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.
  • Trefethen, L. N. (2009; revised 2019). "Rootfinding and Minima and Maxima." Chebfun Guide. Guide page.
  • 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.
  • Bigoni, D., Engsig-Karup, A. P., and Marzouk, Y. M. (2016). "Spectral Tensor-Train Decomposition." SIAM Journal on Scientific Computing, 38(4), A2405-A2439. DOI: 10.1137/15M1036919.
  • 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, 47-51. 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.
  • Goreinov, S. A., Oseledets, I. V., Savostyanov, D. V., Tyrtyshnikov, E. E., and Zamarashkin, N. L. (2010). "How to Find a Good Submatrix." In Matrix Methods: Theory, Algorithms and Applications, World Scientific, 247-256. DOI: 10.1142/9789812836021_0015. (The practical maxvol algorithm; the 1997/2001 Goreinov et al. papers above give the maximal-volume concept.)

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. (Main-effect/first-order indices and the ANOVA decomposition.)
  • Homma, T. and Saltelli, A. (1996). "Importance Measures in Global Sensitivity Analysis of Nonlinear Models." Reliability Engineering & System Safety, 52(1), 1-17. DOI: 10.1016/0951-8320(96)00002-6. (Introduces the total-effect/total-order index.)
  • 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.
  • Zhang, Z., Choi, M., and Karniadakis, G. E. (2011). "Anchor Points Matter in ANOVA Decomposition." In Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering 76, 347-355. DOI: 10.1007/978-3-642-15337-2_32.

Piecewise Smoothness and Numerical Differentiation

  • Chebfun project. "First steps in Chebfun." Chebfun Guide. Guide page.
  • Chebfun project. "Edge detection in Chebfun." Chebfun Examples. Example.
  • Pachón, R., Platte, R. B., and Trefethen, L. N. (2010). "Piecewise-Smooth Chebfuns." IMA Journal of Numerical Analysis, 30(4), 898-916. DOI: 10.1093/imanum/drp008.
  • Gottlieb, D. and Shu, C.-W. (1997). "On the Gibbs Phenomenon and Its Resolution." SIAM Review, 39(4), 644-668. DOI: 10.1137/S0036144596301390.
  • SciPy project. "Finite Difference Differentiation." SciPy API Reference. Documentation.

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.
  • MoCaX Intelligence. "Research & Resources." Public resources on Chebyshev tensors, tensor extension algorithms, and risk-calculation acceleration. Research resources.
  • Ruiz, I. and Zeron, M. (2020). "Dynamic sensitivities and Initial Margin via Chebyshev Tensors." arXiv: 2011.04544.
  • Zeron, M. and Ruiz, I. (2021). "Tensoring Dynamic Sensitivities and Dynamic Initial Margin." Risk Magazine, Cutting Edge. PDF.
  • Zeron-Medina Laris, M. and Ruiz, I. (2021). "Denting the FRTB-IMA Computational Challenge via Orthogonal Chebyshev Sliding Technique." Wilmott Magazine (January 2021). DOI: 10.1002/wilm.10907; arXiv: 1911.10948. (Origin of the "Orthogonal Chebyshev Sliding Technique"; ChebyshevSlider is a related first-order anchored decomposition, not the PCA-coupled construction.)
  • 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.
  • Longstaff, F. A. and Schwartz, E. S. (2001). "Valuing American Options by Simulation: A Simple Least-Squares Approach." The Review of Financial Studies, 14(1), 113-147. DOI: 10.1093/rfs/14.1.113; UCLA PDF.
  • Glau, K., Mahlstedt, M., and Pötz, C. (2019). "A New Approach for American Option Pricing: The Dynamic Chebyshev Method." SIAM Journal on Scientific Computing, 41(1), B153-B180. DOI: 10.1137/18M1193001; arXiv: 1806.05579.
  • Glau, K., Pötz, C., Soloveitchik, M., and Wunderlich, L. (2021). "Efficient Valuation of Callable Bonds: The Dynamic Chebyshev Method." QuantMinds International presentation/article. PDF; summary.
  • Lim, H., Lee, S., and Kim, G. (2014). "Efficient pricing of Bermudan options using recombining quadratures." Journal of Computational and Applied Mathematics, 271, 195-205. DOI: 10.1016/j.cam.2014.04.007.
  • Rabitz, H., and Aliş, Ö. F. (1999). "General foundations of high-dimensional model representations." Journal of Mathematical Chemistry, 25, 197-233. DOI: 10.1023/A:1019188517934.
  • Kan, K. H. (2010). Simulation-based Valuation and Counterparty Exposure Estimation of American Options. PhD thesis, The University of Western Ontario. Repository record.

