Numerical Differentiation and Finite Differences

  • Robert M. Corless
  • Nicolas Fillion
Chapter

Abstract

Taking derivatives of numerical functions is one of the most often performed tasks in computation. Finite differences are a standard way to approximate the derivative of a function, and compact finite differences are especially attractive. We study the conditioning of differentiation, including some structured condition numbers for differentiation of polynomials. We look at differentiation matrices for derivatives of polynomials expressed in a Lagrange or Hermite interpolational basis. We look at regularization or smoothing before taking derivatives, and briefly touch on automatic differentiation. ⊲

Keywords

Convolution Sine Rounding Error 

References

  1. Anderssen, R. S., & Hegland, M. (1999). For numerical differentiation, dimensionality can be a blessing! Mathematics of Computation, 68, 1121–1141.CrossRefMATHMathSciNetGoogle Scholar
  2. Anderssen, R. S., & Hegland, M. (2010). Derivative spectroscopy—an enhanced role for numerical differentiation. Journal of Integral Equations and Applications, 22(3), 355–367.CrossRefMATHMathSciNetGoogle Scholar
  3. Bellman, R., Kashef, B., & Casti, J. (1972). Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics, 10(1), 40–52.CrossRefMATHMathSciNetGoogle Scholar
  4. Boole, G. (1880). A treatise on the calculus of finite differences (1951 ed. edited by J. F. Moulton). Macmillan and Company, 3rd Edition, London.Google Scholar
  5. Corliss, G. F., Faure, C., Griewank, A., Hascoët, L., & Naumann, U. (2002). Automatic differentiation of algorithms: from simulation to optimazation. New York: Springer.CrossRefGoogle Scholar
  6. Cullum, J. (1971). Numerical differentiation and regularization. SIAM Journal on Numerical Analysis, 8(2), 254–265.CrossRefMATHMathSciNetGoogle Scholar
  7. Don, W. S., & Solomonoff, A. (1995). Accuracy and speed in computing the Chebyshev collocation derivative. SIAM Journal of Scientific Computing, 16, 1253–1268.CrossRefMATHMathSciNetGoogle Scholar
  8. Elliot, Ralph WV. Runes: An Introduction Praeger (1981)Google Scholar
  9. Forth, S. A. (2006). An efficient overloaded implementation of forward mode automatic differentiation in MATLAB. ACM Transactions on Mathematical Software, 32(2), 195–222.CrossRefMathSciNetGoogle Scholar
  10. Lanczos, C. (1988). Applied analysis. Dover, New York, NY.Google Scholar
  11. Lele, S. K. (1992). Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1), 16–42.CrossRefMATHMathSciNetGoogle Scholar
  12. Lu, S., & Pereverzev, S. V. (2006). Numerical differentiation from a viewpoint of regularization theory. Mathematics of Computation, 75(256), 1853.CrossRefMATHMathSciNetGoogle Scholar
  13. Lyness, J. N., & Moler, C. B. (1967). Numerical differentiation of analytic functions. SIAM Journal on Numerical Analysis, 4(2), 202–210.CrossRefMATHMathSciNetGoogle Scholar
  14. Milne-Thomson, L. M. (1951). The calculus of finite differences (2nd ed., 1st ed. 1933). MacMillan and Company, London.Google Scholar
  15. Olver, S., & Townsend, A. (2013). A fast and well-conditioned spectral method. SIAM Review, 55(3), 462–489.CrossRefMATHMathSciNetGoogle Scholar
  16. Pettigrew, M. F., & Rasmussen, H. (1996). A compact method for second-order boundary value problems on nonuniform grids. Computers & Mathematics with Applications, 31(9), 1–16.CrossRefMATHMathSciNetGoogle Scholar
  17. Rokicki, J., & Floryan, J. (1995). Domain decomposition and the compact fourth-order algorithm for the Navier–Stokes equations. Journal of Computational Physics, 116(1), 79–96.CrossRefMATHMathSciNetGoogle Scholar
  18. Squire, W., & Trapp, G. (1998). Using complex variables to estimate derivatives of real functions. SIAM Review, 40(1), 110–112.CrossRefMATHMathSciNetGoogle Scholar
  19. Trefethen, L. N. (2000). Spectral methods in MATLAB. Philadelphia: SIAM.CrossRefMATHGoogle Scholar
  20. Trefethen, L. N. (2013). Approximation theory and approximation practice. Philadelphia: SIAM.MATHGoogle Scholar
  21. Weideman, J. A., & Reddy, S. C. (2000). A matlab differentiation matrix suite. ACM Transactions on Mathematical Software, 26, 465–519.CrossRefMathSciNetGoogle Scholar
  22. Zhao, J., Davison, M., & Corless, R. M. (2007). Compact finite difference method for American option pricing. Journal of Computational and Applied Mathematics, 206, 306–321.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Robert M. Corless
    • 1
  • Nicolas Fillion
    • 1
  1. 1.Applied MathematicsUniversity of Western OntarioLondonCanada

Personalised recommendations