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A Sparse Symmetric Indefinite Direct Solver for GPU Architectures

In recent years, there has been considerable interest in the potential for graphics processing units (GPUs) to speed up the performance of sparse... (more)

Replicated Computational Results (RCR) Report for A Sparse Symmetric Indefinite Direct Solver for GPU Architectures

A Sparse Symmetric Indefinite Direct Solver for GPU Architectures includes performance results and... (more)

Sampling Exactly from the Normal Distribution

An algorithm for sampling exactly from the normal distribution is given. The algorithm reads some number of uniformly distributed random digits in a given base and generates an initial portion of the representation of a normal deviate in the same base. Thereafter, uniform random digits are copied directly into the representation of the normal... (more)

Testing Matrix Function Algorithms Using Identities

Algorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit... (more)

ShearLab 3D

Wavelets and their associated transforms are highly efficient when approximating and analyzing one-dimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient in sparsely approximating and also in analyzing in the sense of, for instance, detecting... (more)

Parameterized Complexity of Discrete Morse Theory

Optimal Morse matchings reveal essential structures of cell complexes that lead to powerful tools to study discrete geometrical objects, in... (more)

Algorithm 955

New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector... (more)

Algorithm 956

Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined.... (more)

Algorithm 957

Nonstandard Gaussian quadrature is applied to evaluate the repeated integral inerfc x of the coerror function for n ∈ N0, x ∈ R in an appropriate domain of the (n, x)-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.... (more)

Remark on “Algorithm 673

This remark presents a correction to Algorithm 673 (dynamic Huffman coding) [Vitter 1989] and its translation to MATLAB.... (more)

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The ACM TOMS Replicated Computational Results (RCR) Initiative.

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Forthcoming Articles

General Template Units for the Finite Volume Method in Box-shaped Domains

Algorithm XXX: A Practical Iterative Algorithm for the Art Gallery Problem using Integer Linear Programming

In the last decades, the search for exact algorithms for NP-hard geometric problems has intensified. Many of these solutions use Integer Linear Programming (ILP) modeling and rely on state of the art solvers to find optimal solutions for large instances, in a matter of minutes. Here, we discuss an ILP based algorithm that solves to optimality the Art Gallery Problem (AGP), one of the most studied problems in Computational Geometry. The basic idea of our method is to iteratively generate upper and lower bounds through the resolution of discretized versions of the AGP, which are reduced to instances of the Set Cover Problem. Our algorithm was implemented and tested on almost 3000 instances and attained optimal solutions for the vast majority of them, greatly increasing the set of instances for which exact solutions are known. To our knowledge, in spite of the extensive study of the AGP, no other algorithm has shown the ability to solve the AGP as effectively and efficiently as the one described here. Evidence of its robustness is presented through tests done on a number of classes of polygons of various sizes with and without holes. A sofware implementing the algorithm is made available.

Algorithm xxx: VBF: A Library of C++ Classes for Vector Boolean Functions in Cryptography

Remark on algorithm 916: computing the Faddeyeva and Voigt functions: Efficiency Improvements and Fortran Translation

Stability and Performance of Various Singular Value QR Implementations on Multicore CPU with a GPU

To orthonormalize a set of dense vectors, Singular Value QR (SVQR) requires only one global reduction between the parallel processing units, and uses BLAS-3 kernels to perform most of its local computation. As a result, compared to other orthogonalization schemes, SVQR obtains superior performance on many of the current computers. In this paper, we study the stability and performance of various SVQR implementations on multicore CPUs with a GPU, focusing on the dense triangular solve, which performs half of the total floating-point operations in SVQR. As a part of this study, we examine its adaptive mixed-precision variant that decides if a lower-precision arithmetic can be used for the triangular solution at runtime without increasing the order of its orthogonality error. Since the backward error of this adaptive mixed-precision variant is significantly greater than that of the standard SVQR, we study its effects on the solution convergence of several subspace projection methods for solving a linear system of equations and for computing singular values or eigenvalues of a sparse matrix. Our experimental results indicate that in some cases, the convergence rate of the solver may not be affected by the larger backward errors, while reducing the time to solution.