Reinforcement Learning and Dynamic Programming

  • Bellman, R. (1957). Dynamic Programming. Princeton University Press.
  • Bertsekas, D. P. (2017). Dynamic Programming and Optimal Control, Vol. I, 4th ed. Athena Scientific.
  • Puterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley. ISBN: 978-0-471-61977-2.
  • Sutton, R. S. and Barto, A. G. (2018). Reinforcement Learning: An Introduction, 2nd ed. MIT Press.
  • Judd, K. L. (1998). Numerical Methods in Economics. MIT Press. ISBN: 978-0-262-10071-7.
  • Becker, S., Cheridito, P., and Jentzen, A. (2019). "Deep Optimal Stopping." Journal of Machine Learning Research, 20(74), 1-25. arXiv: 1804.05394.
  • Miranda, M. J. and Fackler, P. L. (2002). Applied Computational Economics and Finance. MIT Press. Publisher page.
  • Lagoudakis, M. G. and Parr, R. (2003). "Least-Squares Policy Iteration." Journal of Machine Learning Research, 4, 1107-1149. JMLR page.
  • Rao, A. (2020). "RL for Optimal Exercise of American Options." Stanford CME 241 lecture note. PDF.
  • Rao, A. (2020). "Pricing American Options with Reinforcement Learning." Stanford RL for Finance notes. PDF.
  • Halperin, I. (2017). "QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds." arXiv: 1712.04609.
  • QuantEcon. ContinuousDPs.jl, a Julia package for continuous-state dynamic programming examples and Bellman-collocation workflows. Repository.
  • Perla, J., Sargent, T. J., and Stachurski, J. "Optimal Growth I: The Stochastic Optimal Growth Model." Quantitative Economics with Julia. Lecture.
  • Perla, J., Sargent, T. J., and Stachurski, J. "Optimal Growth II: Time Iteration." Quantitative Economics with Julia. Lecture.
  • Pascal, J. (2017). "Solving Bellman Equations by the Collocation Method." Blog post.
  • CVXPortfolio project. Portfolio optimization documentation organized around policies, objective terms, constraints, and simulators. Documentation.
  • PyPortfolioOpt project. Portfolio optimization documentation organized around expected returns, risk models, objective functions, and optimizers. Documentation.

