Using Cadabra for Tensor Computations in General Relativity
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The Sagemanifolds Project
Tensor calculus with free softwares: the SageManifolds project Eric´ Gourgoulhon1, Micha l Bejger2 1Laboratoire Univers et Th´eories (LUTH) CNRS / Observatoire de Paris / Universit´eParis Diderot 92190 Meudon, France http://luth.obspm.fr/~luthier/gourgoulhon/ 2Centrum Astronomiczne im. M. Kopernika (CAMK) Warsaw, Poland http://users.camk.edu.pl/bejger/ Encuentros Relativistas Espa~noles2014 Valencia 1-5 September 2014 Eric´ Gourgoulhon, Micha l Bejger (LUTH, CAMK) SageManifolds ERE2014, Valencia, 2 Sept. 2014 1 / 44 Outline 1 Differential geometry and tensor calculus on a computer 2 Sage: a free mathematics software 3 The SageManifolds project 4 SageManifolds at work: the Mars-Simon tensor example 5 Conclusion and perspectives Eric´ Gourgoulhon, Micha l Bejger (LUTH, CAMK) SageManifolds ERE2014, Valencia, 2 Sept. 2014 2 / 44 Differential geometry and tensor calculus on a computer Outline 1 Differential geometry and tensor calculus on a computer 2 Sage: a free mathematics software 3 The SageManifolds project 4 SageManifolds at work: the Mars-Simon tensor example 5 Conclusion and perspectives Eric´ Gourgoulhon, Micha l Bejger (LUTH, CAMK) SageManifolds ERE2014, Valencia, 2 Sept. 2014 3 / 44 In 1969, during his PhD under Pirani supervision at King's College, Ray d'Inverno wrote ALAM (Atlas Lisp Algebraic Manipulator) and used it to compute the Riemann tensor of Bondi metric. The original calculations took Bondi and his collaborators 6 months to go. The computation with ALAM took 4 minutes and yield to the discovery of 6 errors in the original paper [J.E.F. Skea, Applications of SHEEP (1994)] In the early 1970's, ALAM was rewritten in the LISP programming language, thereby becoming machine independent and renamed LAM The descendant of LAM, called SHEEP (!), was initiated in 1977 by Inge Frick Since then, many softwares for tensor calculus have been developed.. -
Using Macaulay2 Effectively in Practice
Using Macaulay2 effectively in practice Mike Stillman ([email protected]) Department of Mathematics Cornell 22 July 2019 / IMA Sage/M2 Macaulay2: at a glance Project started in 1993, Dan Grayson and Mike Stillman. Open source. Key computations: Gr¨obnerbases, free resolutions, Hilbert functions and applications of these. Rings, Modules and Chain Complexes are first class objects. Language which is comfortable for mathematicians, yet powerful, expressive, and fun to program in. Now a community project Journal of Software for Algebra and Geometry (started in 2009. Now we handle: Macaulay2, Singular, Gap, Cocoa) (original editors: Greg Smith, Amelia Taylor). Strong community: including about 2 workshops per year. User contributed packages (about 200 so far). Each has doc and tests, is tested every night, and is distributed with M2. Lots of activity Over 2000 math papers refer to Macaulay2. History: 1976-1978 (My undergrad years at Urbana) E. Graham Evans: asked me to write a program to compute syzygies, from Hilbert's algorithm from 1890. Really didn't work on computers of the day (probably might still be an issue!). Instead: Did computation degree by degree, no finishing condition. Used Buchsbaum-Eisenbud \What makes a complex exact" (by hand!) to see if the resulting complex was exact. Winfried Bruns was there too. Very exciting time. History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) I tried to do \real mathematics" but Dave Bayer (basically) rediscovered Groebner bases, and saw that they gave an algorithm for computing all syzygies. I got excited, dropped what I was doing, and we programmed (in Pascal), in less than one week, the first version of what would be Macaulay. -
Redberry: a Computer Algebra System Designed for Tensor Manipulation
Redberry: a computer algebra system designed for tensor manipulation Stanislav Poslavsky Institute for High Energy Physics, Protvino, Russia SRRC RF ITEP of NRC Kurchatov Institute, Moscow, Russia E-mail: [email protected] Dmitry Bolotin Institute of Bioorganic Chemistry of RAS, Moscow, Russia E-mail: [email protected] Abstract. In this paper we focus on the main aspects of computer-aided calculations with tensors and present a new computer algebra system Redberry which was specifically designed for algebraic tensor manipulation. We touch upon distinctive features of tensor software in comparison with pure scalar systems, discuss the main approaches used to handle tensorial expressions and present the comparison of Redberry performance with other relevant tools. 1. Introduction General-purpose computer algebra systems (CASs) have become an essential part of many scientific calculations. Focusing on the area of theoretical physics and particularly high energy physics, one can note that there is a wide area of problems that deal with tensors (or more generally | objects with indices). This work is devoted to the algebraic manipulations with abstract indexed expressions which forms a substantial part of computer aided calculations with tensors in this field of science. Today, there are many packages both on top of general-purpose systems (Maple Physics [1], xAct [2], Tensorial etc.) and standalone tools (Cadabra [3,4], SymPy [5], Reduce [6] etc.) that cover different topics in symbolic tensor calculus. However, it cannot be said that current demand on such a software is fully satisfied [7]. The main difference of tensorial expressions (in comparison with ordinary indexless) lies in the presence of contractions between indices. -
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix Slaes