Algorithm xxx -- FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems

An experimental exploration of Marsaglia's xorshift generators, scrambled

The BLIS Framework: Experiments in Portability

ADiJaC  Automatic Differentiation of Java Classfiles

This work presents the current design and implementation of ADiJaC, an automatic differentiation tool for Java classfiles. ADiJaC uses source code transformation to generate derivative codes in both the forward and the reverse modes of automatic differentiation. We describe the overall architecture of the tool and present various details and examples for each of the two modes of differentiation. We emphasize the enhancements that have been made over previous versions of ADiJaC and illustrate their influence on the generality of the tool and on the performance of the generated derivative codes. The ADiJaC tool has been used to generate derivatives for a variety of problems, including real-world applications. We evaluate the performance of such codes and compare it to derivatives generated by Tapenade, a well-established automatic differentiation tool for Fortran and C/C++. Additionally, we present more detailed performance analyses of two real-world applications. Apart from being the only general-purpose automatic differentiation tool for Java bytecode, we argue that ADiJaC's features and performance are comparable to those of similar mature tools for other programming languages such as C/C++ or Fortran.

PUMI: Parallel Unstructured Mesh Infrastructure

What are the Correct Results for the Special Values of the Operands of the Power Operation?

Pipelined Iterative Solvers with Kernel Fusion for Graphics Processing Units

We revisit the implementation of iterative solvers on discrete graphics processing units and demonstrate the benefit of implementations using extensive kernel fusion for pipelined formulations over conventional implementations of classical formulations. The proposed implementations with both CUDA and OpenCL are freely available in ViennaCL and are shown to be competitive with or even superior to other solver packages for graphics processing units. Highest performance gains are obtained for small to medium-sized systems, while our implementations are on par with vendor-tuned implementations for very large systems. Our results are especially beneficial for transient problems, where many small to medium-sized systems instead of a single big system need to be solved.

IEEE754 precision-$k$ base-$\beta$ arithmetic inherited by precision-$m$ base-$\beta$ arithmetic for $k<m$

A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization

We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices appear in many applications, e.g., finite element methods, boundary element methods, etc. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The \emph{compression} algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUMPACK (STRUctured Matrices PACKage) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.

A Source Transformation via Operator Overloading Method for the Automatic Differentiation of Mathematical Functions in MATLAB

Analytical Modeling is Enough for High Performance BLIS

We show how the BLAS-like Library Instantiation Software (BLIS) framework, which provides a more detailed layering of the GotoBLAS (now maintained as OpenBLAS) implementation, allows one to analytically determine optimal tuning parameters for high-end instantiations of the matrix-matrix multiplication. This is of both practical and scientific importance, as it greatly reduces the development effort required for the implementation of the level-3 BLAS while also advancing our understanding of how hierarchically layered memories interact with high performance software. This allows the community to move on from valuable engineering solutions (empirically autotuning) to scientific understanding (analytical insight).

Discrete Wavelet Transforms in the Large Time-Frequency Analysis Toolbox for Matlab/GNU Octave

The discrete wavelet transform module is a recent addition to the Large Time-Frequency Analysis Toolbox (LTFAT). It provides implementations of various generalizations of the well-known Mallat's algorithm (iterated filterbank) such that completely general filterbank trees, dual-tree complex wavelet transforms and wavelet packets can be created. The resulting transforms can be equivalently represented as filterbanks and analyzed as filterbank frames using fast algorithms.

