Summary
The functionalities of the chebfun and chebop systems are surveyed. The chebfun system is a collection of Matlab codes to manipulate functions in a manner that resembles symbolic computing. The operations, however, are performed numerically using polynomial representations. Chebops are built with the aid of chebfuns to represent linear operators and allow chebfun solutions of differential equations. In this article we present examples to illustrate the simplicity and effectiveness of the software. Among other problems, we consider edge detection in logistic map functions and the solution of linear and nonlinear differential equations.
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Keywords
- Bifurcation Diagram
- Polynomial Representation
- Edge Detection Algorithm
- Homogeneous Dirichlet Boundary Condition
- Piecewise Smooth Function
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References
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004)
Pachón, R., Platte, R.B., Trefethen, L.N.: Piecewise-smooth chebfuns. IMA J. Numer. Anal. doi:10.1093/imanum/drp008 (2009)
Driscoll, T.A., Bornemann, F., Trefethen, L.N.: The chebop system for automatic solution of differential equations. BIT Numer. Math. 48(4), 701–723 (2008)
Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. Comput. Sci. 1(1), 9–19 (2007)
Boyd, J.P.: Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding. SIAM J. Numer. Anal. 40(5), 1666–1682 (2002)
Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Quart. J. Math. 12, 61–68 (1961)
Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)
Salzer, H.E.: Lagrangian interpolation at the Chebyshev points X n,ν≡cos(νπ∕n), ν=0(1)n; some unnoted advantages. Comput. J. 15, 156–159 (1972)
Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24(4), 547–556 (2004)
Trefethen, L.N., Pachón, R., Platte, R.B., Driscoll, T.A.: Chebfun version 2. http://www.maths.ox.ac.uk/chebfun/ (2008)
Sprott, J.C.: Chaos and Time-Series Analysis. Oxford University Press, New York (2003)
Bresten, C.L., Jung, J.-H.: A study on the numerical convergence of the discrete logistic map. Commun. Nonlinear Sci. 14(7), 3076–3088 (2009)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia, PA (2000)
O’Malley, R. Jr.: Singularly perturbed linear two-point boundary value problems. SIAM Rev. 50(3), 459–482 (2008)
Orszag, S. A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703 (1971)
Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. I. Springer, New York (1999). Reprint of the 1978 original
Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87(4), 567–576 (1982/1983).
Trefethen, L.N.: Householder triangularization of a quasimatrix. IMA J. Numer. Anal. doi:10.1093/imanum/drp018 (2009)
Holmer, J., Roudenko, S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Commun. Math. Phys. 282(2), 435–467 (2008)
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Platte, R.B., Trefethen, L.N. (2010). Chebfun: A New Kind of Numerical Computing. In: Fitt, A., Norbury, J., Ockendon, H., Wilson, E. (eds) Progress in Industrial Mathematics at ECMI 2008. Mathematics in Industry(), vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12110-4_5
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