HyperDualNumbers

HyperDualNumbers

Hyper-dual numbers can be used to compute first and second derivatives numerically without the cancellation errors of finite-differencing schemes.

The initial Julia implementation (and up to v3.0.1) is directly based on the C++ implementation by Jeffrey Fike and Juan J Alonso, both of Stanford University, department of Aeronautics and Astronautics and is described in the paper:

The Development of Hyper-Dual Numbers for Exact Second Derivative Calculations

The early versions have been derived/written by Rob J Goedman (goedman@icloud.com).

HyperDualNumbers.jl v4.0.0 has been completely redone by Benoit Pasquier and follows the structure of the JuliaDiff/DualNumbers package.]

Creation of a HyperDualNumber

HyperDualNumbers.Hyper — Method.

Creation of a HyperDuakNumber

hn = Hyper(2.0, 1.0, 1.0, 0.0)

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Symbolic derivative list

HyperDualNumbers.symbolic_derivative_list — Constant.

Symbolic derivative list

The format is a list of (Symbol,Expr,Expr) tuples.

Entries in each tuple:

* `:f `     : Function symbol
* `:df`    : Symbolic expression for first derivative
* `:d²f`   : Symbolic expression for second derivative

The symbol :x is used within deriv_expr for the point at which the derivative should be evaluated.

Example of a tuple in the list

(:sqrt, :(1/2/sqrt(x)), :(-1/4/x^(3/2)))
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Fields of a HyperDualNumber

HyperDualNumbers.ε₁ — Constant.

ɛ₁ contains the first derivative after the evaluation

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HyperDualNumbers.ε₂ — Constant.

ɛ₂ contains the first derivative after the evaluation

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HyperDualNumbers.ε₁ε₂ — Constant.

ɛ₁ɛ₂ contains the second derivative after the evaluation

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Index