Algorithm xxx: BACOLI: B-spline Adaptive Collocation Software for PDEs with Interpolation-based Spatial Error Control

Algorithm xxx: DISODE45: A Matlab Runge-Kutta solver for Piecewise Smooth IVPs of Filippov type

In this paper an adaptive Runge-Kutta code, based on the DOPRI5(4) pair, for solving Initial Value Problems for differential systems with Piecewise Smooth solutions (PWS) is presented and the algorithms used in the code are described. The code automatically detects and locate accurately the switching points of the PWS, restarting the integration after each discontinuity. Further, in the case of Filippov systems, algorithms to handle properly sliding mode regimes in an automatic way are included. The code requires from the user a description of the IVP and the functions defining the hypersurfaces where the switching points are located, and it gives the discrete approximated solution together with the switching points. Several numerical experiments are presented to illustrate the reliability and efficiency of the code.

Replicated Computational Results (RCR) Report for ``A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization'

In this report we replicate a subset of the performance results in the paper ``A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization''.

Competitive Divide-and-Conquer Algorithm for Unconstrained Large Scale Black-Box Optimization

Algorithm xxx: A Distributed Memory Fast Multipole Method for Volume Potentials

The solution of a constant-coefficient elliptic partial differential equation (PDE) can be computed using an integral transform: a convolution with the fundamental solution of the PDE, also known as a volume potential. We present a Fast Multipole Method (FMM) for computing volume potentials and use them to construct spatially-adaptive solvers for the Poisson, Stokes and Helmholtz problems. Conventional N-body methods apply to discrete particle interactions. With volume potentials, one replaces the sums with volume integrals. In this paper, we discuss the efficient implementation of such an FMM. We use high-order piecewise Chebyshev polynomials and an octree data structure to represent the input and output fields, enable spectrally accurate approximation of the near field, and the kernel independent FMM (KIFMM) for the far field approximation. For distributed memory parallelism, we use space filling curves, locally essential trees, and a hypercube-like communication scheme developed previously in our group. We present new near and far interaction traversals which optimize cache usage and use vectorization, including the AVX instruction set to get over 50% of peak floating point performance. We use task parallelism to employ the Xeon Phi co-processors.

Manycore algorithms for batch scalar and block tridiagonal solvers

Engineering, scientific and financial applications often require the simultaneous solution of a large number of independent tridiagonal systems of equations with varying coefficients. Since the number of systems is large enough to offer considerable parallelism on many-core systems, the choice between different tridiag- onal solution algorithms, such as Thomas, CR (Cyclic Reduction) or PCR (Parallel Cyclic Reduction) needs to be re-examined. This work investigates the optimal choice of tridiagonal algorithm for CPUs and GPUs with a focus on minimizing the amount of data transfer to and from the main memory, and maximizing the achieved bandwidth. It also considers block tridiagonal solutions which are sometimes required in CFD (Computational Fluid Dynamic) applications.

Algorithm xxx: An efficient algorithm to compute the genus of discrete surfaces and applications to turbulent flows

A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects defined in a structured-collocated grid, as those obtained by thresholding a scalar field, and does not require any triangulation of the data. This makes the algorithm fast, memory-efficient and suitable for large datasets. Applications to the characterization of complex surfaces in turbulent flows are presented to illustrate the method.

Implementing multifrontal sparse solvers for multicore architectures with Sequential Task Flow runtime systems

To face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. This paper evaluates the usability and effectiveness of runtime systems based on the Sequential Task Flow model for complex applications, namely, sparse matrix multifrontal factorizations which feature extremely irregular workloads, with tasks of different granularities and characteristics and with a variable memory consumption. Most importantly, it shows how this parallel programming model eases the development of complex features that benefit the performance of sparse, direct solvers as well as their memory consumption. We illustrate our discussion with the multifrontal QR factorization running on top of the StarPU runtime system.

\texttt{PSelInv}--A Distributed Memory Parallel Algorithm for Selected Inversion : the Symmetric Case

A Parallel Geometric Multifrontal Solver Using Hierarchically Semiseparable Structure

Matslise 2.0: a Matlab toolbox for Sturm-Liouville computations

Modular SIMD arithmetic in Mathemagix

Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular integer arithmetic, and present detailed vectorized counterparts. We also present several applications, such as fast modular Fourier transforms and multiplication of integer polynomials and matrices. The vectorized algorithms have been implemented in C++ inside the free computer algebra and analysis system Mathemagix. The performance of our implementation is illustrated by various benchmarks.

