Books and papers by Lloyd N. Trefethen

Coauthorship graph organized by Michael Overton at the LNT60 conference, Oxford, August 2015.

Books are listed in green.
Additional essays can be found here.
For one-line summaries of each publication, click here.

Not yet appeared

Vandermonde with Arnoldi, with P. D. Brubeck and Y. Nakatsukasa. SIAM Rev., submitted.
An algorithm for real and complex rational minimax approximation, with Y. Nakatsukasa. SIAM J. Sci. Comp., submitted.
Numerical conformal mapping with rational functions. Comp. Meth. Funct. Th., to appear.

Eight perspectives on an exponentially ill-conditioned ODE. SIAM Rev., 2020.
Approximation Theory and Approximation Practice, extended edition. SIAM, 2020.
Quantifying the ill-conditioning of analytic continuation, BIT Numer. Math., 2020.

Solving Laplace problems with corner singularities via rational functions, with A. Gopal. SIAM J. Numer. Anal., 57 (2019), 2074-2094.
New Laplace and Helmholtz solvers, with A. Gopal. Proc. Nat. Acad. Sci. USA, 116 (2019), 10223.
Representation of conformal maps by rational functions, with A. Gopal. Numer. Math., 142 (2019), 359-382.
Smooth random functions, random ODEs, and Gaussian processes. SIAM Rev., 61 (2019), 185-205.

The AAA algorithm for rational approximation, with Y. Nakatsukasa and O. Sète. SIAM J. Sci. Comp., 40 (2018), A1494-A1522.
Series solution of Laplace problems. ANZIAM J., 60 (2018), 1-26.
Rational minimax approximation via adaptive barycentric representations, with S.-I. Filip and Y. Nakatsukasa. SIAM J. Sci. Comp, 40 (2018), A2427-A2455.
Rational approximation of xn. Proc. Amer. Math. Soc., 146 (2018), 5219-5224.

Exploring ODEs, with A. Birkisson and T. A. Driscoll. SIAM, 2017.
Block operators and spectral discretizations, with J. L. Aurentz. SIAM Rev., 59 (2017), 423-446.
Chopping a Chebyshev series, with J. L. Aurentz. ACM Trans. Math. Softw., 43 (2017), 33.
Trigonometric interpolation and quadrature in perturbed points, with A. P. Austin. SIAM J. Numer. Anal., 55 (2017), 2113-2122.
Cubature, approximation, and isotropy in the hypercube. SIAM Rev., 59 (2017), 469-491.
Multivariate polynomial approximation in the hypercube. Proc. AMS, 145 (2017), 4837-4844.
Chebfun in three dimensions, with B. Hashemi. SIAM J. Sci. Comp., 39 (2017), C341-C363.

Euler-Maclaurin and Gregory interpolants, with M. Javed. Numer. Math. 132 (2016), 201-216.
Inverse Yogiisms. Notices AMS, 63 (2016).

Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic, with A. P. Austin. SIAM J. Sci. Comp., 37 (2015), A1365-A1387.
Mathematics of the Faraday Cage, with S. J. Chapman and D. P. Hewett. SIAM Rev., 57 (2015), 398-417.
Extension of Chebfun to periodic functions, with G. B. Wright, M. Javed, and H. Montanelli. SIAM J. Sci. Comp, 37 (2015), C554-C573.
Continuous analogues of matrix factorizations, with A. Townsend. Proc. Roy. Soc. A, 471 (2015), 20140585.
Computing numerically with functions instead of numbers. Comm. ACM, 58 (2015), 91-97.

A trapezoidal rule error bound unifying the Euler-Maclaurin formula and geometric convergence for periodic functions, with M. Javed. Proc. R. Soc. A, 470 (2014), xx-xx.
Chebfun Guide, with T. A. Driscoll and N. Hale. Pafnuty Press, 2014.
The exponentially convergent trapezoidal rule, with J. A. C. Weideman. SIAM Rev., 56 (2014), 385-458.
Numerical algorithms based on analytic function values at roots of unity, with A. P. Austin and P. Kravanja. SIAM J. Numer. Anal., 52 (2014), 1795-1821.

