NOTE: This list is still missing a large number of important papers

Pseudospectra
=============

Single-input single-pass algorithm for shifted Hessenberg system solves.
Cited as the impetus for BischofDattaPurkayastha-1994
@article{Datta-1989,
  author={Biswa N. Datta},
  title={Parallel and large-scale matrix computations in control: some ideas},
  journal={Linear Algebra and its Applications},
  volume=121,
  pages={243--264},
  year=1989
}

Introduced single-pass shifted Hessenberg solves for A X - X H = C and a
blocked extension. This computational kernel is the basis for Elemental's 
proposed interleaved Lancozs pseudospectra algorithm.
@article{BischofDattaPurkayastha-1994,
  author={Christian H. Bischof and Biswa N. Datta and Avijit Purkayastha},
  title={A parallel algorithm for the Sylvester-Observer Equation},
  journal={SIAM Journal on Scientific Computing}, 
  volume=17,
  number=3,
  pages={686--698},
  year=1994
}

Numerically-robust extensions of BischofDattaPurkayastha-1994's 
multiplication-based shifted Hessenberg solver (discussed in the 
upper-Hessenberg setting)
@techreport{Henry-1994,
  author={Greg Henry},
  title={The shifted Hessenberg system solve computation},
  type={{T}echnical {R}eport},
  institution={Cornell University}, 
  year=1994
}

Introduced the triangularization + inverse iteration approach
@article{Lui-1997,
  author={Shiu-Hong Lui},
  title={Computation of pseudospectra by continuation},
  journal={SIAM Journal on Scientific Computing},
  volume=18,
  number=2,
  pages={567--573},
  year=1997
}

A comprehensive review paper on computing pseudospectra.
@article{Trefethen-1999,
  author={Lloyd N. Trefethen},
  title={Computation of pseudospectra},
  journal={Acta Numerica}, 
  volume=8,
  pages={247--295},
  year=1999
}

Spectral Divide and Conquer
===========================

Elemental contains preliminary implementations of spectral divide and conquer
algorithms derived from the following paper:
@article{BaiEtAl-1997,
  author={Zhaojun Bai and James Demmel and Jack Dongarra and Antoine Petitet 
          and Howard Robinson and Ken Stanley},
  title={The spectral decomposition of nonsymmetric matrices on distributed
         memory parallel computers},
  journal={SIAM Journal on Scientific Computing},
  volume=18,
  number=5,
  pages={1446--1461}, 
  year=1997
}

The randomized approach from the following paper is used in order to avoid a
pivoted QR decomposition:
@article{DemmelDumitriuHoltz-2007,
  author={James Demmel and Ioana Dumitriu and Olga Holtz},
  title={Fast linear algebra is stable},
  journal={Numerische Mathematik},
  volume=108,
  number=1,
  pages={59--91},
  year=2007
}

Subsequent developments and refinement of the randomized approach from 
"Fast linear algebra is stable"
@techreport{BallardDemmelDumitiu-2011,
  author={Grey Ballard and James Demmel and Ioana Dumitiu},
  title={Minimizing communication for eigenproblems and the {S}ingular {V}alue
         {D}ecomposition},
  type={{T}echnical {R}eport},
  institution={University of California at Berkeley}, 
  number={UCB/EECS-2011-14},
  year=2011
}

SDC algorithms based upon QWDH
@article{NakatsukasaHigham-2013,
  author={Yuji Nakatsukasa and Nicholas J. Higham},
  title={Stable and efficient {S}pectral {D}ivide and {C}onquer algorithms for
         the symmetric eigenvalue decomposition and the {SVD}},
  journal={SIAM Journal on Scientific Computing},
  volume=35,
  number=3,
  pages={A1325--A1349},
  year=2013
}

Hessenberg QR algorithm
=======================

Cited by HenryWatkinsDongarra-2002 as the inspiration for Watkins-1994
@techreport{Dubrulle-1992,
  author={A. Dubrulle},
  title={The multishift QR algorithm -- Is it worth the trouble?},
  type={{T}echnical {R}eport},
  institution={IBM Scientific Center, Palo Alto, CA},
  year=1992
}

@article{Watkins-1994,
  author={David S. Watkins},
  title={Shifting strategies for the parallel QR algorithm},
  journal={SIAM Journal on Scientific Computing},
  volume=15,
  pages={953--958},
  year=1994
}