A High Performance QDWH-SVD Solver using Hardware Accelerators

This paper describes a new high performance implementation of the QR-based Dynamically Weighted Halley Singular Value Decomposition (QDWH-SVD) solver on multicore architecture enhanced with multiple GPUs. The standard QDWH-SVD algorithm was introduced by Nakatsukasa and Higham (SIAM SISC, 2013) and combines three successive computational stages: (1) the polar decomposition calculation using the QDWH algorithm, (2) the symmetric eigendecomposition of the resulting polar factor to obtain the singular values and the right singular vectors and (3) the matrix-matrix multiplication to get the associated left singular vectors. A comprehensive test suite highlights the numerical robustness of the QDWH-SVD solver. Although it performs up to two times more flops when computing all singular vectors compared to the standard SVD solver algorithm, our new high performance implementation results in up to 4x improvements for asymptotic matrix sizes, compared to the equivalent routines from existing state-of-the-art open-source and commercial libraries. However, when only singular values are needed, QDWH-SVD is penalized by performing up to 14 times more flops but can still run up to 18% faster than the best existing equivalent routines. Integrating mixed precision techniques in the solver can additionally provide up to 40% improvement at the price of losing few digits of accuracy.

Algorithm xxx: A General Software Tool for Constructing Rank-1 Lattice Rules

On BLAS Level-3 Implementations of Common Solvers for (Quasi-) Triangular Generalized Lyapunov Equations

The solutions of Lyapunov and generalized Lyapunov equations are a key player in many applications in systems and control theory. Their stable numerical computation, when the full solution is sought, is considered solved since the seminal work of Bartels and Stewart. A number of variants of their algorithm have been proposed, but none of them goes beyond BLAS level-2 style implementation. On modern computers, however, the formulation of level-3 BLAS type implementations is crucial to enable optimal usage of cache hierarchies and modern block scheduling methods based on directed acyclic graphs describing the interdependence of single block computations. Our contribution closes this gap by a transformation of the aforementioned level-2 variants to level-3 versions and a comparison on a standard multicore machine.

A radix-independent error analysis of the Cornea-Harrison-Tang method

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

This paper considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP is developed for solving this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is computed using the variable fixing method or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the dual function, and computing the solution of the dual problem. Also, a new version of the variable fixing algorithm is developed that is convergent even when the objective Hessian is not strictly positive definite. The hybrid algorithm NAPHEAP that uses a Newton-type method (variable fixing method, secant method, or Newton's method) to bracket a root, followed by a heap-based monotone break point search, can be faster than a Newton-type method by itself, as demonstrated in the numerical experiments.

A Nonlinear QR Algorithm for Banded Nonlinear Eigenvalue Problems

A robust and scalable implementation of the Parks-McClellan algorithm for designing FIR filters

With a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well-known alternative for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where such codes fail to work. In this paper, we introduce a new implementation of this algorithm. It is based on three main ingredients: (1) a new heuristic initialization strategy that generally improves the convergence properties of the Parks-McClellan routine, (2) numerically stable barycentric Lagrange interpolation formulas, and (3) colleague matrix-based rootfinding algorithms. We argue that our approach is very robust in practice, even for hard to design problems. The result, an open source C++ library, is capable of constructing filters where the final degree is more than 50000, outperforming other implementations.

Algorithm xxx: Estimation of Stochastic Covariance Models using a Continuum of Moment Conditions.

