refs.bib

% References for Dissertation

@book{LAaction,
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    year={2013},
    publisher={American Mathematical Soc.}
}

@book{gvl4,
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    year={2012},
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}

@Book{HighamASNA,
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    year = "2002",
    pages = "xxx + 680",
    ISBN = "0-89871-521-0"
}

@book{baby_rudin,
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@book{mhaskar2000fundamentals,
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}

@book{ATAP,
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}

@book{davis_interpolation,
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}

@book{rivlin2003introduction,
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}

@book{interpFunctionsBook,
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}

@book{powell1981approximation,
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}

@article{Knuth_Concrete,
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}

@book{knopp_series,
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@article{erdos1943some,
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}

@article{trefethen1991two,
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    volume={65},
    number={3},
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    year={1991},
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}

@article{runge_example,
    title={On the Runge example},
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    volume={94},
    number={4},
    pages={329--341},
    year={1987},
    publisher={Taylor \& Francis}
}

@article{runge1901,
    title={{\"U}ber empirische Funktionen und die Interpolation zwischen
        {\"a}quidis\-tanten Ordinaten},
    author={Runge, Carl},
    journal={Zeitschrift f{\"u}r Mathematik und Physik},
    volume={46},
    number={224-243},
    pages={20},
    year={1901}
}

@article{chan1987rank,
    title={Rank revealing QR factorizations},
    author={Chan, Tony F},
    journal={Linear algebra and its applications},
    volume={88},
    pages={67--82},
    year={1987},
    publisher={Elsevier}
}

@article{chandrasekaran1994rank,
    title={On rank-revealing factorisations},
    author={Chandrasekaran, Shivkumar and Ipsen, Ilse CF},
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    volume={15},
    number={2},
    pages={592--622},
    year={1994},
    publisher={SIAM}
}

@article{msnInterp,
    title = "Minimum Sobolev norm interpolation with trigonometric
        polynomials on the torus",
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    volume = "249",
    number = "Supplement C",
    pages = "96 - 112",
    year = "2013",
    issn = "0021-9991",
    doi = "https://doi.org/10.1016/j.jcp.2013.03.041",
    url = "http://www.sciencedirect.com/science/article/pii/S0021999113002192",
    author = "S. Chandrasekaran and K.R. Jayaraman and H.N. Mhaskar",
    keywords = "Multivariate interpolation, Uniformly bounded interpolatory
        operators, Minimum Sobolev norm interpolation, Localized kernels"
}

@article{msnBirkhoff,
    title={Minimum Sobolev norm interpolation of scattered derivative data},
    author={Chandrasekaran, Shivkumar and Gorman, CH and
        Mhaskar, Hrushikesh Narhar},
    journal={Journal of Computational Physics},
    volume={365},
    pages={149--172},
    year={2018},
    publisher={Elsevier}
}

@article{msnPDE,
    title={A minimum Sobolev norm technique for the numerical discretization
        of PDEs},
    author={Chandrasekaran, Shivkumar and Mhaskar, Hrushikesh Narhar},
    journal={Journal of Computational Physics},
    volume={299},
    pages={649--666},
    year={2015},
    publisher={Elsevier}
}

@article{Varah_1975,
    title = "A lower bound for the smallest singular value of a matrix",
    journal = "Linear Algebra and its Applications",
    volume = "11",
    number = "1",
    pages = "3 - 5",
    year = "1975",
    issn = "0024-3795",
    doi = "https://doi.org/10.1016/0024-3795(75)90112-3",
    url = "http://www.sciencedirect.com/science/article/pii/0024379575901123",
    author = "J.M. Varah"
}

@article{RZApprox,
    title={Fast algorithms for multiple evaluations of the
        Riemann zeta function},
    author={Odlyzko, Andrew M and Sch{\"o}nhage, Arnold},
    journal={Transactions of the American Mathematical Society},
    volume={309},
    number={2},
    pages={797--809},
    year={1988}
}


@article{IEEE_interp_history, 
    author={E. Meijering}, 
    journal={Proceedings of the IEEE}, 
    title={A chronology of interpolation: from ancient astronomy
        to modern signal and image processing}, 
    year={2002}, 
    volume={90}, 
    number={3}, 
    pages={319-342}, 
    keywords={interpolation;reviews;image processing;signal processing;
        astronomy;chronology;interpolation;ancient astronomy;
        signal processing;image processing;interpolation theory;
        Interpolation;Astronomy;Signal processing;Image processing;History;
        Read only memory;Polynomials;Biomedical signal processing;
        Accidents;Art}, 
    doi={10.1109/5.993400}, 
    ISSN={0018-9219}, 
    month={March},
}