Gaussian elimination as an iterative algorithm, with A. Townsend. SIAM News, 2013.
An extension of Chebfun to two dimensions, with A. Townsend. SIAM J. Sci. Comp., 35 (2013), C495-C518.
Approximation Theory and Approximation Practice, SIAM, 2013.
Robust Padé approximation via SVD, with P. Gonnet and S. Güttel. SIAM Rev., 55 (2013), 101-117.

Chebfun and numerical quadrature, with N. Hale. Sci. China Math., 55 (2012), 1749-1760.
Stability of barycentric interpolation formulas for extrapolation, with M. Webb and P. Gonnet. SIAM J. Sci. Comp., 34 (2012), A3009-A3015.

A sinc function analogue of Chebfun, with M. Richardson. SIAM J. Sci. Comp., 33 (2011), 2519-2535.
Robust rational interpolation and least-squares, with P. Gonnet and R. Pachón. Elect. Trans. Numer. Anal., 38 (2011), 146-167.
Trefethen's Index Cards: Forty Years of Notes about People, Words and Mathematics. World Scientific Publishing, 2011.
Six myths of polynomial interpolation and quadrature. Maths. Today, 47 (2011), 184-188. (Persian translation)
A robust implementation of the Carathéodory-Fejér method for rational approximation, with J. Van Deun. BIT Numer. Math., 51 (2011), 1039-1050.
Ten digit problems. In D. Schleicher and M. Lackmann, eds., Invitation to Mathematics: From Competitions to Research, Springer, 2011.
Impossibility of fast stable approximation of analytic functions from equispaced samples, with R. B. Platte and A. B. J. Kuijlaars. SIAM Rev., 53 (2011), 308-318.

Four bugs on a rectangle, with S. J. Chapman and J. Lottes. Proc. Roy. Soc. A, 467 (2010), 881-896.
Piecewise-smooth chebfuns, with R. Pachón and R. B. Platte. IMA J. Numer. Anal., 30 (2010), 898-916.
Householder triangularization of a quasimatrix. IMA J. Numer. Anal., 30 (2010), 887-897.
Chebfun: A new kind of numerical computing, with R. B. Platte. In Progress in Industrial Mathematics at ECMI 2008, Springer, 2010.

Barycentric-Remez algorithms for best polynomial approximation in the chebfun system, with R. Pachón. BIT Numer. Anal., 49 (2009), 721-741.

The chebop system for automatic solution of differential equations, with T. A. Driscoll and F. Bornemann. BIT Numer. Math., 48 (2008), 701-723.
Numerical analysis. In W. T. Gowers, ed., Princeton Companion to Mathematics, Princeton U. Press, 2008.
Computing Aα, log(A), and related matrix functions by contour integrals, with N. Hale and N. J. Higham. SIAM J. Numer. Anal., 46 (2008), 2505-2523.
G.H. Hardy, applied mathematician. SIAM News, 41, no. 3 (2008), pp. 1 and 3.
New quadrature formulas from conformal maps, with N. Hale. SIAM J. Numer. Anal., 46 (2008), 930-948.
Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev., 50 (2008), 67-87.

Obituary Gene H. Golub (1932-2007). Nature, 450 (2007), 962.
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals, with T. Schmelzer. Elect. Trans. Numer. Anal., 29 (2007), 1-18.
The kink phenomenon in Fejér and Clenshaw-Curtis quadrature, with J. A. C. Weideman. Numer. Math., 107 (2007), 707-727.
Computing numerically with functions instead of numbers. Math. Comput. Sci., 1 (2007), 9-19.
Computing the gamma function using contour integrals and rational approximations, with T. Schmelzer. SIAM J. Numer. Anal., 45 (2007), 558-571.
Parabolic and hyperbolic contours for computing the Bromwich integral, with J. A. C. Weideman. Math. Comp., 76 (2007), 1341-1356.

A rational spectral collocation method with adaptively transformed Chebyshev grid points, with T.-W. Tee. SIAM J. Sci. Comp., 28 (2006), No. 5, 1798-1811.
Talbot quadratures and rational approximations with J. A. C. Weideman and T. Schmelzer. BIT Numer. Math., 46 (2006) 46, 653-670.
Computed eigenmodes of planar regions, with T. Betcke. In N. Chernov, et al., eds., Recent Advances in Differential Equations and Mathematical Physics, Contemp. Math. 412, Amer. Math. Soc., 2006, 297-314.

Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, with M. Embree. Princeton U. Press, 2005.
Ten Digit Algorithms. Numerical Analysis Report 05/13, Oxford University, 2005. (From the 2005 A. R. Mitchell Lecture, Dundee.)
Wave packet pseudomodes of variable coefficient differential operators. Proc. Roy. Soc. Lond. A, 461 (2005), 3099-3122.
Reviving the method of particular solutions, with T. Betcke. SIAM Rev., 47 (2005), 469-491.
Fourth-order time-stepping for stiff PDEs, with A. K. Kassam. SIAM J. Sci. Comp., 26 (2005), 1214-1233.

Computations of eigenvalue avoidance in planar domains, with T. Betcke. Proc. Appl. Math. Mech., 4 (2004), 634-635.
Wave packet pseudomodes of twisted Toeplitz matrices, with S. J. Chapman. Comm. Pure Appl. Math., 57 (2004), 1233-1264.
An extension of Matlab to continuous functions and operators, with Z. Battles. SIAM J. Sci. Comp. 25 (2004), 1743-1770.
Barycentric Lagrange interpolation, with J.-P. Berrut. SIAM Rev., 46 (2004), 501-517.

An American at Oxford. Oxford Magazine, May 2003. (This essay was discussed in the Times Higher Education Supplement and elsewhere.)
Linearized pipe flow to Reynolds number 107, with A. Meseguer. J. Comp. Phys., 186 (2003), 178-197.
A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides, with L. Banjai. SIAM J. Sci. Comp., 25 (2003), 1042-1065.

Schwarz-Christoffel Mapping, with Tobin A. Driscoll. Cambridge U. Press, 2002.
Pseudospectra of rectangular matrices, with T. G. Wright. IMA J. Numer. Anal., 22 (2002), 501-519.
Piecewise continuous Toeplitz matrices and operators: slow approach to infinity, with A. Böttcher and M. Embree. SIAM J. Matrix Anal. Applics., 24 (2002), 484-489.
A hundred-dollar, hundred-digit challenge. SIAM News, 35, No. 1, Jan./Feb. 2002. (This led to two books by Bornemann et al.)
Chastened challenge sponsor: "I misjudged". SIAM News, 35, No. 6, July/Aug. 2002. (follow-up article).

Numerical solution of the omitted area problem of univalent function theory, with L. Banjai. Comp. Meth. Func. Th., 1 (2001), 259-273.
Spectra, pseudospectra and localization for random bidiagonal matrices, with M. Contedini and M. Embree. Comm. Pure Appl. Math., 54 (2001), 595-623.
Large-scale computation of pseudospectra using ARPACK and eigs, with T. G. Wright. SIAM J. Sci. Comput., 23 (2001), 591-605.
Generalizing eigenvalue theorems to pseudospectra theorems, with M. Embree. SIAM J. Sci. Comput., 23 (2001), 583-590.
Eigenvalues and musical instruments, with V. E. Howle. J. Comp. Appl. Math., 135 (2001), 23-40.
Computing Lyapunov constants for random recurrences with smooth coefficients, with T. G. Wright. J. Comp. Appl. Math., 132 (2001), 331-340.

Spectral Methods in MATLAB. SIAM, 2000.
Threshold amplitudes for transition to turbulence in a pipe, with S. J. Chapman, D. S. Henningson, A. Meseguer, T. Mullin, and F. T. M. Nieuwstadt. Num. Anal. Group Report NA 00/17, Oxford U. Computing Lab., 2000.
Predictions for scientific computing fifty years from now. Mathematics Today, April 2000, 53-57. (This essay won the Catherine Richards Prize of the Inst. for Maths. and Applics., 2000.)
How many shuffles to randomize a deck of cards?, with L. M. Trefethen. Proc. Roy. Soc. London A, 456 (2000), 2561-2568. (This article led to dozens of press reports in newspapers, magazines, radio and television.)