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs Milena Veneva ∗1 and Alexander Ayriyan y1 1Joint Institute for Nuclear Research, Laboratory of Information Technologies, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Abstract This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pen- tadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed. 1 Introduction Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engineering prob- lems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. On the other hand, special matrix’s char- acteristics like diagonal dominance, positive definiteness, etc. are not always feasible. The latter two points explain why there is a need of methods for solving of SLAEs which take into account the band structure of the matrices and do not have any other special requirements to them. One possible approach to this problem is the symbolic algorithms. -
CAS (Computer Algebra System) Mathematica
CAS (Computer Algebra System) Mathematica- UML students can download a copy for free as part of the UML site license; see the course website for details From: Wikipedia 2/9/2014 A computer algebra system (CAS) is a software program that allows [one] to compute with mathematical expressions in a way which is similar to the traditional handwritten computations of the mathematicians and other scientists. The main ones are Axiom, Magma, Maple, Mathematica and Sage (the latter includes several computer algebras systems, such as Macsyma and SymPy). Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963. Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. -
SNC: a Cloud Service Platform for Symbolic-Numeric Computation Using Just-In-Time Compilation
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCC.2017.2656088, IEEE Transactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 SNC: A Cloud Service Platform for Symbolic-Numeric Computation using Just-In-Time Compilation Peng Zhang1, Member, IEEE, Yueming Liu1, and Meikang Qiu, Senior Member, IEEE and SQL. Other types of Cloud services include: Software as a Abstract— Cloud services have been widely employed in IT Service (SaaS) in which the software is licensed and hosted in industry and scientific research. By using Cloud services users can Clouds; and Database as a Service (DBaaS) in which managed move computing tasks and data away from local computers to database services are hosted in Clouds. While enjoying these remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud great services, we must face the challenges raised up by these applications on powerful machines. The key drivers towards this unexploited opportunities within a Cloud environment. paradigm for the scientific computing field include the substantial Complex scientific computing applications are being widely computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific studied within the emerging Cloud-based services environment computing, however, we need to design an application-specific [4-12]. Traditionally, the focus is given to the parallel scientific platform to help the users efficiently migrate their applications. In HPC (high performance computing) applications [6, 12, 13], this, we propose a Cloud service platform for symbolic-numeric where substantial effort has been given to the integration of computation– SNC. -
Singular Value Decomposition and Its Numerical Computations
Singular Value Decomposition and its numerical computations Wen Zhang, Anastasios Arvanitis and Asif Al-Rasheed ABSTRACT The Singular Value Decomposition (SVD) is widely used in many engineering fields. Due to the important role that the SVD plays in real-time computations, we try to study its numerical characteristics and implement the numerical methods for calculating it. Generally speaking, there are two approaches to get the SVD of a matrix, i.e., direct method and indirect method. The first one is to transform the original matrix to a bidiagonal matrix and then compute the SVD of this resulting matrix. The second method is to obtain the SVD through the eigen pairs of another square matrix. In this project, we implement these two kinds of methods and develop the combined methods for computing the SVD. Finally we compare these methods with the built-in function in Matlab (svd) regarding timings and accuracy. 1. INTRODUCTION The singular value decomposition is a factorization of a real or complex matrix and it is used in many applications. Let A be a real or a complex matrix with m by n dimension. Then the SVD of A is: where is an m by m orthogonal matrix, Σ is an m by n rectangular diagonal matrix and is the transpose of n ä n matrix. The diagonal entries of Σ are known as the singular values of A. The m columns of U and the n columns of V are called the left singular vectors and right singular vectors of A, respectively. Both U and V are orthogonal matrices. -
New Directions in Symbolic Computer Algebra 3.5 Year Doctoral Research Student Position Leading to a Phd in Maths Supervisor: Dr
Cadabra: new directions in symbolic computer algebra 3:5 year doctoral research student position leading to a PhD in Maths supervisor: Dr. Kasper Peeters, [email protected] Project description The Cadabra software is a new approach to symbolic computer algebra. Originally developed to solve problems in high-energy particle and gravitational physics, it is now attracting increas- ing attention from other disciplines which deal with aspects of tensor field theory. The system was written from the ground up with the intention of exploring new ideas in computer alge- bra, in order to help areas of applied mathematics for which suitable tools have so far been inadequate. The system is open source, built on a C++ graph-manipulating core accessible through a Python layer in a notebook interface, and in addition makes use of other open maths libraries such as SymPy, xPerm, LiE and Maxima. More information and links to papers using the software can be found at http://cadabra.science. With increasing attention and interest from disciplines such as earth sciences, machine learning and condensed matter physics, we are now looking for an enthusiastic student who is interested in working on this highly multi-disciplinary computer algebra project. The aim is to enlarge the capabilities of the system and bring it to an ever widening user base. Your task will be to do research on new algorithms and take responsibility for concrete implemen- tations, based on the user feedback which we have collected over the last few years. There are several possibilities to collaborate with people in some of the fields listed above, either in the Department of Mathematical Sciences or outside of it. -