@article{platte2011impossibility,
    title={Impossibility of fast stable approximation of analytic functions
        from equispaced samples},
    author={Platte, Rodrigo B and Trefethen, Lloyd N and Kuijlaars, Arno BJ},
    journal={SIAM review},
    volume={53},
    number={2},
    pages={308--318},
    year={2011},
    publisher={SIAM}
}

@article{vertesi1990optimal,
    title={Optimal Lebesgue constant for Lagrange interpolation},
    author={V{\'e}rtesi, P},
    journal={SIAM Journal on Numerical Analysis},
    volume={27},
    number={5},
    pages={1322--1331},
    year={1990},
    publisher={SIAM}
}

@inproceedings{smith2006lebesgue,
    title={Lebesgue constants in polynomial interpolation},
    author={Smith, Simon J},
    booktitle={Annales Mathematicae et Informaticae},
    volume={33},
    number={109-123},
    pages={1787--5021},
    year={2006},
    organization={Eszterh{\'a}zy K{\'a}roly College, Institute of Mathematics and Computer Science}
}

@article{Brutman1978,
    author = {Brutman, L.},
    title = {On the Lebesgue Function for Polynomial Interpolation},
    journal = {SIAM Journal on Numerical Analysis},
    volume = {15},
    number = {4},
    pages = {694-704},
    year = {1978},
    doi = {10.1137/0715046},
    URL = {https://doi.org/10.1137/0715046},
    eprint = {https://doi.org/10.1137/0715046}
}

@article{gu1996efficient,
    title={Efficient algorithms for computing a strong rank-revealing
        QR factorization},
    author={Gu, Ming and Eisenstat, Stanley C},
    journal={SIAM Journal on Scientific Computing},
    volume={17},
    number={4},
    pages={848--869},
    year={1996},
    publisher={SIAM}
}

@incollection{gasca2001history,
    title={On the history of multivariate polynomial interpolation},
    author={Gasca, Mariano and Sauer, Thomas},
    booktitle={Numerical Analysis: Historical Developments in the 20th Century},
    pages={135--147},
    year={2001},
    publisher={Elsevier}
}

@article{gasca2000polynomial,
    title={Polynomial interpolation in several variables},
    author={Gasca, Mariano and Sauer, Thomas},
    journal={Advances in Computational Mathematics},
    volume={12},
    number={4},
    pages={377},
    year={2000},
    publisher={Springer}
}

@article{lorentz2000multivariate,
    title={Multivariate Hermite interpolation by algebraic polynomials:
        A survey},
    author={Lorentz, RA},
    journal={Journal of computational and applied mathematics},
    volume={122},
    number={1-2},
    pages={167--201},
    year={2000},
    publisher={Elsevier}
}

@book{rudy,
    title={Multivariate Birkhoff Interpolation},
    author={Lorentz, Rudolph A},
    year={1992},
    publisher={Springer}
}

@article{karlin1972hermite,
    title={On Hermite-Birkhoff interpolation},
    author={Karlin, Samuel and Karon, John M},
    journal={Journal of Approximation Theory},
    volume={6},
    number={1},
    pages={90--115},
    year={1972},
    publisher={Academic Press}
}

@article{lorentz1971birkhoff,
    title={Birkhoff interpolation},
    author={Lorentz, Georg Gunther and Zeller, KL},
    journal={SIAM Journal on Numerical Analysis},
    volume={8},
    number={1},
    pages={43--48},
    year={1971},
    publisher={SIAM}
}

@misc{driscoll2014chebfun,
    title={Chebfun guide},
    author={Driscoll, Tobin A and Hale, Nicholas and Trefethen, Lloyd N},
    year={2014},
    publisher={Pafnuty Publications, Oxford}
}

@article{guide1998mathworks,
    title={The mathworks},
    author={Guide, {MATLAB User's}},
    journal={Inc., Natick, MA},
    volume={5},
    pages={333},
    year={1998}
}

@book{ExpODEs,
    title={Exploring ODEs},
    author={Trefethen, Lloyd N and Birkisson, {\'A}sgeir and Driscoll, Tobin A},
    volume={157},
    year={2017},
    publisher={SIAM}
}

@article{driscoll2015rectangular,
    title={Rectangular spectral collocation},
    author={Driscoll, Tobin A and Hale, Nicholas},
    journal={IMA Journal of Numerical Analysis},
    volume={36},
    number={1},
    pages={108--132},
    year={2015},
    publisher={Oxford University Press}
}

@article{aurentz2017block,
    title={Block operators and spectral discretizations},
    author={Aurentz, Jared L and Trefethen, Lloyd N},
    journal={SIAM Review},
    volume={59},
    number={2},
    pages={423--446},
    year={2017},
    publisher={SIAM}
}