Growth and decay of random Fibonacci sequences, with M. Embree. Proc. Roy. Soc. Lond. A, 455 (1999), 2471-2485.
Large-amplitude transient growth in the linear evolution of equatorial spread F with a sheared zonal flow, with J. P. Flaherty and C. E. Seyler. J. Geophys. Res., 104 (1999), 6843.
Spectra and pseudospectra for pipe Poiseuille flow, with A. E. Trefethen and P. J. Schmid. Comp. Meth. Appl. Mech. Engr., 1926 (1999), 413-420.
Computation of pseudospectra. Acta Numer., 8 (1999), 247-295.
The Kreiss matrix theorem on a general complex domain, with K.-C. Toh. SIAM J. Matrix Anal. Applics., 21 (1999), 145-165.
Green's functions for multiply connected domains via conformal mapping, with M. Embree. SIAM Rev., 41 (1999), 721-744.

Schwarz-Christoffel mapping in the computer era, with T. A. Driscoll. Proc. Int. Congr. Math. 1998, 533-542.
From potential theory to matrix iterations in six steps, with T. A. Driscoll and K.-C. Toh. SIAM Rev., 40 (1998), 547-578.
Condition numbers of random triangular matrices, with D. Viswanath. SIAM J. Matrix Anal. Applics., 19 (1998), 564-581.
Maxims about numerical mathematics, computers, science, and life. SIAM News v. 31, no. 1 (1998), p. 4.
The Chebyshev polynomials of a matrix, with K.-C. Toh. SIAM J. Matrix Anal. Applics., 20 (1998), 400-419.
A numerical analyst looks at the 'cutoff phenomenon' in card shuffling and other Markov chains, with G. F. Jonsson. In D. F. Griffiths, D. J. Higham, and G. A. Watson, eds., Numerical Analysis 1997, Addison Wesley Longman, 1998.

Numerical Linear Algebra, with David Bau III. SIAM, 1997 (SIAM's all-time bestseller.)
Pseudospectra of linear operators. In ICIAM '95: Proceedings of the Third International Congress on Industrial and Applied Mathematics, Akademie-Verlag, Berlin, 1996, pp. 401-434; also SIAM Rev. 39 (1997), 383-406.
Low-dimensional models of subcritical transition to turbulence, with J. S. Baggett. Phys. Fluids, 9 (1997), 1043-1053.

Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Graduate textbook, unpublished, 1996.
MultiMATLAB: MATLAB on multiple processors, with A. E. Trefethen, V. S. Menon, C.-C. Chang, G. J. Czajkowski, and C. Myers. Tech. Rep. CTC96TR293, Cornell Theory Center, 1996.
Matrix behaviour, unitary reducibility, and Hadamard products, with D. Viswanath. Tech. Rep. TR96-1596, Dept. of Comp. Sci., Cornell U., July, 1996.
Pseudospectra for the wave operator with an absorbing boundary, with T. A. Driscoll. J. Comp. Appl. Math., 69 (1996), 125-142.
Calculation of pseudospectra by the Arnoldi iteration, with K.-C. Toh. SIAM J. Sci. Comp. 17, (1996), 1-15.

A mostly linear model of transition to turbulence, with J. S. Baggett and T. A. Driscoll. Phys. Fluids, 7 (1995), 833-838.
Do the pseudospectra of a matrix determine its behavior?, with A. Greenbaum. Unpublished technical report, 1995.

Pseudospectra of the convection-diffusion operator, with S. C. Reddy . SIAM J. Appl. Math., 54 (1994), 1634-1649.
Pseudozeros of polynomials and pseudospectra of companion matrices, with K.-C. Toh. Numer. Math., 68 (1994), 403-425.
GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, with A. Greenbaum. SIAM J. Sci. Comput., 15 (1994), 359-368.
From the Buffon needle problem to the Kreiss Matrix Theorem, with E. Wegert. Amer. Math. Monthly, 101 (1994), 132-139.

