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.

/pdf/scipy-1-0-fundamental-algorithms-for-scientific-computing-in-111lr790ol.pdf

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.

FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with hand-optimized libraries, and describes the software structure that makes our current FFTW3 version flexible and adaptive. We further discuss a new algorithm for real-data DFTs of prime size, a new way of implementing DFTs by means of machine-specific single-instruction, multiple-data (SIMD) instructions, and how a special-purpose compiler can derive optimized implementations of the discrete cosine and sine transforms automatically from a DFT algorithm.

/pdf/the-design-and-implementation-of-fftw3-jfdn0nf0le.pdf

The Design and Implementation of FFTW3

Preface How to Get the Software Part I. Linear Equations. 1. Basic Concepts and Stationary Iterative Methods 2. Conjugate Gradient Iteration 3. GMRES Iteration Part II. Nonlinear Equations. 4. Basic Concepts and Fixed Point Iteration 5. Newton's Method 6. Inexact Newton Methods 7. Broyden's Method 8. Global Convergence Bibliography Index.

/pdf/iterative-methods-for-linear-and-nonlinear-equations-1ayr7dq087.pdf

Iterative Methods for Linear and Nonlinear Equations

http://ab-initio.mit.edu/~oskooi/papers/Oskooi10.pdf

Meep: A flexible free-software package for electromagnetic simulations by the FDTD method

https://digital.library.unt.edu/ark:/67531/metadc1069483/m2/1/high_res_d/5232139.pdf

A standard test set for numerical approximations to the shallow water equations in spherical geometry

This paper extends the direct method of cyclic reduction to linear systems which result from the discretization of separable elliptic equations with Dirichlet, Neumann, or periodic boundary conditions. For an $m \times n$ net, the operation count is proportional to $mn\log _2 n$ and $mn$ storage locations are required.

A direct Method for the Discrete Solution of Separable Elliptic Equations

Computer models of geophysical processes often require the numerical solution of elliptic partial differential equations. This is particularly true for models that make use of stream functions, velocity potentials, or vorticity equations and for models that compute the pressure of an incompressible fluid. The numerical solution of elliptic equations can be a formidable programming task. Moreover, the equations are often time dependent, requiring repeated solutions and, hence, considerable computing resources. Recent advances in computing methods [1, 2] made it possible to solve a very large class of elliptic equations (the separable ones) rapidly and with minimal storage. And as a result of work on singular problems [6, 8], this class is free of special cases for which solutions cannot be obtained numerically. This paper describes a package of computer programs that make use of current methods for solving elliptic partial differential equations. The package is fully documented in [7]. We fLrst became involved in implementing the Buneman algorithm [2] and its extensions via the capacitance matrix approach [1] for solving Poisson's equation

Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3]

The adaptation of the Cooley-Tukey, the Pease and the Stockham FFT's to vector computers is discussed. Each of these algorithms computes the same result namely, the discrete Fourier transform. They differ only in the way that intermediate computations are stored. Yet it is this difference that makes one or the other more appropriate depending on the application. This difference also influences the computational efficiency on a vector computer and motivates the development of methods to improve efficiency. Each of the FFT's is defined rigorously by a short expository FORTRAN program which provides the basis for discussions about vectorization. Several methods for lengthening vectors are discussed, including the case of multiple and multi-dimensional transforms where M sequences of length N can be transformed as a single sequence of length MN using a 'truncated' FFT. The implementation of an in place FFT on a computer with memory-to-memory architecture is made possible by in place matrix-vector multiplication.

FFT algorithms for vector computers

The purpose of this paper is to describe technical note, NCAR TN/IA-109, which is intended to provide scientists with a package of computer programs which make use of current methods for solving elliptic partial differential equations. Computer models of geophysical processes often require the numerical solution of elliptic partial differential equations. This is particularly true for models which make use of stream functions, velocity potentials, or vorticity equations, or in which the pressure of an incompressible fluid is computed. The numerical solution of elliptic equations can be a formidable programming task. Also, the equations are often time-dependent, requiring repeated solutions and, hence, considerable computing resources. Efficient, reliable, and well-documented computer subroutines for solving such equations are needed. With recent advances in computing methods, it became apparent to the authors that a very large class of elliptic equations (separable) could be solved rapidly and with minimal storage. Of particular importance was the fact that, as a result of work on singular problems, this class was free of special cases for which solutions could not be obtained numerically.

Paul N. Swarztrauber

Papers

A standard test set for numerical approximations to the shallow water equations in spherical geometry

A direct Method for the Discrete Solution of Separable Elliptic Equations

Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3]

FFT algorithms for vector computers

Efficient FORTRAN subprograms for the solution of elliptic partial differential equations.