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.

A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that

Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates

We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gived credence to our results.

Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates

A new procedure for gridding elevation and stream line data with automatic removal of spurious pits

In a 1980 paper [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] the authors developed a univariate piecewise cubic interpolation algorithm which produces a monotone interpolant to monotone data. This paper is an extension of those results to monotone $\mathcal{C}^1 $ piecewise bicubic interpolation to data on a rectangular mesh. Such an interpolant is determined by the first partial derivatives and first mixed partial (twist) at the mesh points. Necessary and sufficient conditions on these derivatives are derived such that the resulting bicubic polynomial is monotone on a single rectangular element. These conditions are then simplified to a set of sufficient conditions for monotonicity. The latter are translated to a system of linear inequalities, which form the basis for a monotone piecewise bicubic interpolation algorithm.

/pdf/monotone-piecewise-cubic-interpolation-asgei7e152.pdf

R. E. Carlson

Papers

Monotone Piecewise Cubic Interpolation