Some time ago I was involved in a discussion of a doctoral dissertation
J. A. Fike,
“Multi-Objective Optimization Using Hyper-Dual Numbers. Ph.D. Dissertation, Stanford University, 2013)
and subsequent publications related thereto. One of the key points in this work was the use of hyper-dual numbers
in the procedure of automatic differentiation. The latter made it possible (with computer implementation) to obtain
the values of the studied function, its first and second derivatives with machine accuracy. Relatively speaking,
a hyper-dual function was presented as an object with properties that include: the values of the function and its
derivatives calculated by exact analytical formulas. When trying to implement this approach for their own purposes,
I came across some redundancy of the hyper-dual number (function) itself: the value of the first derivative
was repeated twice, which led to an increase in memory for storing the hyper-dual number (function). Moreover,
if the concept of a hyper-dual number is expanded (taking into account derivatives of a higher order),
the indicated redundancy will increase. This circumstance prompted me to create truncated hyper-dual numbers (functions)
that are free from duplication of the values of derivatives. There was also a need to build hyper-dual numbers with varying
degrees of expansion, called super-dual numbers and their functions.
This new type of data has been used in a number of computational tasks that require multiple calculations of the values of the function being studied and
its derivatives are of different orders.
There has recently been a growing interest in dual-numbers issues. However, in most cases, these studies
are related to the theory of numbers, and practical application refers to the simplest iterration procedures. Moreover, in Russian-speaking sources
hyper-dual numbers have little or no lighting.
Therefore, in order to fill this gap, I have decided to make public my some private research related to
application of hyper-dual and super-dual numbers in computational processes (see submitted materials).
When using and quoting submitted materials, please respect the copyrights of VIO Solutions Inc.