@article{HenryVanDeGeijn-1997,
  author={Greg Henry and Robert van de Geijn},
  title={Parallelizing the QR algorithm for the unsymmetric algebraic 
         eigenvalue problem: Myths and reality},
  journal={SIAM Journal on Scientific Computing},
  volume=17,
  pages={870--883},
  year=1997
}

@article{HenryWatkinsDongarra-2002,
  author={Greg Henry and David S. Watkins and Jack J. Dongarra},
  title={A parallel implementation of the nonsymmetric QR algorithm for 
         distributed memory architectures},
  journal={SIAM Journal on Scientific Computing},
  volume=24,
  number=1,
  pages={284--311},
  year=2002
}

The resulting TOMS publication from LAWN 153 on extending ScaLAPACK's
pdlahqr to complex arithmetic (pzlahqr)
@article{Fahey-2003,
  author={Mark R. Fahey},
  title={A parallel eigenvalue routine for complex Hessenberg matrices},
  journal={ACM Transactions on Mathematical Software},
  volume=29,
  number=3,
  pages={326--336},
  year=2003
}

Introduced a parallel and high-performance "computational window" scheme for 
reordering eigenvalues in Schur form
@article{GranatKagstromKressner-2009,
  author={Robert Granat and Bo Kagstrom and Daniel Kressner},
  title={Parallel eigenvalue reordering in real Schur forms},
  journal={Concurrency and Computation: Practice and Experience},
  volume=21,
  number=9,
  pages={1225--1250},
  year=2009
}

The first major publication for the parallel QR algorithm with 
Aggressive Early Deflation and computational windows
@article{GranatKagstromKressner-2010,
  author={Robert Granat and Bo Kagstrom and Daniel Kressner},
  title={A novel parallel QR algorithm for hybrid distributed memory HPC 
         systems},
  journal={SIAM Journal on Scientific Computing},
  volume=32,
  number=4,
  pages={2345--2378},
  year=2010
}

Pivoted QR
==========

Introduced the Businger-Golub algorithm for column-pivoted QR decompositions.
@article{BusingerGolub-1965,
  author={Peter A. Businger and Gene H. Golub},
  title={Linear least squares solutions by {H}ouseholder transformations},
  journal={Numerische Mathematik},
  volume=7,
  number=3,
  pages={269--276},
  year=1965
}

Introduced GKS matrix, which the greedy RRQR fails on.
@techreport{GolubKlemaStewart-1976,
  author={Gene H. Golub and Virginia Klema and G.W. Stewart},
  title={Rank degeneracy and least squares problems},
  institution={Stanford University}, 
  number={STAN-CS-76-559},
  year=1976
}

Standard reference for (strong) RRQR factorizations, which will hopefully be 
added to Elemental in the near future.
@article{GuEisenstat-1996,
  author={Ming Gu and Stanley Eisenstat},
  title={Efficient algorithms for computing a strong rank-revealing {QR}
         factorization},
  journal={SIAM Journal on Scientific Computing},
  volume=17,
  number=4,
  pages={848--869},
  year=1996
}

Elemental uses the same norm updating strategy as this paper and the 
corresponding LAPACK implementation of dgeqpf.f
@article{DrmacBujanovic-2008,
  author={Zlatko Drmac and Zvonimir Bujanovic},
  title={On the failure of {R}ank-{R}evealing {QR} factorization software --
         a case study},
  journal={ACM Transactions on Mathematical Software},
  volume=35,
  number=2,
  pages={12:1--12:28},
  year=2008
}

Up-/downdating LU factorizations
================================

Elemental's LUMod closely follows the discussion of Algorithm I from the 
following paper, which is attributed to the textbook 
"Numerische lineare Algebra" by A. Kielbasinski and H. Schwetlick.
@article{StangeGriewankBollhofer-2007,
  author={Peter Stange and Andreas Griewank and Matthias Bollhofer},
  title={On the efficient update of rectangular {LU}-factorizations subject
         to low rank modifications},
  journal={Electronic Transactions on Numerical Analysis},
  volume=26,
  pages={161--177},
  year=2007
}

Up-/downdating Cholesky factorizations
======================================

It is demonstrated that downdated Cholesky factorizations can be an
ill-conditioned function of the original Cholesky factor and the update vector.
@article{Stewart-1979,
  author={G.W. Stewart},
  title={The effects of rounding error on an algorithm for downdating a 
         Cholesky factorization},
  journal={IMA Journal of Applied Mathematics},
  volume=23,
  number=2,
  pages={203--213},
  year=1979
}

