There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

SciPy is an open source scientific computing library for the Python programming language. SciPy 1.0 was released in late 2017, about 16 years after the original version 0.1 release. SciPy has become a de facto standard for leveraging scientific algorithms in the Python programming language, with more than 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories, and millions of downloads per year. This includes usage of SciPy in almost half of all machine learning projects on GitHub, and usage by high profile projects including LIGO gravitational wave analysis and creation of the first-ever image of a black hole (M87). The library includes functionality spanning clustering, Fourier transforms, integration, interpolation, file I/O, linear algebra, image processing, orthogonal distance regression, minimization algorithms, signal processing, sparse matrix handling, computational geometry, and statistics. In this work, we provide an overview of the capabilities and development practices of the SciPy library and highlight some recent technical developments.

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SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

/pdf/scipy-1-0-fundamental-algorithms-for-scientific-computing-in-45othhotbu.pdf

SciPy 1.0: fundamental algorithms for scientific computing in Python.

We describe the University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications The Collection is widely used by the numerical linear algebra community for the development and performance evaluation of sparse matrix algorithms It allows for robust and repeatable experiments: robust because performance results with artificially generated matrices can be misleading, and repeatable because matrices are curated and made publicly available in many formats Its matrices cover a wide spectrum of domains, include those arising from problems with underlying 2D or 3D geometry (as structural engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, economic and financial modeling, theoretical and quantum chemistry, chemical process simulation, mathematics and statistics, power networks, and other networks and graphs) We provide software for accessing and managing the Collection, from MATLAB™, Mathematica™, Fortran, and C, as well as an online search capability Graph visualization of the matrices is provided, and a new multilevel coarsening scheme is proposed to facilitate this task

The university of Florida sparse matrix collection

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierls's depth-first search with Eisenstat and Liu's symmetric structural reductions to speed up symbolic factorization. 
We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

/pdf/a-supernodal-approach-to-sparse-partial-pivoting-30rkhd31hw.pdf

A Supernodal Approach to Sparse Partial Pivoting

We present the main algorithmic features in the software package SuperLU_DIST, a distributed-memory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software's parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on large-scale distributed machines.

/pdf/superlu-dist-a-scalable-distributed-memory-sparse-direct-34h5mah2x2.pdf

SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems

We give an overview of the algorithms, design philosophy, and implementation techniques in the software SuperLU, for solving sparse unsymmetric linear systems. In particular, we highlight the differences between the sequential SuperLU (including its multithreaded extension) and parallel SuperLU_DIST. These include the numerical pivoting strategy, the ordering strategy for preserving sparsity, the ordering in which the updating tasks are performed, the numerical kernel, and the parallelization strategy. Because of the scalability concern, the parallel code is drastically different from the sequential one. We describe the user interfaces of the libraries, and illustrate how to use the libraries most efficiently depending on some matrix characteristics. Finally, we give some examples of how the solver has been used in large-scale scientific applications, and the performance.

/pdf/an-overview-of-superlu-algorithms-implementation-and-user-2nk2impxly.pdf

An overview of SuperLU: Algorithms, implementation, and user interface

We give an overview of the algorithms, design philosophy, and implementation techniques in the software SuperLU, for solving sparse unsymmetric linear systems. In particular, we highlight the differences between the sequential SuperLU (including its multithreaded extension) and parallel SuperLU_DIST. These include the numerical pivoting strategy, the ordering strategy for preserving sparsity, the ordering in which the updating tasks are performed, the numerical kernel, and the parallelization strategy. Because of the scalability concern, the parallel code is drastically different from the sequential one. We describe the user interfaces ofthe libraries, and illustrate how to use the libraries most efficiently depending on some matrix characteristics. Finally, we give some examples of how the solver has been used in large-scale scientific applications, and the performance.

Semiseparable matrices and many other rank-structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast-structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright q 2009 John Wiley & Sons, Ltd.

https://www.math.purdue.edu/~xiaj/work/fasthss.pdf

Xiaoye S. Li

Papers

A Supernodal Approach to Sparse Partial Pivoting

SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems

An overview of SuperLU: Algorithms, implementation, and user interface

An overview of SuperLU: Algorithms, implementation, and user interface

Fast algorithms for hierarchically semiseparable matrices