Algorithm ???: POLYNOMIAL: An object-oriented Matlab library of fast and efficient algorithms for polynomials

ALGORITHM xxx: MINRES-QLP for Singular Symmetric and Hermitian Linear Equations and Least-Squares Problems

Matrix multiplication over word-size modular fields using approximate formulae

WorkStream -- a design pattern for multicore-enabled finite element computations

KBLAS: An Optimized Library for Dense Matrix-Vector Multiplication on GPU Accelerators

KBLAS is a new open source high performance library that provides optimized kernels for a subset of Level 2 BLAS functionalities on CUDA-enabled GPUs. Since performance of dense matrix-vector multiplication is hindered by the overhead of memory accesses, a double-buffering optimization technique is employed to overlap data motion with computation. After identifying a proper set of tuning parameters, KBLAS is able to efficiently run on various GPU architectures across different generations, avoiding the time-consuming step of code rewriting, while still being compliant with the standard BLAS API. Another advanced optimization technique allows to ensure coalesced memory access when dealing with submatrices, especially in the context of high level dense linear algebra algorithms. All four precisions KBLAS kernels have been leveraged to multi-GPUs environment, which requires the introduction of new APIs to ease users experiences on these challenging systems. The KBLAS performance outperforms existing state-of-the-art implementations on all matrix sizes, achieves asymptotically up to 50% and 60% speedup on single GPU and multi-GPUs systems, respectively, and validates our performance model. A subset of KBLAS high performance kernels has been integrated into NVIDIAs standard BLAS implementation (cuBLAS) for larger dissemination, starting version 6.0.

About TOMS

The purpose of the ACM Transactions on Mathematical Software (TOMS) is to communicate important research results addressing the development, evaluation and use of mathematical software...

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Bibliometrics

Publication Years 1975-2016
Publication Count 1430
Citation Count 12917
Available for Download 1430
Downloads (6 weeks) 5058
Downloads (12 Months) 45250
Downloads (cumulative) 967493
Average downloads per article 677
Average citations per article 9
First Name Last Name Award
David H Bailey ACM Gordon Bell Prize (2008)
Ronald Boisvert ACM Distinguished Member (2006)
Bruno Buchberger ACM Paris Kanellakis Theory and Practice Award (2007)
Edmond Chow ACM Gordon Bell Prize
Special Category (2009) ACM Gordon Bell Prize
Special Category (2009)
James Demmel ACM Paris Kanellakis Theory and Practice Award (2014)
Jack Dongarra ACM-IEEE CS Ken Kennedy Award (2013)
Stuart Feldman ACM Software System Award (2003)
John Gunnels ACM Gordon Bell Prize (2007)
John A. Gunnels ACM Gordon Bell Prize (2006)
Michael A Heroux ACM Distinguished Member (2009)
William Kahan A. M. Turing Award (1989)
Matthew Knepley SIAM/ACM Prize in Computational Science and Engineering (2014)
David J Kuck ACM-IEEE CS Ken Kennedy Award (2010)
ACM-IEEE CS Eckert-Mauchly Award (1993)
Byounghak Lee ACM Gordon Bell Prize (2008)
Lois C McInnes SIAM/ACM Prize in Computational Science and Engineering (2014)
Cleve Moler SIAM/ACM Prize in Computational Science and Engineering (2009)
Dianne P O'Leary ACM Distinguished Member (2006)
Naren Ramakrishnan ACM Distinguished Member (2009)
Sartaj K Sahni ACM Karl V. Karlstrom Outstanding Educator Award (2003)
Robert Schreiber ACM Distinguished Member (2006)
Horst D Simon ACM Gordon Bell Prize
Special Category (2009) ACM Gordon Bell Prize
Special Category (2009)
Barry Smith SIAM/ACM Prize in Computational Science and Engineering (2014)
James E. Smith ACM-IEEE CS Eckert-Mauchly Award (1999)
Kris Stewart ACM Senior Member (2015)
Nathan Tallent ACM-IEEE CS George Michael Memorial HPC Fellowships (2009)
Robert E Tarjan ACM Paris Kanellakis Theory and Practice Award (1999)
A. M. Turing Award (1986)
Jeffrey S Vetter ACM Distinguished Member (2012)
ACM Gordon Bell Prize
Performance (2010)
Robert A. Walker Outstanding Contribution to ACM Award (2007)
ACM Distinguished Member (2006)
Bruce W Weide ACM Senior Member (2009)
Kesheng Wu ACM Distinguished Member (2010)
ACM Senior Member (2007)
Andrew C Yao A. M. Turing Award (2000)
Hong Zhang SIAM/ACM Prize in Computational Science and Engineering (2014)
Benjamin G Zorn ACM Distinguished Member (2011)