A general analysis of the stability of triangularizing matrices via 
hyperbolic Householder transformations.
@article{StewartStewart-1998,
  author={Michael Stewart and G.W. Stewart},
  title={On hyperbolic triangularization: stability and pivoting},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume=19,
  number=4,
  pages={847--860},
  year=1998
}

A review of generalized/hyperbolic Householder transforms which also introduces
blocked algorithms for up-/downdating via accumulated generalized Householder
transforms.
@article{VanDeGeijnVanZee-2011,
  author={Robert A. van de Geijn and Field G. van Zee},
  title={High-performance up-and-downdating via Householder-like 
    transformations},
  journal={ACM Transactions on Mathematical Software},
  volume=38,
  number=1,
  pages={4:1--4:17},
  year=2011
}

Singular Value Decomposition
============================

This paper introduced the standard algorithm for computing the SVD.
@article{GolubReinsch-1970,
  author={Gene H. Golub and Christian Reinsch},
  title={Singular value decomposition and least squares solutions},
  journal={Numerische Mathematik},
  volume=14,
  number=5,
  pages={403--420},
  year=1970
}

This paper introduced the idea of using a QR decomposition as a first
step in the SVD of a non-square matrix in order to accelerate the 
computation (well, an earlier Golub paper mentioned it as well).
@article{Chan-1982,
  author={Tony F. Chan},
  title={An improved algorithm for computing the {S}ingular {V}alue
         {D}ecomposition},
  journal={ACM Transactions on Mathematical Software},
  volume=8,
  number=1,
  pages={72--83},
  year=1982
}

This could serve as a foundation for achieving high absolute accuracy
in a cross-product based algorithm for computing the SVD. Such an
approach should be more scalable than the current 
bidiagonalization-based approach.
@article{Jia-2006,
  author={Zhongxiao Jia},
  title={Using cross-product matrices to compute the {SVD}},
  journal={Numerical Algorithms},
  volume=42,
  number=1,
  pages={31--61},
  year=2006
}

Symmetric positive-definite inversion
=====================================

The variant 2 single-sweep algorithm from Fig. 9 was parallelized for
Elemental's HPD inversion.
@article{BientinesiGunterVanDeGeijn-2008,
  author={Paolo Bientinesi and Brian Gunter and Robert A. van de Geijn},
  title={Families of algorithms related to the inversion of a {S}ymmetric
         {P}ositive {D}efinite matrix},
  journal={ACM Transactions on Mathematical Software},
  volume=35,
  number=1,
  pages={3:1--3:22},
  year=2008
}

Interpolative and skeleton decompositions
=========================================

Standard reference for (pseudo-)skeleton approximations, which are also referred
to as CUR decompositions, especially when the center matrix is non-square.
@article{GoreinovTyrtyshnikovZamarashkin-1997,
  author={S.A. Goreinov and E.E. Tyrtyshnikov and N.L. Zamarashkin},
  title={A theory of pseudoskeleton approximations},
  journal={Linear Algebra and Appl},
  volume=261,
  number=1--3,
  pages={1--21}, 
  year=1997
}

Introduced effective randomized approximations of interpolative decompositions
@article{LibertyEtAl-2007,
  author={Edo Liberty and Franco Woolfe and Per-Gunnar Martinsson and 
          Vladimir Rokhlin and Mark Tygert},
  title={Randomized algorithms for the low-rank approximation of matrices},
  journal={Proceedings of the National Academy of Sciences, USA},
  volume=104,
  pages={20167--20172},
  year=2007
}

Contains a thorough analysis of many randomized algorithms for (pseudoskeleton)
decompositions using RRQR factorizations.
@article{ChiuDemanet-2013,
  author={Jiawei Chiu and Laurent Demanet},
  title={Sublinear randomized algorithms for skeleton decompositions},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume=34, 
  number=3,
  pages={1361--1383},
  year=2013
}

Householder tridiagonalization
==============================

Contains the algorithm used for Elemental's square-grid tridiagonalization.
@techreport{Stanley-1997,
  author={Ken Stanley},
  title={Execution time of symmetric eigensolvers},
  type={{Ph.D.} {D}issertation},
  institution={University of California at Berkeley},
  number={CSD-99-1039},
  pages=183,
  year=1997
}

