A Hyper-Dual Number Approach to Higher-Order Derivative Computation

This paper develops a theoretical framework for the computation of higher-order derivatives based on the algebra of hyper-dual numbers. Extending the classical dual number system, hyper-dual numbers provide a natural and rigorous mechanism for encoding and propagating derivative information through function composition and arithmetic operations. We formalize the underlying algebraic structure, define generalized hyper-dual extensions of scalar functions via multidimensional Taylor expansions, and establish consistency with standard differential calculus. The proposed approach enables exact computation of partial derivatives and mixed higher-order derivatives without resorting to symbolic manipulation or approximation schemes. We further investigate the algebraic properties and closure under differentiable operations, illustrating the method’s potential for unifying aspects of automatic differentiation with multivariable calculus. This study contributes to the theoretical foundation of algorithmic differentiation and highlights hyper-dual numbers as a precise and elegant tool in computational differential analysis.