Numerical construction of conformal maps. Appendix to E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis, 2nd ed., Prentice Hall, 1993.
Hydrodynamic stability without eigenvalues, with A. E. Trefethen, S. C. Reddy, and T. A. Driscoll. Science, 261 (1993), 578-584.
Stiffness of ODEs, with D. J. Higham. BIT Numer. Math., 33 (1993), 285-303.
The definition of numerical analysis. SIAM News 25, 6 Nov. 1992. Reprinted in Bull. Inst. Maths. and Applics., 1993 and again as an appendix to L. N. Trefethen and D. A. Bau, III, Numerical Linear Algebra, SIAM, 1997.

Pseudospectra of matrices. In D. F. Griffiths and G. A. Watson, eds., Numerical Analysis 1991, Longman, 1992.
Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, with L. Reichel. Lin. Alg. Applics., 162-164 (1992), 153-185.
How fast are nonsymmetric matrix iterations?, with N. M. Nachtigal and S. C. Reddy. SIAM J. Matrix Anal. Applics., 13 (1992), 778-795.
A hybrid GMRES algorithm for nonsymmetric linear systems, with N. M. Nachtigal and L. Reichel. SIAM J. Matrix Anal. Applics., 13 (1992), 796-825.
Stability of the method of lines, with S. C. Reddy. Numer. Math., 62 (1992), 235-267.

Two results on polynomial interpolation in equally spaced points, with J. A. C. Weideman. J. Approx. Theory, 65 (1991), 247-260.

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, with S. C. Reddy. Comp. Meth. Appl. Mech. Engr., 80 (1990), 147-164.
The CF table, with E. Hayashi and M. H. Gutknecht. Constr. Approx., 6 (1990), 195-223.
A modified Schwarz-Christoffel transformation for elongated regions with L. H. Howell. SIAM J. Sci. Stat. Comput., 11 (1990), 928-949.
Approximation theory and numerical linear algebra. In J. C. Mason and M. G. Cox, eds., Algorithms for Approximation II, Chapman and Hall, 1990.
Average-case stability of Gaussian elimination, with R. S. Schreiber. SIAM J. Matrix Anal. Applics., 11 (1990), 335-360.

Schwarz-Christoffel mapping in the 1980's. Numerical Analysis Report 89-1, Dept. of Mathematics, MIT, 1989.
SCPACK User's Guide. Numerical Analysis Report 89-2, Dept. of Mathematics, MIT, 1989. (An earlier edition appeared as an ICASE internal report in 1983.)

Lax-stability vs. eigenvalue stability of spectral methods. In K. W. Morton and M. J. Baines, eds., Clarendon Press, 1988.
The eigenvalues of second-order spectral differentiation matrices, with J. A. C. Weideman. SIAM J. Numer. Anal., 25 (1988), 1279-1298.
Ill-posedness of absorbing boundary conditions for migration, with L. H. Howell. Geophysics, 53 (1988), 593-603.
Fourier analysis of the SOR iteration, with R. J. LeVeque. IMA J. Numer. Anal., 8 (1988), 273-279.
Wide-angle one-way wave equations, with L. Halpern. J. Acoust. Soc. Amer., 84 (1988), 1397-1404.

Padé, stable Padé, and Chebyshev-Padé approximation with M. H. Gutknecht. In J. C. Mason and M. G. Cox, eds., Algorithms for Approximation, Clarendon Press, 1987.
An instability phenomenon in spectral methods, with M. R. Trummer. SIAM J. Numer. Anal., 24 (1987), 1008-1023.
Ideal jet flow in two dimensions, with F. Dias and A. R. Elcrat. J. Fluid Mech., 185 (1987), 275-288.

Numerical Conformal Mapping, editor. Elsevier, 1986.
Classical free-streamline flow over a polygonal obstacle, with A. R. Elcrat. J. Comp. Appl. Math., 14 (1986), 251-265.
MATLAB programs for CF approximation. In C. K. Chui, et al., eds., Approximation Theory V, Academic Press, 1986. (This is probably the first research paper ever published containing a MATLAB program.)
Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions, with R. J. Williams. J. Comp. Appl. Math., 14 (1986), 227-249.
Well-posedness of one-way wave equations and absorbing boundary conditions, with L. Halpern. Math. Comp., 47 (1986), 421-435.
Dispersion, dissipation, and stability. In D. F. Griffiths and G. A. Watson, eds., Numerical Analysis, Longman, 1986.

