It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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The Exponentially Convergent Trapezoidal Rule

As both CPUs and GPUs become employed in a wide range of applications, it has been acknowledged that both of these Processing Units (PUs) have their unique features and strengths and hence, CPU-GPU collaboration is inevitable to achieve high-performance computing. This has motivated a significant amount of research on heterogeneous computing techniques, along with the design of CPU-GPU fused chips and petascale heterogeneous supercomputers. In this article, we survey Heterogeneous Computing Techniques (HCTs) such as workload partitioning that enable utilizing both CPUs and GPUs to improve performance and/or energy efficiency. We review heterogeneous computing approaches at runtime, algorithm, programming, compiler, and application levels. Further, we review both discrete and fused CPU-GPU systems and discuss benchmark suites designed for evaluating Heterogeneous Computing Systems (HCSs). We believe that this article will provide insights into the workings and scope of applications of HCTs to researchers and motivate them to further harness the computational powers of CPUs and GPUs to achieve the goal of exascale performance.

https://dl.acm.org/doi/pdf/10.1145/2788396

A Survey of CPU-GPU Heterogeneous Computing Techniques

We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.

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NLEVP: A Collection of Nonlinear Eigenvalue Problems

An integral method for solving nonlinear eigenvalue problems

A contour integral method is proposed to solve nonlinear eigenvalue problems numerically. The target equation is F (λ)x = 0, where the matrix F (λ) is an analytic matrix function of λ. The method can extract only the eigenvalues λ in a domain defined by the integral path, by reducing the original problem to a linear eigenvalue problem that has identical eigenvalues in the domain. Theoretical aspects of the method are discussed, and we illustrate how to apply of the method with some numerical examples.

A numerical method for nonlinear eigenvalue problems using contour integrals

We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. When the contour integral is approximated numerically, eigenvalues on the periphery of the path are also obtained. Error analysis shows that, even though condition numbers of those exterior eigenvalues can be huge, the interior eigenvalues are calculated less erroneously. Four numerical examples are presented, which confirm the theoretical predictions.

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A numerical method for polynomial eigenvalue problems using contour integral

An Evaluation of Singular Value Computation by the Discrete Lotka--Volterra System

Some numerical algorithms are known to be related to discrete-time integrable systems, where it is essential that quantities to be computed (for example, eigenvalues and singular values of a matrix, poles of a continued fraction) are conserved quantities. In this paper, a new application of conserved quantities of integrable systems to numerical algorithms is presented. For an N × N (N ≥ 2) real upper bidiagonal matrix B where all the diagonals and the upper subdiagonals are positive, conserved quantities Tr(((BTB)M)−1) (M = 1, 2, ...) of the discrete finite Toda equation give a sequence of lower bounds of the minimal singular value of B. Recurrence relations for computing higher order conserved quantities Tr(((BTB)M)−1) are also derived.

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Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices

Solution of large-scale dense nonsymmetric eigenvalue problem is required in many areas of scientific and engineering computing, such as vibration analysis of automobiles and analysis of electronic diffraction patterns. In this study, we focus on the Hessenberg reduction step and consider accelerating it in a hybrid CPU-GPU computing environment. Considering that the Hessenberg reduction algorithm consists almost entirely of BLAS (Basic Linear Algebra Subprograms) operations, we propose three approaches for distributing the BLAS operations between CPU and GPU. Among them, the third approach, which assigns small-size BLAS operations to CPU and distributes large-size BLAS operations between CPU and GPU in some optimal manner, was found to be consistently faster than the other two approaches. On a machine with an Intel Core i7 processor and an NVIDIA Tesla C1060 GPU, this approach achieved 3.2 times speedup over the CPU-only case when computing the Hessenberg form of a 8,192 × 8,192 real matrix.

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Kinji Kimura

Papers

A numerical method for nonlinear eigenvalue problems using contour integrals

A numerical method for polynomial eigenvalue problems using contour integral

An Evaluation of Singular Value Computation by the Discrete Lotka--Volterra System

Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices

Acceleration of Hessenberg Reduction for Nonsymmetric Eigenvalue Problems in a Hybrid CPU-GPU Computing Environment