First Name Last Name Paper Counts
Iain Duff 28
Robert Renka 22
Fred Krogh 20
Robert Van De Geijn 19
Jennifer Scott 18
John Reid 17
Jack Dongarra 16
Bo Kågström 15
Donald Amos 14
Lawrence Shampine 14
Timothy Davis 14
Wayne Enright 14
Richard Hanson 13
Jorge Moré 12
John Rice 12
Layne Watson 12
Burton Garbow 12
James Demmel 12
Joseph Liu 11
Fred Gustavson 11
William Cody 10
Timothy Hopkins 9
Walter Gautschi 9
Elias Houstis 9
Jeff Cash 9
Linda Kaufman 9
Nicholas Higham 8
Nico Temme 8
Ping Tang 8
Amparo Gil 8
Xiaoyesherry Li 8
Javier Segura 8
John Pryce 8
Patrick Keast 8
Michael Heroux 7
Terje Espelid 7
Ronald Boisvert 7
Field Van Zee 7
Gregorio Quintana-Ortí 7
Ralph Kearfott 7
Enrique Quintana-Ortí 7
Hiroshi Akima 7
John Lewis 7
A Buckley 7
Almerico Murli 7
Thomas Hull 6
Albrecht Preusser 6
James Lyness 6
Roger Grimes 6
Richard Bartels 6
Robert Kirby 6
David Gay 6
Anders Logg 6
Sven Hammarling 6
Robert Ward 6
Ian Gladwell 6
Nicholas Gould 6
Ronald Cools 6
Panos Pardalos 6
Annie Cuyt 6
Gilbert Stewart 6
Geoffrey Hill 5
Robert Schnabel 5
Tamar Schlick 5
Thomas Coleman 5
Patrick Gaffney 5
Jerzy Waśniewski 5
Philippe Toint 5
Graeme Fairweather 5
Tony Chan 5
Kendall Atkinson 5
Van Van Snyder 5
Jarle Berntsen 5
Sivan Toledo 5
Richard Sincovec 5
Pierre L'Ecuyer 5
Alexander Morgan 5
Krzysztof Sikorski 5
William Mitchell 5
Yu Kuznetsov 5
Paolo Bientinesi 5
Mauricio Resende 5
Jeremy Du Croz 5
Paolo Toth 4
Ian Robinson 4
David Dodson 4
Wayne Dyksen 4
Daniel Kressner 4
Alan George 4
Paul Muir 4
Valerio Parisi 4
Francesco Zirilli 4
Jean Muller 4
Richard Lehoucq 4
Christof Vömel 4
Paul Bailey 4
Uwe Naumann 4
Norman Schryer 4
Naren Ramakrishnan 4
Christian Bischof 4
Werner Rheinboldt 4
Calvin Ribbens 4
Jan Verwer 4
John Monahan 4
Zhaojun Bai 4
Daniel Lozier 4
Anil Rao 4
Lothar Reichel 4
Osni Marques 4
Charles Fulton 4
David Kahaner 4
Jorge Nocedal 4
Ahmed Sameh 4
Michael Saunders 4
Filippo Aluffi-Pentini 4
Carl De Boor 4
Joke Blom 4
Michael Patterson 4
Tamara Kolda 4
Beresford Parlett 4
David Smith 4
Richard Brent 4
Phyllis Fox 3
Fayez Alhargan 3
Granville Sewell 3
Kenneth Hillstrom 3
Andrew Conn 3
Steven Benson 3
John Gunnels 3
Margaret Wright 3
Charles Gear 3
Silvano Martello 3
Richard Simard 3
Wolfgang Hörmann 3
Willy Govaerts 3
Shaun Forth 3
Norman Gibbs 3
Alan Genz 3
Jon Bentley 3
Sanjiva Weerawarana 3
Tobin Driscoll 3
David Kincaid 3
Thomas Aird 3
Yifan Hu 3
Ron Dembo 3
Francesco Romani 3
Armido Didonato 3
Alan Jennings 3
Jonathan Hogg 3
R Brankin 3
William Brown 3
Alfred Morris 3
Philip Sharp 3
Joachim Ahrens 3
Paola Favati 3
Juan Meza 3
William Kahan 3
Marzio Sala 3
David Bailey 3
Desmond Higham 3
Maria Sosonkina 3
Bennett Fox 3
Aaron Fogelson 3
Webb Miller 3
David Shanno 3
Martin Berzins 3
Mariarosaria Rizzardi 3
Yozo Hida 3
Kenneth Neves 3
Anton Zettl 3
Richard Fateman 3
Stan Cabay 3
Anshul Gupta 3
Ian Barrodale 3
Ulrich Dieter 3
William Symes 3