Fixed-Income Baseline Libraries

  • QLNet project. "QLNet C# Library." GitHub repository and documentation site. Repository; project site.
  • QLNet quick-start guide. Installation instructions for NuGet and source builds. Quick-start.
  • QLNet VanillaOption source. Vanilla option instrument used by the American-option case study. Source.
  • QLNet finite-difference Black-Scholes vanilla engine source. Numerical engine used as the American-option reference pricer. FdBlackScholesVanillaEngine.cs.
  • QLNet binomial vanilla engine source. Cox-Ross-Rubinstein cross-check engine for American-option references. BinomialEngine.cs.
  • QuantLib documentation examples. American vanilla options with AmericanExercise, FdBlackScholesVanillaEngine, and binomial engines. QuantLib repository.
  • QLNet FixedRateBond source. Public constructors for fixed-rate bonds with schedules, coupon rates, day counters, payment conventions, redemption, and issue dates. Source.
  • QLNet DiscountingBondEngine source. Bond engine that discounts future bond cashflows with a YieldTermStructure. Source.
  • QLNet callable-bond source. QLNet's C# implementation of CallableFixedRateBond and callable-bond arguments. CallableBond.cs.
  • QLNet callable tree engine source. Numerical lattice engine wiring for callable fixed-rate bonds. TreeCallableBondEngine.cs.
  • QLNet discretized callable-bond source. Coupon/call event ordering and call-date snapping used by the tree engine. DiscretizedCallableFixedRateBond.cs.
  • QLNet Hull-White source. One-factor Hull-White model and trinomial-tree fitting parameter implementation. HullWhite.cs.
  • QLNet trinomial lattice source. Recombining tree and lattice rollback mechanics used by short-rate engines. TrinomialTree.cs.
  • QuantLib example. "CallableBonds.cpp." Reference example using CallableFixedRateBond, Hull-White, and a tree callable-bond engine. Example source.
  • RQuantLib documentation. CallableBond description using a Hull-White model and TreeCallableFixedBondEngine. Documentation.
  • QuantLib.js API. TreeCallableFixedRateBondEngine class documentation describing the numerical lattice engine for callable fixed-rate bonds. API docs.
  • QuantLib-Python documentation. Bond pricing engine examples including DiscountingBondEngine(discountCurve). Bond pricing engines.
  • QuantLib Guide. "Vanilla bonds." Examples of fixed-rate bond pricing, yield, duration, and convexity calculations. Guide page.
  • NuGet Gallery. QuantLib package metadata and thread-safety notes for the C# wrapper. QuantLib package.
  • OpenGamma Strata API. FixedCouponBond definition describing fixed periodic coupon payments and final nominal payment. API docs.
  • OpenGamma Strata API. DiscountingFixedCouponBondProductPricer present-value and sensitivity methods for fixed coupon bonds. API docs.
  • OpenGamma Strata API. FixedCouponBondTradeCalculations present value and PV01 entry points for fixed coupon bond trades. API docs.
  • OpenGamma. "Strata Analytics: Curve Calibration & Bucketed PV01 Calculation." Public terminology for bucketed PV01, bucketed delta, rate sensitivities, and key-rate duration. Article.
  • OpenGamma Strata API. pv01CalibratedBucketed and pv01MarketQuoteBucketed definitions distinguishing calibrated-curve-node and market-quote one-basis-point sensitivity. API docs.
  • QuantLib Guide. "Cash-flow analysis." Examples of inspecting bond cashflows, coupon amounts, accrual dates, and payment dates. Guide page.

Public Market Data Sources

  • Board of Governors of the Federal Reserve System. "Yield Curve Models and Data: Nominal Yield Curve." Data page.
  • Gürkaynak, R. S., Sack, B., and Wright, J. H. (2006). "The U.S. Treasury Yield Curve: 1961 to the Present." Federal Reserve FEDS 2006-28. Paper.
  • U.S. Department of the Treasury. "Treasury Auction Results: 2-Year Note, CUSIP 91282CQL8." April 27, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Auction Results: 3-Year Note, CUSIP 91282CQR5." May 11, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Auction Results: 5-Year Note, CUSIP 91282CQK0." April 27, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Auction Results: 7-Year Note, CUSIP 91282CQN4." April 28, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Auction Results: 10-Year Note, CUSIP 91282CQQ7." May 12, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Auction Results: 30-Year Bond, CUSIP 912810UU0." May 13, 2026. Auction result.
  • U.S. Department of the Treasury. "Treasury Daily Interest Rate XML Feed." Feed documentation.
  • Federal Reserve Bank of New York. "SOFR Averages and Index Data." Data page.

Node Conventions

  • ChebyshevSharp uses Type I roots with n nodes and no endpoints. This is the same point set documented by NumPy chebpts1, and the C# implementation converts values to coefficients with a DCT-II convention.
  • MoCaX source-package checks 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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