Note on usage and theory of hyper-dual numbers for first and second order automatic differentiation
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Similar to dual numbers, hyper-dual numbers are extended numbers that can be used for automatic differentiation. In contrast to dual numbers, hyper-dual numbers can be directly used to compute derivatives of second order. In this note we motivate hyper-dual numbers from finite differences and dual numbers. In this note we explain in an illustrative way how to use hyper-dual numbers. We convey our exposition with mathematical theorems, that validate the correctness of the derivatives as computed with hyper-dual numbers. Eventually we refer to freely available implementations of hyper-dual numbers.
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