Alan Laub 3
Giorgio Carpaneto 3
Christoph Lauter 3
Roscoe Bartlett 3
Steven Pruess 3
Salvatore Filippone 3
Julien Langou 3
Arnold Neumaier 3
Grazia Lotti 3
Bruce Schmeiser 3
Andrew Sommese 3
Chao Yang 3
Lars Karlsson 3
Danny Sorensen 3
John Rice 3
Ron Brown 3
Robert Granat 3
Kristján Jónasson 3
Bruno Lang 3
Niel Madsen 3
Toyat Cheung 3
Stavros Zenios 3
Giulio Giunta 3
Masao Kodama 3
Kanghoh Phua 3
J Diaz 3
William Hager 3
Garth Wells 3
Eric Grosse 3
Michael Berry 3
David Stoutemyer 3
Wolfgang Bangerth 3
Luisa D'Amore 3
Frank Stenger 3
Heidi Thornquist 2
Vivek Sarin 2
Ashok Srinivasan 2
Ove Skovgaard 2
Guohua Jin 2
Tzemeng Low 2
Will Wright 2
Jorn Springer 2
Bryan Ford 2
George Marsaglia 2
Bruce Fabijonas 2
Arno Rasch 2
Scott Sarra 2
Colin Cryer 2
John Burkardt 2
Francisco Sayas 2
Benjamin Burton 2
Joaquim Martins 2
George Miminis 2
Bart Waanders 2
Kazushige Goto 2
Thomas Foley 2
Michele Goano 2
I Hill 2
J Weideman 2
José Morales 2
H Knoble 2
Michel Cosnard 2
James Ball 2
Nelson Beebe 2
Duncan, Lawrie 2
Voratas Kachitvichyanukul 2
George Davis 2
Christopher Darby 2
Ilyssa Sanders 2
Geoffrey Huntington 2
Jean L'excellent 2
Alex Pothen 2
Alan Hindmarsh 2
Wlodzimierz Proskurowski 2
D Le 2
Paolo D'Alberto 2
Karin Bennett 2
Paolo Costantini 2
Kjell Gustafsson 2
Yihua Bai 2
Robert Piessens 2
Michael Engquist 2
Leon Lasdon 2
Richard O'Neill 2
John Nazareth 2
John Edwards 2
Hayato Waki 2
Richard Wang 2
Walter Anderson 2
Herman Watts 2
Nabih Abdelmalek 2
William Poole 2
Markus Neher 2
Michael Vrahatis 2
Roy Wampler 2
John Butcher 2
Martin Alnaes 2
Paul Zimmermann 2
David Eyre 2
Timo Betcke 2
William Eddy 2
T Lam 2
Jose Román 2
Randall Shirts 2
Jan Vlček 2
Michael Liepelt 2
Brian Gunter 2
Geza Schrauf 2
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2000
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1992
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1991
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1990
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1989
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1988
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1987
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1986
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1985
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1984
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1983
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1982
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1981
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1980
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1979
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1978
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1977
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1976
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1975
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