@article{xu2015explicit,
    title={Explicit construction of rectangular differentiation matrices},
    author={Xu, Kuan and Hale, Nicholas},
    journal={IMA Journal of Numerical Analysis},
    volume={36},
    number={2},
    pages={618--632},
    year={2015},
    publisher={Oxford University Press}
}

@book{boyd2001chebyshev,
    title={Chebyshev and Fourier spectral methods},
    author={Boyd, John P},
    year={2001},
    publisher={Courier Corporation}
}

@book{zygmund,
    title={Trigonometric series},
    author={Zygmund, Antoni},
    volume={1},
    year={2002},
    publisher={Cambridge university press}
}

@book{petzoldODEs,
    title={Computer methods for ordinary differential equations and
        differential-algebraic equations},
    author={Ascher, Uri M and Petzold, Linda R},
    volume={61},
    year={1998},
    publisher={Siam}
}

@book{ascherBVPs,
    title={Numerical solution of boundary value problems for
        ordinary differential equations},
    author={Ascher, Uri M and Mattheij, Robert MM and Russell, Robert D},
    volume={13},
    year={1994},
    publisher={Siam}
}

@book{levequeFDMs,
    title={Finite difference methods for ordinary and partial
        differential equations: steady-state and time-dependent problems},
    author={LeVeque, Randall J},
    volume={98},
    year={2007},
    publisher={Siam}
}

@book{brennerFEMs,
    title={The mathematical theory of finite element methods},
    author={Brenner, Susanne and Scott, Ridgway},
    volume={15},
    year={2007},
    publisher={Springer Science \& Business Media}
}



@article{RandomReview2011,
    author={Halko, N. and Martinsson, P. G. and Tropp, J. A.},
    title={Finding structure with randomness: probabilistic algorithms
        for constructing approximate matrix decompositions},
    journal={SIAM Rev.},
    volume={53},
    year={2011},
    number={2},
    pages={217--288},
    issn={0036-1445}
}

@book{prob_book,
    title={{Discrete Probability Models and Methods: Probability on Graphs
        and Trees, Markov Chains and Random Fields, Entropy and Coding}},
    author={Br{\'e}maud, P.},
    isbn={978331943\-4766},
    series={Probability Theory and Stochastic Modelling},
    year={2017},
    publisher={Springer International Publishing}
}

@book{bhatia2013matrix,
    title={Matrix analysis},
    author={Bhatia, Rajendra},
    volume={169},
    year={2013},
    publisher={Springer Science \& Business Media}
}

@Article{Chandrasekaran2005HSS,
    author={Chandrasekaran, S. and Gu, M. and Lyons, W.},
    title={A fast adaptive solver for hierarchically semiseparable
        representations},
    journal={CALCOLO},
    year={2005},
    month={Dec},
    day={01},
    volume={42},
    number={3},
    pages={171--185},
    issn={1126-5434}
}

@article{bjorck1994GS,
    title={{Numerics of Gram-Schmidt orthogonalization}},
    author={\r{A}. Bj\"{o}rck},
    journal={Linear Algebra and its Applications},
    volume={197-198},
    pages={297 - 316},
    year={1994},
    issn={0024-3795}
}

@article{stewart2008,
    title={{Block Gram-Schmidt Orthogonalization}},
    author={G.W. Stewart},
    journal={SIAM Journal on Scientific Computing},
    volume={31},
    number={1},
    pages={761-775},
    year={2008}
}

@article{yu2018efficient,
    title={Efficient Randomized Algorithms for the Fixed-Precision
        Low-Rank Matrix Approximation},
    author={Yu, Wenjian and Gu, Yu and Li, Yaohang},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={39},
    number={3},
    pages={1339--1359},
    year={2018},
    publisher={SIAM}
}

@article{martinsson2016randomized,
    title={A randomized blocked algorithm for efficiently computing
        rank-revealing factorizations of matrices},
    author={Martinsson, Per-Gunnar and Voronin, Sergey},
    journal={SIAM Journal on Scientific Computing},
    volume={38},
    number={5},
    pages={S485--S507},
    year={2016},
    publisher={SIAM}
}

@article{gu2015subspace,
    title={Subspace iteration randomization and singular value problems},
    author={Gu, Ming},
    journal={SIAM Journal on Scientific Computing},
    volume={37},
    number={3},
    pages={A1139--A1173},
    year={2015},
    publisher={SIAM}
}

