@techreport{NA-08/20,
  abstract = "Variants of the Remez algorithm for best polynomial approximation are presented based on two key features: the use of the barycentric interpolation formula to represent the trial polynomials, and the setting of the whole computation in the chebfun system, where the determination of local and global extrema at each iterative step becomes trivial. The new algorithms make it a routine matter to compute approximations of degrees in the hundreds, and as an example, we report approximation of |x| up to degree 10,000. Since barycentric formulas can also represent rational functions, the algorithms we introduce may also point the way to new methods for computing best rational approximations.",
  author = "Ricardo Pach\'{o}n and Nick Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "December",
  number = "NA-08/20",
  title = "Barycentric-Remez algorithms for best polynomial approximation in the chebfun system",
  year = "2008",
}


    
      @techreport{NA-08/13,
  abstract = "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 resambles 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.",
  author = "Rodrigo Platte and Nick Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "October",
  number = "NA-08/13",
  title = "Chebfun: A New Kind of Numerical Computing",
  year = "2008",
}


    
      @misc{NA-LNT-all,
  author = "L. N. Trefethen",
  title = "All LNT publications, list from 1980",
  url = "http://web.comlab.ox.ac.uk/people/Nick.Trefethen/publication/publication.html",
  year = "2008",
}


    
      @techreport{NA-08/07,
  abstract = "Algorithms are described that make it possible to manipulate piecewise-smooth functions on real intervals numerically with close to machine precision. Breakpoints are introduced in some such calculations at points determined by numerical rootfinding, and in others by recursive subdivision or automatic edge detection. Functions are represented on each smooth subinterval by Chebyshev series or interpolants. The algorithms are implemented in object-oriented MATLAB in an extension of the chebfun system, which was previously limited to smooth functions on [-1, 1].",
  author = "Ricardo Pach\'{o}n and Rodrigo Platte and Nick Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "May",
  number = "NA-08/07",
  title = "Piecewise smooth chebfuns",
  year = "2008",
}


    
      @techreport{NA-08/11,
  abstract = "In MATLAB, it would be good to be able to solve a linear differential equation by typing&nbsp;<strong>u = L\f</strong>, where&nbsp;<strong>f</strong>,&nbsp;<strong>u</strong>, and&nbsp;<strong>L</strong>&nbsp;are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with&nbsp;<strong>expm(L)</strong>&nbsp;or determine eigenvalues and eigenfunctions with&nbsp;<strong>eigs(L)</strong>. A system is described in which such calculations are indeed possible, based on the previously developed chebfun system in object-oriented MATLAB. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.",
  author = "Tobin A. Driscoll and Folkmar Bornemann and Nick Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-08/11",
  title = "The chebop system for automatic solution of differential equations",
  year = "2008",
}


    
      @techreport{NA-08/12,
  abstract = "A standard algorithm for computing the QR factorization of a matrix&nbsp;<em>A</em>&nbsp;is Householder triangularization. Here this idea is generalized to the situation in which&nbsp;<em>A</em>&nbsp;is a quasimatrix, that is, a &ldquo;matrix&rdquo; whose &ldquo;columns&rdquo; are functions defined on an interval [<em>a,b</em>]. Applications are mentioned to quasimatrix leastsquares fitting, singular value decomposition, and determination of ranks, norms, and condition numbers, and numerical illustrations are presented using the chebfun system.",
  author = "Nick Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "July",
  number = "NA-08/12",
  title = "Householder triangularization of a quasimatrix",
  year = "2008",
}


    
      @techreport{NA-07/15,
  abstract = "Gauss and Clenshaw-Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may &quot;waste&quot; a factor of &pi;/2 with respect to each space dimension. We propose new non-polynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight-sided domain. The new methods are compared with related ideas of Bakhvalov, Kosloff and Tal-Ezer, Rokhlin and Alpert, and others. An advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands. For example, one of the formulas presented is proved to converge 50% faster than Gauss quadrature for functions analytic in an &epsilon;-neighborhood of [-1,1].",
  author = "Nicholas Hale and Lloyd N Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-07/15",
  title = "New quadrature formulas from conformal maps",
  year = "2007",
}


    
      @techreport{NA-07/17,
  abstract = "New methods are proposed for the numerical evaluation of&nbsp;<em>f</em>(<strong>A</strong>)$ or&nbsp;<em>f</em>(<strong>A</strong>)&nbsp;<em>b</em>, where&nbsp;<em>f</em>(<strong>A</strong>) is a function such as&nbsp;<strong>A</strong><sup>&frac12;</sup>&nbsp;or log(<strong>A</strong>) with singularities in (-&infin;,0] and&nbsp;<strong>A</strong>&nbsp;is a matrix with eigenvalues on or near (0,&infin;). The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of&nbsp;<em>f</em>(<strong>A</strong>)<em>b</em>&nbsp;is typically reduced to one or two dozen linear system solves, which can be carried out in parallel.",
  author = "Nicholas Hale and Nicholas J Higham and Lloyd N Trefethen",
  institution = "Oxford University Computing Laboratory",
  month = "August",
  number = "NA-07/17",
  title = "Computing A^{\alpha}, log(A) and related matrix functions by contour integrals",
  year = "2007",
}