Dual Algorithms

The cubic spline interpolation method, the Runge–Kutta method, and the Newton–Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated compositions of functions which are not necessarily defined by a closed form expression. The code for the algorithms has been written in Fortran and some examples are presented. Among them, we use the dual Newton–Raphson method to obtain the derivatives of the output angle in the RRRCR spatial mechanism; we use the dual normal cubic spline interpolation algorithm to obtain the thermal diffusivity using photothermal techniques; and we use the dual Runge–Kutta method to obtain the derivatives of functions depending on the solution of the Duffing equation.

[1]  Xiang-Tuan Xiong,et al.  Fourier truncation method for high order numerical derivatives , 2006, Appl. Math. Comput..

[2]  Maxwell Blair,et al.  DNAD, a simple tool for automatic differentiation of Fortran codes using dual numbers , 2013, Comput. Phys. Commun..

[3]  R. Ohba,et al.  New finite difference formulas for numerical differentiation , 2000 .

[4]  M. Ramachandran.  Fast derivative computation using smooth X-splines , 2012 .

[5]  Carlos Alberto Cruz Villar,et al.  A Dual Number Approach for Numerical Calculation of Velocity and Acceleration in the Spherical 4R Mechanism , 2013, ArXiv.

[6]  Riaz A. Usmani,et al.  Inversion of a tridiagonal jacobi matrix , 1994 .

[7]  Beer S Bhadauria,et al.  Finite Difference Formulae for Unequal Sub- Intervals Using Lagrange's Interpolation Formula , 2009 .

[8]  H. Cohen.  Numerical Approximation Methods , 2011 .

[9]  A. A. Mohamad,et al.  An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy , 2012, J. Comput. Appl. Math..

[10]  A. Wightman,et al.  Mathematical Physics. , 1930, Nature.

[11]  William H. Press,et al.  Numerical recipes in Fortran 77 : the art of scientificcomputing. , 1992 .

[12]  J. Alonso,et al.  The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations , 2011 .

[13]  Vladimir I. Dmitriev,et al.  Numerical differentiation using spline functions , 2012 .

[14]  Chu-Li Fu,et al.  A wavelet-Galerkin method for high order numerical differentiation , 2010, Appl. Math. Comput..

[15]  W. Gander.  On Halley's Iteration Method , 1985 .

[16]  Dan Piponi,et al.  Automatic Differentiation, C++ Templates, and Photogrammetry , 2004, J. Graphics, GPU, & Game Tools.

[17]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[18]  J. Y. S. Luh,et al.  Dual-number transformation and its applications to robotics , 1987, IEEE Journal on Robotics and Automation.

[19]  Bengt Fornberg,et al.  Numerical Differentiation of Analytic Functions , 1981, TOMS.

[20]  Harry H. Cheng.  Programming with dual numbers and its applications in mechanisms design , 1994, Engineering with Computers.

[21]  Griewank,et al.  On automatic differentiation , 1988 .

[22]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[23]  H. Nijmeijer,et al.  On Lyapunov control of the Duffing equation , 1995 .

[24]  Jianping Li.  General explicit difference formulas for numerical differentiation , 2005 .

[25]  Natanael Karjanto,et al.  Newton's method's basins of attraction revisited , 2009, Appl. Math. Comput..

[26]  Clifford,et al.  Preliminary Sketch of Biquaternions , 1871 .

[27]  Stephan Freytag.  An Introduction To Splines For Use In Computer Graphics And Geometric Modeling , 2016 .

[28]  Peter Uwer,et al.  EasyNData: A simple tool to extract numerical values from published plots , 2007, 0710.2896.

[29]  Zewen Wang,et al.  Numerical differentiation for high orders by an integration method , 2010, J. Comput. Appl. Math..

[30]  J. Jesús Cervantes-Sánchez,et al.  Function generation with the RRRCR spatial linkage , 2014 .

[31]  Elçin Yusufoglu,et al.  Numerical solution of Duffing equation by the Laplace decomposition algorithm , 2006, Appl. Math. Comput..

[32]  S. Hassani.  Mathematical physics : a modern introduction to Its foundations , 2013 .

[33]  E. Pennestrì,et al.  Linear algebra and numerical algorithms using dual numbers , 2007 .

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[35]  Richard D. Neidinger,et al.  Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming , 2010, SIAM Rev..

[36]  Karsten Ahnert,et al.  Numerical differentiation of experimental data: local versus global methods , 2007, Comput. Phys. Commun..

[37]  M. Depriester,et al.  New methodology for thermal parameter measurements in solids using photothermal radiometry , 2005 .

[38]  Hans-Hellmut Nagel,et al.  Automatic differentiation facilitates OF-integration into steering-angle-based road vehicle tracking , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).