Three mysteries of Gaussian elimination. ACM SIGNUM Newsletter, October 1985.
Stability of hyperbolic finite-difference models with one or two boundaries. In B. E. Engquist, et al., eds., Large-Scale Computations in Fluid Mechanics, v. 2, Amer. Math. Soc., 1985.
Stability of finite-difference models containing two boundaries or interfaces. Math. Comp., 45 (1985), 279-300.
On convergence and degeneracy in rational Padé and Chebyshev approximation, with M. H. Gutknecht. SIAM J. Math. Anal., 16 (1985), 198-210.

On the resolvent condition in the Kreiss Matrix Theorem, with R. J. LeVeque. BIT Numer. Math., 24 (1984), 584-591.
The asymptotic accuracy of rational best approximations to ez on a disk. J. Approx. Th., 40 (1984), 380-383.
Square blocks and equioscillation in the Padé, Walsh, and CF tables. In P. R. Graves-Morris, et al., eds., Rational Approximation and Interpolation, Lect. Notes in Math, v. 1105, Springer, 1984.
Real vs. complex rational Chebyshev approximation on complex domains, with M. H. Gutknecht. In L. Collatz et al., eds., Numerical Methods of Approximation Theory, v. 7, Birkhäuser, 1984.
Analysis and design of polygonal resistors by conformal mapping, Z. Angew. Math. Phys., 35 (1984), 692-704.
Instability of finite difference models for hyperbolic initial boundary value problems. Comm. Pure Appl. Math., 37 (1984), 329-367. (This article won the first Fox Prize in Numerical Analysis.)
On Lp-instability and dispersion at discontinuities in finite difference schemes. In R. Vichnevetsky and R. Stepleman, eds., Proc. Fifth IMACS Int. Symp. on Computer Methods for Partial Diffl. Eqs., IMACS, 1984.
Fourier Analysis of Numerical Approximations of Hyperbolic Equations, by R. Vichnevetsky and J. B. Bowles, SIAM Rev., 26 (1984), 439-441 (book review).

Circularity of the error curve and sharpness of the CF method in complex Chebyshev approximation. SIAM J. Numer. Anal., 20 (1983), 1258-1263.
The Carathéodory-Fejér (CF) method for recursive digital filter design, with M. H. Gutknecht and J. O. Smith. IEEE Trans. Acoust., Speech, Sign. Proc. ASSP-31 (1983), 1417-1426.
The Carathéodory-Fejér method for real rational approximation, with M. H. Gutknecht. SIAM J. Numer. Anal. 20 (1983), 420-436.
Real vs. complex rational Chebyshev approximation on an interval, with M. H. Gutknecht. Trans. Amer. Math. Soc., (1983), 555-561.
Nonuniqueness of best rational Chebyshev approximations on the unit disk, with M. H. Gutknecht. J. Approx. Th., 39 (1983), 275-288.
Real and complex Chebyshev approximation on the unit disk and interval, with M. H. Gutknecht. Bull. Amer. Math. Soc., 8 (1983), 455-458.
Chebyshev approximation on the unit disk. In Computational Aspects of Complex Analysis, H. Werner, et al., eds., D. Reidel, 1983.
Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström. J. Comp. Phys., 49 (1983), 199-217.

Group velocity in finite difference schemes. SIAM Rev., 24 (1982), 113-136.
Real polynomial Chebyshev approximation by the Carathéodory-Fejér method, with M. H. Gutknecht. SIAM J. Numer. Anal., 19 (1982), 358-371.

Rational Chebyshev approximation on the unit disk. Numer. Math., 37 (1981), 297-320.
Near-circularity of the error curve in complex Chebyshev approximation. J. Approx. Theory, 31 (1981), 344-367.

Computation and application of Schwarz-Christoffel transformations. In Proceedings of the 1980 Army Numerical Analysis and Computers Conference, Army Research Office.
Numerical computation of the Schwarz-Christoffel transformation and erratum. SIAM J. Sci. Stat. Comput., 1 (1980), 82-102.

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