VIO Solutions ® 
June 18, 2021

LIST OF SUBMITTED MATERIALS


1. Olifer V.I. A BRIEF HISTORY OF DUAL AND HYPER-DUAL NUMBERS

Briefly describes the history of the emergence of dual numbers, their definition and use. The concept of hyper-dual numbers as an extension of dual numbers is given. The basic operations on hyper-dual numbers and the concept of the function of a hyper-dual argument are considered. The use of hyper-dual numbers and functions from them for solving some practical problems that require multiple accurate calculation of the gradient and Hessian values in the computational processes of automatic differentiation is indicated.

updated: 4/07/2020

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2. Olifer V.I. TO THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM USING HYPER-DUAL NUMBERS

In this paper, we consider the use of extended hyper-dual numbers in the numerical integration of the equations of the Cauchy problem using Taylor series expansion and computer-aided differentiation. A new type of numbers is introduced - dual numbers of the third class. The basic operations and basic functions of this class of dual numbers are described. The procedure of sequential pointwise integration using automatic differentiation and dual numbers of the third class is presented. A computer implementation for the SWIFT language of the macOS operating system is given. Numerical experiments based on the obtained software were carried out.

updated: 4/01/2020

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3. Olifer V.I. AUTOMATIC DIFFERENTIATION BASED ON SUPER-DUAL NUMBERS

The issues related to the use of special dual numbers (super-dual numbers) in the method of computer automatic differentiation are considered. New types of dual numbers of different classes are introduced. The basic operations and basic functions of the space of some classes of super-dual numbers are considered. Appendices 1, 2 and 3 provide computer implementation of compact dual class third numbers for the SWIFT language of the macOS operating system is given.

updated: 4/07/2020

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4. Olifer V.I. TRUNCATED HYPER-DUAL NUMBERS IN AUTOMATIC DIFFERENTIATION

This paper discusses issues related to the use of special numbers (hyper-dual numbers) in the method of computer automatic differentiation. A new type of numbers is introduced (truncated hyper-dual numbers), which are free of the redundancy inherent in hyper-dual numbers. Basic operations and basic functions are given. spaces of truncated hyper-dual numbers. Given their matrix representation, examples of implementation Newton and Newton-Chebyshev iterative methods based on truncated hyper-dual functions of one and two variables, and also the use of truncated hyper-dual functions in the Pareto optimization procedure is given. Appendices 1, 2, and 3 provide computer implementation of truncated hyper-dual numbers for the SWIFT language of the macOS operating system.

updated: 4/07/2020

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5. Olifer V.I. SOLUTION OF NONLINEAR EQUATIONS ON THE BASIS OF AUTOMATIC DIFFERENTIATION AND TRUNCATED HYPER-DUAL NUMBERS

This article discusses issues related to the use of special numbers (hyper-dual numbers) in the method of computer automatic differentiation. A new type of numbers (truncated hyper-dual numbers) are introduced that are free from the redundancy inherent in hyper-dual numbers. The main operations and basic functions of the space of truncated hyper-dual numbers are given. Given their matrix form of presentation, examples of the implementation of iterative methods Newton, Chebyshev, Rafson and the continuous analogue of Newton's method with a special procedure of acceleration of convergence based on truncated hyper-dual functions. Appendices 1, 2 and 3 provide computer implementation truncated hyper-dual numbers for the SWIFT macOS operating system language.

updated: 4/07/2020

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6. Olifer V.I. TO CALCULATING RESPONSE CURVES USING KRIGING INTERPOLATION, AUTOMATIC DIFFERENTIATION AND TRUNCATED HYPER-DUAL NUMBERS

The article discusses issues related to the use of special numbers (hyper-dual numbers) and computer automatic differentiation in kriging interpolation. The matrix-block representations of the resolving equations of kriging interpolation are given taking into account the first and second derivatives. A new type of numbers (truncated hyper-dual numbers) is introduced, which are free of redundancy inherent in hyper-dual numbers. The basic operations and basic functions of the space of truncated hyper-dual numbers are given. A computer implementation of Kriging interpolation is presented taking into account truncated hyper-dual numbers for the SWIFT language of the macOS operating system.

updated: 4/07/2020

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7. Olifer V.I. HYPER-DUAL MATRIX EQUATIONS AND THEIR SENSITIVITY

This article presents a special type of numbers (truncated hyper-dual numbers). The basic algebraic operations on these numbers are given. Matrices and matrix equations are considered whose components are truncated hyper-dual numbers. Appendices 1, 2, and 3 show the computer implementation of truncated hyper-dual numbers and matrices for the SWIFT language of the macOS operating system. Appendix 4 gives the results of calculating the sensitivity of the matrix equation with truncated hyper-dual components.

updated: 4/07/2020

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8. Olifer V.I. BALLISTIC METHOD USING HYPER-DUAL NUMBERS

This publication describes the use of extended hyper-dual numbers in the numerical implementation of the ballistic method. To solve the intermediate Cauchy problem, the method of expansion into Taylor series and computer automatic differentiation based on dual numbers of the third class are used. The basic operations and basic functions of this class of dual numbers are described. The procedure of sequential pointwise integration of the initial two-point boundary value problem of the second order using automatic differentiation and dual numbers of the third class is presented. A computer implementation for the SWIFT language of the macOS operating system is given. Numerical experiments based on the obtained software were carried out.

updated: 04/07/2020

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9. Olifer V.I. NUMERICAL SOLUTION OF THE NONLINEAR CAUCHY PROBLEM OF THE 2ND ORDER USING TRACED HYPERDUAL NUMBERS

This publication discusses a numerical method for solving the second-order nonlinear Cauchy problem, based on the use of Taylor series expansion and automatic differentiation based on special dual numbers (truncated hyper-dual numbers). Computer implementation introduced this method for the SWIFT language of the macOS operating system on the basis of which numerical experiments.

updated: 04/07/2020

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10. Olifer V.I. CALCULATION OF THE GRADIENT AND HESSIAN BY THE AUTOMATIC DIFFERENTIATION METHOD WITH THE USE OF TRUNNED HYPER-DUAL NUMBERS

The article discusses the use of truncated hyper-dual numbers, functions of them and computer automatic differentiation for accurate (with machine precision) calculation of the values of the gradient components and the Hessian of functions of many variables. A computer code is given in the SWIFT language of the macOS operating system that implements the proposed algorithm.

updated: 07/21/2020

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11. Olifer V.I. TO NUMERICAL SIMULATION OF THE DZHANIBEKOV EFFECT

This publication discusses the numerical modeling of the Dzhanibekov effect, also known as the tennis racket effect or the intermediate axis theorem. The basis is the dynamic Euler equations, the solution of which is carried out by the Cauchy method using the Taylor series. To take into account various forms of functions of external torques, the method of automatic differentiation based on truncated hyper-dual numbers is applied. A computer implementation of the described model for the SWIFT language of the macOS operating system is presented, on the basis of which numerical experiments have been carried out.

updated: 09/07/2020

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12. Olifer V.I. NUMERICAL INTEGRATION USING HYPER-DUAL NUMBERS

This publication discusses a numerical integration method based on the Taylor series expansion and automatic differentiation using special dual numbers (truncated hyper-dual numbers). The computer implementation of this method for the SWIFT language operating macOS systems. Numerical experiments have been carried out.

updated: 02/20/2021

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13. Olifer V.I. DUAL NUMBERS IN NUMERICAL INTEGRATION

This publication discusses a numerical integration method based on Taylor series expansion and automatic differentiation using classical dual numbers. A computer implementation of this method is presented for macOS SWIFT language.

updated: 03/20/2021

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