Derivation of Augmented Arithmetic for Computing Gradient, Hessian and Jacobian through Forward Mode AD Using Dual Numbers

P.Senthil Vadivu, S. Ponnusamy


This paper presents a new approach to Automatic Differentiation (AD) for a scalar valued and twice continuously
differentiable function f : R^n - R. A new arithmetic is obtained based on the chain rule and using
augmented algebra of real numbers. The chain rule based differentiation arithmetic is used to find the Gradient
and Hessian. Jacobian is evaluated using Gradient arithmetic by computing Gradient for components and is arranged
in matrix form to give Jacobian value. The resulting derivative evaluation uses the operator overloading
concept which uses computer programs written in C++.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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