Table of Contents

Citations

This page lists the references used by ChebyshevSharp documentation and algorithms.

Chebyshev Interpolation

  • Berrut, J.-P. and Trefethen, L. N. (2004). "Barycentric Lagrange Interpolation." SIAM Review, 46(3), 501-517. DOI: 10.1137/S0036144502417715.
  • Trefethen, L. N. (2013; extended edition 2019). Approximation Theory and Approximation Practice. SIAM. Author page; extended-edition DOI: 10.1137/1.9781611975949.
  • Trefethen, L. N. (2011). "Six Myths of Polynomial Interpolation and Quadrature." Mathematics Today, 47(4), 184-188. Author bibliography.
  • Trefethen, L. N. (2000). Spectral Methods in MATLAB. SIAM. DOI: 10.1137/1.9780898719598.
  • Trefethen, L. N. (2017). "Lecture 3: Chebyshev Series." Oxford University. PDF.
  • Runge, C. (1901). "Uber empirische Funktionen und die Interpolation zwischen aquidistanten Ordinaten." Zeitschrift fur Mathematik und Physik, 46, 224-243. Internet Archive scan.
  • Clenshaw, C. W. and Curtis, A. R. (1960). "A Method for Numerical Integration on an Automatic Computer." Numerische Mathematik, 2, 197-205. DOI: 10.1007/BF01386223.
  • Waldvogel, J. (2006). "Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules." BIT Numerical Mathematics, 46(1), 195-202. DOI: 10.1007/s10543-006-0045-4.
  • Good, I. J. (1961). "The Colleague Matrix, a Chebyshev Analogue of the Companion Matrix." The Quarterly Journal of Mathematics, 12(1), 61-68. DOI: 10.1093/qmath/12.1.61.
  • Salzer, H. E. (1972). "Lagrangian Interpolation at the Chebyshev Points x_{n,v} = cos(v pi/n), v = 0(1)n; Some Unnoted Advantages." The Computer Journal, 15(2), 156-159. DOI: 10.1093/comjnl/15.2.156.
  • Smoktunowicz, A. (2002). "Backward Stability of Clenshaw's Algorithm." BIT Numerical Mathematics, 42(3), 600-610. DOI: 10.1023/A:1022001931526.

Tensor Train Algorithms

  • Oseledets, I. V. (2011). "Tensor-Train Decomposition." SIAM Journal on Scientific Computing, 33(5), 2295-2317. DOI: 10.1137/090752286.
  • Oseledets, I. V. and Tyrtyshnikov, E. E. (2010). "TT-cross approximation for multidimensional arrays." Linear Algebra and its Applications, 432(1), 70-88. DOI: 10.1016/j.laa.2009.07.024.
  • Goreinov, S. A., Tyrtyshnikov, E. E., and Zamarashkin, N. L. (1997). "A theory of pseudoskeleton approximations." Linear Algebra and its Applications, 261(1-3), 1-21. DOI: 10.1016/S0024-3795(96)00301-1.
  • Goreinov, S. A. and Tyrtyshnikov, E. E. (2001). "The Maximal-Volume Concept in Approximation by Low-Rank Matrices." Contemporary Mathematics, 280, 47-52. DOI: 10.1090/conm/280/4620.
  • Goreinov, S. A., Zamarashkin, N. L., and Tyrtyshnikov, E. E. (1997). "Pseudo-skeleton Approximations by Matrices of Maximal Volume." Mathematical Notes, 62(4), 515-519. DOI: 10.1007/BF02358985.
  • Savostyanov, D. V. and Oseledets, I. V. (2011). "Fast Adaptive Interpolation of Multi-dimensional Arrays in Tensor Train Format." 7th International Workshop on Multidimensional Systems, 1-8. DOI: 10.1109/nDS.2011.6076873.

Sensitivity Analysis

  • Sobol, I. M. (2001). "Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates." Mathematics and Computers in Simulation, 55(1-3), 271-280. DOI: 10.1016/S0378-4754(00)00270-6.
  • Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., and Tarantola, S. (2010). "Variance Based Sensitivity Analysis of Model Output. Design and Estimator for the Total Sensitivity Index." Computer Physics Communications, 181(2), 259-270. DOI: 10.1016/j.cpc.2009.09.018.

Finance Context

  • Ruiz, I. and Zeron, M. (2022). Machine Learning for Risk Calculations: A Practitioner's View. Wiley Finance. ISBN: 978-1-119-79138-6.
  • Gaß, M., Glau, K., Mahlstedt, M., and Mair, M. (2018). "Chebyshev Interpolation for Parametric Option Pricing." Finance and Stochastics, 22, 701-731. DOI: 10.1007/s00780-018-0361-y.
  • Glau, K., Kressner, D., and Statti, F. (2019). "Low-Rank Tensor Approximation for Chebyshev Interpolation in Parametric Option Pricing." arXiv: 1902.04367.

Node Conventions

  • ChebyshevSharp follows PyChebyshev / NumPy chebpts1: Type I roots, n nodes, no endpoints, DCT-II coefficient convention.
  • MoCaX C and Ruiz--Zeron examples use Chebyshev--Lobatto/extrema nodes: N+1 nodes including endpoints, typically paired with DCT-I / Clenshaw--Curtis conventions.
  • Do not mix value tensors sampled on these grids. Resample or rebuild when moving data between conventions.
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