Sage? History Community Some Useful Features
What is Sage? History Community Some useful features Sage : mathematics software based on Python Sébastien Labbé [email protected] Franco Saliola [email protected] Département de Mathématiques, UQÀM 29 novembre 2010 What is Sage? History Community Some useful features Outline 1 What is Sage? 2 History 3 Community 4 Some useful features What is Sage? History Community Some useful features What is Sage? What is Sage? History Community Some useful features Sage is . a distribution of software What is Sage? History Community Some useful features Sage is a distribution of software When you install Sage, you get: ATLAS Automatically Tuned Linear Algebra Software BLAS Basic Fortan 77 linear algebra routines Bzip2 High-quality data compressor Cddlib Double Description Method of Motzkin Common Lisp Multi-paradigm and general-purpose programming lang. CVXOPT Convex optimization, linear programming, least squares Cython C-Extensions for Python F2c Converts Fortran 77 to C code Flint Fast Library for Number Theory FpLLL Euclidian lattice reduction FreeType A Free, High-Quality, and Portable Font Engine What is Sage? History Community Some useful features Sage is a distribution of software When you install Sage, you get: G95 Open source Fortran 95 compiler GAP Groups, Algorithms, Programming GD Dynamic graphics generation tool Genus2reduction Curve data computation Gfan Gröbner fans and tropical varieties Givaro C++ library for arithmetic and algebra GMP GNU Multiple Precision Arithmetic Library GMP-ECM Elliptic Curve Method for Integer Factorization -
An Alternative Algorithm for Computing the Betti Table of a Monomial Ideal 3
AN ALTERNATIVE ALGORITHM FOR COMPUTING THE BETTI TABLE OF A MONOMIAL IDEAL MARIA-LAURA TORRENTE AND MATTEO VARBARO Abstract. In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal. 1. Introduction Since many years syzygies, and more generally free resolutions, are central in purely theo- retical aspects of algebraic geometry; more recently, after the connection between algebra and statistics have been initiated by Diaconis and Sturmfels in [DS98], free resolutions have also become an important tool in statistics (for instance, see [D11, SW09]). As a consequence, it is fundamental to have efficient algorithms to compute them. The usual approach uses Gr¨obner bases and exploits a result of Schreyer (for more details see [Sc80, Sc91] or [Ei95, Chapter 15, Section 5]). The packages for free resolutions of the most used computer algebra systems, like [Macaulay2, Singular, CoCoA], are based on these techniques. In this paper, we introduce a new algorithmic method to compute the minimal graded free resolution of any finitely generated graded module over a polynomial ring such that some (possibly non- minimal) graded free resolution is known a priori. We describe this method and we present the resulting algorithm in the case of monomial ideals in a polynomial ring, in which situ- ation we always have a starting nonminimal graded free resolution. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
Singular in Sage L-38 22November2019 1/40 Computing with Polynomials in Singular
Computing with Polynomials in Singular 1 Polynomials, Resultants, and Factorization computer algebra for polynomial computations resultants to eliminate variables 2 Gröbner Bases and Multiplication Matrices ideals of polynomials and Gröbner bases normal forms and multiplication matrices multiplicity as the dimension of the local quotient ring 3 Formulas for the 4-bar Coupler Point formulation of the problem processing a Gröbner basis MCS 507 Lecture 38 Mathematical, Statistical and Scientific Software Jan Verschelde, 22 November 2019 Scientific Software (MCS 507) Singular in Sage L-38 22November2019 1/40 Computing with Polynomials in Singular 1 Polynomials, Resultants, and Factorization computer algebra for polynomial computations resultants to eliminate variables 2 Gröbner Bases and Multiplication Matrices ideals of polynomials and Gröbner bases normal forms and multiplication matrices multiplicity as the dimension of the local quotient ring 3 Formulas for the 4-bar Coupler Point formulation of the problem processing a Gröbner basis Scientific Software (MCS 507) Singular in Sage L-38 22November2019 2/40 Singular Singular is a computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory, under the GNU General Public License. Singular’s core algorithms handle polynomial factorization and resultants characteristic sets and numerical root finding Gröbner, standard bases, and free resolutions. Its development is directed by Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann, within the Dept. of Mathematics at the University of Kaiserslautern. Scientific Software (MCS 507) Singular in Sage L-38 22November2019 3/40 Singular in Sage Advanced algorithms are contained in more than 90 libraries, written in a C-like programming language.