One of the origins for the square-grid tridiagonalization algorithm used in
Elemental (which was later refined by Stanley et al.).
@article{HendricksonJessupSmith-1999,
  author={Bruce Hendrickson and Elizabeth Jessup and Christopher Smith},
  title={Towards an efficient parallel eigensolver for dense symmetric 
         matrices},
  journal={SIAM Journal on Scientific Computing},
  volume=20,
  number=3,
  pages={1132--1154},
  year=1999
}

Two-sided triangular transformations
====================================

Contains the main algorithm used for Elemental's two-sided triangular solves.
@inproceedings{SearsStanleyHenry-1998,
  author={Mark P. Sears and Ken Stanley and Greg Henry},
  title={Application of a high performance parallel eigensolver to electronic
         structure calculation},
  booktitle={Proceedings of the ACM/IEEE Conference on Supercomputing},
  publisher={IEEE Computer Society},
  year=1998
}

Matrix functions
================

Heavily used for Elemental's Sign implementation
TODO: Cite paper(s) instead
@book{Higham-2008,
  author={Nicholas J. Higham},
  title={Functions of {M}atrices: {T}heory and {C}omputation}, 
  publisher={SIAM},
  year=2008
}

Algorithm for the polar decomposition which typically converges in less than
seven iterations
@article{NakatsukasaBaiGygi-2010,
  author={Yuji Nakatsukasa and Zhaojun Bai and Francois Gygi},
  title={Optimizing {H}alley's iteration for computing the matrix polar 
         decomposition},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume=31,
  number=5,
  pages={2700--2720},
  year=2010
}

Fast Haar generation
====================

Useful for randomized rank-revealing factorizations of rank-deficient matrices
@article{Stewart-1980,
  author={G.W. Stewart},
  title={The efficient generation of random orthogonal matrices with an 
         application to condition estimators},
  journal={SIAM Journal on Numerical Analysis},
  volume=17,
  number=3,
  pages={403--409},
  year=1980
}

Convex optimization
===================

The line search reference for the L-BFGS algorithm of ByrdEtAl-1995
@article{MoreThuente-1994,
  author={Jorge J. Mor\'e and David J. Thuente},
  title={Line search algorithms with guaranteed sufficient decrease},
  journal={ACM Transactions on Mathematical Software},
  volume=20,
  number=3,
  pages={286--307},
  year=1994
} 

Introduced the lasso
@article{Tibshirani-1995,
  author={Robert Tibshirani},
  title={Regression shrinkage and selection via the lasso},
  journal={Journal of the Royal Statistical Society, Series B (Methodological)},
  volume=58,
  number=1,
  pages={267--288},
  year=1996
} 

The basis for Elemental's (upcoming) L-BFGS implementation
@article{ByrdEtAl-1995,
  author={Richard H. Byrd and Peihuang Lu and Jorge Nocedal and Ciyou Zhu},
  title={A limited memory algorithm for bound constrained optimization},
  journal={SIAM Journal on Scientific Computing},
  volume=16,
  number=5,
  pages={1190--1208},
  year=1995
}

Section 4 contains the sparse covariance selection experiment used in 
examples/convex/SparseInvCov.cpp
@article{dAspremontBanerjeeElGhaoui-2008,
  author={Alexandre d'Aspremont and Onureena Banerjee and Laurent El Ghaoui},
  title={First-order methods for sparse covariance selection},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume=30,
  number=1,
  pages={56--66},
  year=2008
}

Introduces techniques for L+S decompositions and proves that exact recovery
is often possible
@article{CandesEtAl-2011,
  author={Emmanuel J. Cand\`es and Xiaodong Li and Yi Ma and John Wright},
  title={Robust principal component analysis?},
  journal={Journal of the ACM},
  volume=58,
  number=3,
  pages={11:1--11:37},
  year=2011
}

Contains a wide variety of ADMM solvers which can be mapped to parallel
architectures via parallel factorizations/SVD/etc.
@article{BoydEtAl-2011,
  author={Stephen Boyd and Neal Parikh and Eric Chu and Borja Peleato and
          Jonathan Eckstein},
  title={Distributed optimization and statistical learning via the 
         Alternating Direction Method of Multipliers},
  journal={Foundations and Trends in Machine Learning},
  volume=3,
  number=1,
  pages={1--122},
  year=2011
}

Miscellaneous
=============

Added for definition of Kahan matrix (pg. 260)
@book{GolubVanLoan-1996,
  author={Gene H. Golub and Charles F. van Loan},
  title={Matrix {C}omputations},
  edition={3rd}, 
  publisher={Johns Hopkins University Press},
  address={Baltimore},
  year=1996
}