@article{saibaba2019randomized,
    author = {Saibaba, A.},
    title = {Randomized Subspace Iteration: Analysis of Canonical Angles
        and Unitarily Invariant Norms},
    journal = {SIAM Journal on Matrix Analysis and Applications},
    volume = {40},
    number = {1},
    pages = {23-48},
    year = {2019},
    doi = {10.1137/18M1179432},
    URL = {https://doi.org/10.1137/18M1179432},
    eprint = {https://doi.org/10.1137/18M1179432}
}

@article{liu2016parallel,
    title={Parallel randomized and matrix-free direct solvers for large
        structured dense linear systems},
    author={Liu, Xiao and Xia, Jianlin and De Hoop, Maarten V},
    journal={SIAM Journal on Scientific Computing},
    volume={38},
    number={5},
    pages={S508--S538},
    year={2016},
    publisher={SIAM}
}

@article{xi2014fast,
    title={A fast randomized eigensolver with structured LDL factorization
        update},
    author={Xi, Yuanzhe and Xia, Jianlin and Chan, Raymond},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={35},
    number={3},
    pages={974--996},
    year={2014},
    publisher={SIAM}
}

@article{saibaba2016randomized,
    title={Randomized algorithms for generalized Hermitian eigenvalue problems
        with application to computing Karhunen--Lo{\`e}ve expansion},
    author={Saibaba, Arvind K and Lee, Jonghyun and Kitanidis, Peter K},
    journal={Numerical Linear Algebra with Applications},
    volume={23},
    number={2},
    pages={314--339},
    year={2016},
    publisher={Wiley Online Library}
}

@article{gottlieb1997gibbs,
    title={On the Gibbs phenomenon and its resolution},
    author={Gottlieb, David and Shu, Chi-Wang},
    journal={SIAM review},
    volume={39},
    number={4},
    pages={644--668},
    year={1997},
    publisher={SIAM}
}

@inproceedings{donfack2010adapting,
    title={Adapting communic\-tion-avoiding LU and QR factorizations
        to multicore architectures},
    author={Donfack, Simplice and Grigori, Laura and Gupta, Alok Kumar},
    booktitle={2010 IEEE International Symposium on Parallel \&
        Distributed Processing (IPDPS)},
    pages={1--10},
    year={2010},
    organization={IEEE}
}

@article{khabou2013lu,
    title={LU factorization with panel rank revealing pivoting and
        its communication avoiding version},
    author={Khabou, Amal and Demmel, James W and Grigori, Laura and Gu, Ming},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={34},
    number={3},
    pages={1401--1429},
    year={2013},
    publisher={SIAM}
}

@inproceedings{solomonik2011communication,
    title={Communication-optimal parallel 2.5 D matrix multiplication
        and LU factorization algorithms},
    author={Solomonik, Edgar and Demmel, James},
    booktitle={European Conference on Parallel Processing},
    pages={90--109},
    year={2011},
    organization={Springer}
}

@article{kahan1966numerical,
    title={Numerical linear algebra},
    author={Kahan, William},
    journal={Canadian Mathematical Bulletin},
    volume={9},
    number={5},
    pages={757--801},
    year={1966},
    publisher={Cambridge University Press}
}

% Place holder for Randomized LBL article (Arxiv article)
@article{randomHSSLBL,
    author    = {Christopher Gorman and Gustavo Ch{\'{a}}vez and
        Pieter Ghysels and Th{\'{e}}o Mary and 
        Fran{\c{c}}ois{-}Henry Rouet and Xiaoye Sherry Li},
    title     = {Matrix-free construction of {HSS} representation
        using adaptive randomized
    sampling},
    journal   = {CoRR},
    volume    = {abs/1810.04125},
    year      = {2018},
    url       = {http://arxiv.org/abs/1810.04125},
    archivePrefix = {arXiv},
    eprint    = {1810.04125},
    timestamp = {Fri, 14 Dec 2018 09:15:53 +0100},
    biburl    = {https://dblp.org/rec/bib/journals/corr/abs-1810-04125},
    bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{martinsson2011fast,
    title={A fast randomized algorithm for computing a hierarchically
        semiseparable representation of a matrix},
    author={Martinsson, Per-Gunnar},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={32},
    number={4},
    pages={1251--1274},
    year={2011},
    publisher={SIAM}
}

@article{chandrasekaran2006fast,
    title={A fast ULV decomposition solver for hierarchically semiseparable
        representations},
    author={Chandrasekaran, Shiv and Gu, Ming and Pals, Timothy},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={28},
    number={3},
    pages={603--622},
    year={2006},
    publisher={SIAM}
}

