## Finite difference weights, spectral differentiation, and superconvergence

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- by Burhan Sadiq and Divakar Viswanath PDF
- Math. Comp.
**83**(2014), 2403-2427 Request permission

## Abstract:

Let $z_{1},z_{2},\ldots ,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f\left (z_{k}\right )$, with $w_{k}$ being the weight at $z_{k}$. We derive an algorithm for finding the weights $w_{k}$ which uses fewer arithmetic operations and less memory than the algorithm in current use (Fornberg, *Mathematics of Computation*, vol. 51 (1988), pp. 699-706). The algorithm we derive uses fewer arithmetic operations by a factor of $(5m+5)/4$ in the limit of large $N$. The optimized C++ implementation we describe is a hundred to five hundred times faster than MATLAB. The method of Fornberg is faster by a factor of five in MATLAB, however, and thus remains the most attractive option for MATLAB users.

The algorithm generalizes easily to the calculation of spectral differentiation matrices, or equivalently, finite difference weights at several different points with a fixed grid. Unlike the algorithm in current use for the calculation of spectral differentiation matrices, the algorithm we derive suffers from no numerical instability.

The order of accuracy of the finite difference formula for $f^{(m)}(0)$ with grid points $hz_{k}$, $1\leq k\leq N$, is typically $\mathcal {O}\left (h^{N-m}\right )$. However, the most commonly used finite difference formulas have an order of accuracy that is higher than normal. For instance, the centered difference approximation $\left (f(h)\!-\!2f(0)\!+\!f(-h)\right )$ $/h^{2}$ to $f”(0)$ has an order of accuracy equal to $2$ not $1$. Even unsymmetric finite difference formulas can exhibit such superconvergence or boosted order of accuracy, as shown by the explicit algebraic condition that we derive. If the grid points are real, we prove a basic result stating that the order of accuracy can never be boosted by more than $1$.

## References

- Jean-Paul Berrut and Lloyd N. Trefethen,
*Barycentric Lagrange interpolation*, SIAM Rev.**46**(2004), no. 3, 501–517. MR**2115059**, DOI 10.1137/S0036144502417715 - Daniela Calvetti and Lothar Reichel,
*On the evaluation of polynomial coefficients*, Numer. Algorithms**33**(2003), no. 1-4, 153–161. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001). MR**2005559**, DOI 10.1023/A:1025555803588 - R.M. Corless and S.M. Watt,
*Bernstein bases are optimal, but, sometimes, Lagrange bases are better*, In Proceedings of SYNASC, Timisoara, pages 141–153. MIRTON Press, 2004. - Philip J. Davis,
*Interpolation and approximation*, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR**0380189** - Bengt Fornberg,
*Generation of finite difference formulas on arbitrarily spaced grids*, Math. Comp.**51**(1988), no. 184, 699–706. MR**935077**, DOI 10.1090/S0025-5718-1988-0935077-0 - Bengt Fornberg,
*A practical guide to pseudospectral methods*, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR**1386891**, DOI 10.1017/CBO9780511626357 - Bengt Fornberg,
*Calculation of weights in finite difference formulas*, SIAM Rev.**40**(1998), no. 3, 685–691. MR**1642772**, DOI 10.1137/S0036144596322507 - G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR**944909** - Nicholas J. Higham,
*Accuracy and stability of numerical algorithms*, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR**1927606**, DOI 10.1137/1.9780898718027 - Nathan Jacobson,
*Basic algebra. I*, 2nd ed., W. H. Freeman and Company, New York, 1985. MR**780184** - B. Sadiq and D. Viswanath,
*Barycentric Hermite interpolation*, Arxiv preprint, 2011. - D. Viswanath and L. N. Trefethen,
*Condition numbers of random triangular matrices*, SIAM J. Matrix Anal. Appl.**19**(1998), no. 2, 564–581. MR**1614019**, DOI 10.1137/S0895479896312869 - J. A. C. Weideman and S. C. Reddy,
*A MATLAB differentiation matrix suite*, ACM Trans. Math. Software**26**(2000), no. 4, 465–519. MR**1939962**, DOI 10.1145/365723.365727 - Bruno D. Welfert,
*Generation of pseudospectral differentiation matrices. I*, SIAM J. Numer. Anal.**34**(1997), no. 4, 1640–1657. MR**1461800**, DOI 10.1137/S0036142993295545

## Additional Information

**Burhan Sadiq**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: bsadiq@umich.edu
**Divakar Viswanath**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: divakar@umich.edu
- Received by editor(s): July 20, 2012
- Received by editor(s) in revised form: December 22, 2012, and January 17, 2013
- Published electronically: January 6, 2014
- Additional Notes: The authors were supported by NSF grants DMS-0715510, DMS-1115277, and SCREMS-1026317.
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**83**(2014), 2403-2427 - MSC (2010): Primary 65D05, 65D25
- DOI: https://doi.org/10.1090/S0025-5718-2014-02798-1
- MathSciNet review: 3223337