-dimensional geometries generated by hypercomplex numbers. Adv. Appl. Clifford Algebr. 15(1), 1–25 (2005)
[7] Catoni, F., Cannata, R., Nichelatti, E.: The parabolic analytic functions and the derivative of real functions. Adv. Appl. Clifford Algebr. 14(2), 185–190 (2004)
[8] Fjelstad, P., Gal, S.G.: Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers. Adv. Appl. Clifford Algebras 11(1), 81–107 (2001)
[9] Gentili, G., Stoppato, C., Struppa, D.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer (2022)
[10] Hovland, P.D., et al: Sensitivity analysis and design optimization through automatic differentiation. J. Phys.: Conf. Ser. 16, 466–470 (2005)
[11] Kedem, G.: Automatic differentiation of computer programs. ACM Trans. Math. Software 6(2), 150–165 (1980)
[12] Mehmood, S., Ochs, P.: Automatic differentiation of some first-order methods in parametric optimization. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, vol. 108, pp. 1584–1594. PMLR (2020)
[13] Pe˜nu nuri, F., Pe´on, R., Gonz´alez-S´anchez, D., Escalante Soberanis, M.A.: Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Appl. Math. 170, 649–659 (2020)
[14] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in Fortran 77: The art of scientific computing. Cambridge University Press (1992)
[15] Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education (2007)
[16] Yaglom, I.M.: A simple non-Euclidean geometry and its physical basis. Heidelberg Science Library. Springer-Verlag, New York-Heidelberg (1979). An elementary account of Galilean geometry and the Galilean principle of relativity, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon.