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MultiZ: A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers

Authors:

Andres M. Aguirre-Mesa,

Manuel J. Garcia,

Harry MillwaterAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 46, Issue 3

Article No.: 23, Pages 1 - 30

https://doi.org/10.1145/3378538

Published: 13 July 2020 Publication History

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Abstract

Multicomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. The similarities between multicomplex and multidual algebras allowed us to create a unified library to use either one for sensitivity analysis. This library can be used to compute arbitrary order derivates of functions of a single variable or multiple variables. The storage of matrix representations of multicomplex and multidual numbers is avoided using a combination of one-dimensional resizable arrays and an indexation method based on binary bitwise operations. To provide high computational efficiency and low memory usage, the multiplication of hypercomplex numbers up to sixth order is carried out using a hard-coded algorithm. For higher hypercomplex orders, the library uses by default a multiplication method based on binary bitwise operations. The computation of algebraic and transcendental functions is achieved using a Taylor series approximation. Fortran and Python versions were developed, and extensions to other languages are self-evident.

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Cited By

Qian HLi ATian YWang YCao ZLiu QJiang D(2025)Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbationApplied Mathematical Modelling10.1016/j.apm.2024.115669137(115669)Online publication date: Jan-2025
https://doi.org/10.1016/j.apm.2024.115669
Navarro JVelasquez-Gonzalez JAristizabal MJarmer GKessler SMontoya AMillwater HRestrepo D(2024)Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element MethodAIAA Journal10.2514/1.J06283462:4(1447-1460)Online publication date: Apr-2024
https://doi.org/10.2514/1.J062834
Peng ZYang YShao TJiang CZhou K(2024)X-SLAM: Scalable Dense SLAM for Task-aware Optimization using CSFDACM Transactions on Graphics10.1145/365823343:4(1-15)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658233
Show More Cited By

Index Terms

MultiZ: A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Automatic differentiation
      2. Numerical differentiation

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 46, Issue 3

September 2020

267 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3410509

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

Issue’s Table of Contents

Copyright © 2020 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 13 July 2020

Online AM: 07 May 2020

Accepted: 01 January 2020

Revised: 01 December 2019

Received: 01 October 2018

Published in TOMS Volume 46, Issue 3

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U.S. Department of Defense
Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS)

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Cited By

Qian HLi ATian YWang YCao ZLiu QJiang D(2025)Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbationApplied Mathematical Modelling10.1016/j.apm.2024.115669137(115669)Online publication date: Jan-2025
https://doi.org/10.1016/j.apm.2024.115669
Navarro JVelasquez-Gonzalez JAristizabal MJarmer GKessler SMontoya AMillwater HRestrepo D(2024)Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element MethodAIAA Journal10.2514/1.J06283462:4(1447-1460)Online publication date: Apr-2024
https://doi.org/10.2514/1.J062834
Peng ZYang YShao TJiang CZhou K(2024)X-SLAM: Scalable Dense SLAM for Task-aware Optimization using CSFDACM Transactions on Graphics10.1145/365823343:4(1-15)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658233
Rincon-Tabares JAristizabal MBalcer MMontoya AMillwater HRestrepo D(2024)Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element methodAdditive Manufacturing10.1016/j.addma.2024.10448894(104488)Online publication date: Aug-2024
https://doi.org/10.1016/j.addma.2024.104488
Millwater HBalcer MMillwater N(2024)Calculation of machine precision second order derivatives using dual-complex numbersNumerical Algorithms10.1007/s11075-024-01937-2Online publication date: 4-Oct-2024
https://doi.org/10.1007/s11075-024-01937-2
Li AQian HMa YYan XCao ZZhu RJiang D(2024)Sensitivity analysis of the rotor-bearing system with fractional power nonlinearity using multicomplex variable derivationNonlinear Dynamics10.1007/s11071-024-09449-3112:10(8071-8088)Online publication date: 6-Apr-2024
https://doi.org/10.1007/s11071-024-09449-3
Velasquez-Gonzalez JNavarro JAristizabal MMillwater HMontoya ARestrepo D(2023)Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)Applied Sciences10.3390/app1312712513:12(7125)Online publication date: 14-Jun-2023
https://doi.org/10.3390/app13127125
Balcer MAristizabal MRincon-Tabares JMontoya ARestrepo DMillwater H(2023)HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element MethodJournal of Verification, Validation and Uncertainty Quantification10.1115/1.40624598:2Online publication date: 13-Jun-2023
https://doi.org/10.1115/1.4062459
Acosta CBhalla AGuo R(2023)Sensitivity analysis of electro-mechanical properties of soft and hard piezoelectrics using the complex finite element methodFerroelectrics10.1080/00150193.2023.2201781611:1(188-204)Online publication date: 17-Jul-2023
https://doi.org/10.1080/00150193.2023.2201781
Aguirre-Mesa ARestrepo-Velasquez SRamirez-Tamayo DMontoya AMillwater H(2023)Computation of two dimensional mixed-mode stress intensity factor rates using a complex-variable interaction integralEngineering Fracture Mechanics10.1016/j.engfracmech.2022.108981277(108981)Online publication date: Jan-2023
https://doi.org/10.1016/j.engfracmech.2022.108981
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