@article{rouet2016distributed,
    title={A distributed-memory package for dense hierarchically semi-separable
        matrix computations using randomization},
    author={Rouet, Fran{\c{c}}ois-Henry and Li, Xiaoye S and Ghysels, Pieter
        and Napov, Artem},
    journal={ACM Transactions on Mathematical Software (TOMS)},
    volume={42},
    number={4},
    pages={27},
    year={2016},
    publisher={ACM}
}

@inproceedings{ghysels2017robust,
    title={A robust parallel preconditioner for indefinite systems using
        hierarchical matrices and randomized sampling},
    author={Ghysels, Pieter and Li, Xiaoye Sherry and Gorman, Christopher
        and Rouet, Fran{\c{c}}ois-Henry},
    booktitle={2017 IEEE International Parallel and Distributed Processing
        Symposium (IPDPS)},
    pages={897--906},
    year={2017},
    organization={IEEE}
}

@article{chandrasekaran2005some,
    title={Some fast algorithms for sequentially semiseparable representations},
    author={Chandrasekaran, Shiv and Dewilde, Patrick and Gu, Ming and
        Pals, T and Sun, Xiaorui and van der Veen, Alle-Jan and White, Daniel},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={27},
    number={2},
    pages={341--364},
    year={2005},
    publisher={SIAM}
}

@article{hackbusch1999sparse,
    title={A Sparse Matrix Arithmetic Based on H-Matrices.
        Part I: Introduction to H-Matrices},
    author={Hackbusch, Wolfgang},
    journal={Computing},
    volume={62},
    number={2},
    pages={89--108},
    year={1999},
    publisher={Springer}
}


@article{demmel2015communication,
    title={Communication avoiding rank revealing QR factorization
        with column pivoting},
    author={Demmel, James W and Grigori, Laura and Gu, Ming and Xiang, Hua},
    journal={SIAM Journal on Matrix Analysis and Applications},
    volume={36},
    number={1},
    pages={55--89},
    year={2015},
    publisher={SIAM}
}

@article{quintana1998blas,
    title={A BLAS-3 version of the QR factorization with column pivoting},
    author={Quintana-Ort{\'i}, Gregorio and Sun, Xiaobai and
        Bischof, Christian H},
    journal={SIAM Journal on Scientific Computing},
    volume={19},
    number={5},
    pages={1486--1494},
    year={1998},
    publisher={SIAM}
}

@book{anderson1999lapack,
    title={LAPACK Users' guide},
    author={Anderson, Edward and Bai, Zhaojun and Bischof, Christian and
        Blackford, Susan and Dongarra, Jack and Du Croz, Jeremy and
        Greenbaum, Anne and Hammarling, Sven and McKenney, Alan and
        Sorensen, D},
    volume={9},
    year={1999},
    publisher={Siam}
}

@article{bindel2002computing,
    title={On computing Givens rotations reliably and efficiently},
    author={Bindel, David and Demmel, James and Kahan, William
        and Marques, Osni},
    journal={ACM Transactions on Mathematical Software (TOMS)},
    volume={28},
    number={2},
    pages={206--238},
    year={2002},
    publisher={ACM}
}

@article{julialang,
    title={Julia: A fresh approach to numerical computing},
    author={Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and
        Shah, Viral B},
    journal={SIAM review},
    volume={59},
    number={1},
    pages={65--98},
    year={2017},
    publisher={SIAM}
}

@article{vanLoan2000ubiquitous,
    title={The ubiquitous Kronecker product},
    author={Van Loan, Charles F},
    journal={Journal of computational and applied mathematics},
    volume={123},
    number={1-2},
    pages={85--100},
    year={2000},
    publisher={Elsevier}
}

@article{shen2016fast,
    title={Fast structured direct spectral methods for differential equations
        with variable coefficients, I. The one-dimensional case},
    author={Shen, Jie and Wang, Yingwei and Xia, Jianlin},
    journal={SIAM Journal on Scientific Computing},
    volume={38},
    number={1},
    pages={A28--A54},
    year={2016},
    publisher={SIAM}
}

@book{abramowitz1965handbook,
    title={Handbook of mathematical functions: with formulas, graphs,
        and mathematical tables},
    author={Abramowitz, Milton and Stegun, Irene A},
    volume={55},
    year={1965},
    publisher={Courier Corporation}
}

@article{vogel2016superfast,
    title={Superfast divide-and-conquer method and perturbation analysis
        for structured eigenvalue solutions},
    author={Vogel, James and Xia, Jianlin and Cauley, Stephen and
        Balakrishnan, Venkataramanan},
    journal={SIAM Journal on Scientific Computing},
    volume={38},
    number={3},
    pages={A1358--A1382},
    year={2016},
    publisher={SIAM}
}