Computer Algebra in Scientific Computing

Name: Computer Algebra in Scientific Computing | 21st International Workshop, CASC 2019, Moscow, Russia, August 26-30, 2019, Proceedings
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21st International Workshop, CASC 2019, Moscow, Russia, August 26-30, 2019, Proceedings

Matthew England Wolfram Koepf Timur M. Sadykov Werner M. Seiler Evgenii V. Vorozhtsov(Editor)

Springer (Publisher)

Published on 15. August 2019

XIII, 479 pages

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978-3-030-26831-2 (ISBN)

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Content

Intro
Preface
Organization
Contents
Rings: An Efficient JVM Library for Commutative Algebra (Invited Talk)*-12pt
1 Overview
2 Basic Concepts
3 Polynomials, GCDs, and Factorization
3.1 Polynomial Types
3.2 Example: Working with Polynomials
3.3 Example: Writing Algorithms
3.4 Benchmarks
3.5 Implementation Notes
4 Ideals and Gröbner Bases
4.1 Implementation and Benchmarks
5 Conclusion and Future Work
References
Towards Soft Exact Computation (Invited Talk)
1 Introduction
1.1 From Zero Problems to Predicates
2 Two Illustrative Classes of Problems
3 Effectivity of the Subdivision Framework
4 Numerical Precision in Subdivision Framework
5 On Oracle Objects
5.1 Soft Tests
5.2 Whence Number Oracles?
5.3 Cluster Oracles
6 Conclusion and Open Problems
References
An Arithmetic-Geometric Mean of a Third Kind!*-12pt
1 Introduction
2 An Historical Overview of Elliptic Integrals
3 The Generalized Arithmetic-Geometric Mean
3.1 Construction and Definition
3.2 Basic Properties
3.3 Quadratic Convergence
3.4 Alternative Calculations
4 Calculating Three Kinds and Three Types of CEI
4.1 Three Formulas for Calculating Three Kinds of CEI
4.2 An Unifying Formula for Calculating Three Types of CEI
4.3 The Formula for Calculating the Complementary CEI
5 Few Explicit Calculations of CEI via GAGM
6 Conclusion
References
Obtaining and Analysis of the Necessary Conditions of Stability of Orbital Gyrostat by Means of Computer Algebra
1 Introduction
2 Construction of a Symbolical Model and Stability Conditions
3 Relative Equilibria
4 Parametric Analysis of Stability Conditions
4.1 Necessary Stability Conditions of Equilibrium (10).
4.2 About the Gyroscopic Stabilization of the Equilibrium (9)
5 Conclusion
References
Efficient Exact Algorithm for Count Distinct Problem*-12pt
1 Introduction
2 Note and Issues
2.1 Methods
2.2 Solution to the Problem of Fast and Efficient Count Distinct Calculation
2.3 Key Results
3 Conclusions and Future Work
References
Implementing HuPf Algorithm for the Inverse Kinematics of General 6R/P Manipulators
1 Introduction
2 The Hyperplanes
2.1 RRR Hyperplanes, T(v2)
2.2 RRP Hyperplanes
2.3 The Right Chain
3 The Inverse Kinematic Algorithm with an Example
References
Symbolic-Numerical Algorithm for Large Scale Calculations the Orthonormal SU(3) BM Basis
1 Introduction
2 Symbolic-Numerical Orthonormalization Algorithm
3 Benchmark for Symbolic Numerical Algorithm
4 Generation and Solution of SU(3) algebraic problem
5 Conclusion
References
The Implementation of the Symbolic-Numerical Method for Finding the Adiabatic Waveguide Modes of Integrated Optical Waveguides in CAS Maple
1 Introduction
2 Setting of the Problem
3 Asymptotic Method for the Derivation of AWM Equations in Symbolic Form
4 Construction of Fundamental System of Solutions (FSS) for the System of Ordinary Differential Equations (SODE) in Symbolic Form
5 Statement of the Problem of Finding Guided Modes in a Symbolic Form
6 Numerical Calculation of Guided Modes of a Planar Open Three-Layer Waveguide
7 Symbolic Formulation of the Problem with Symbolic Generation of System of Linear Algebraic Equations (SLAE)
8 Numerical Calculations
9 Verification
10 Conclusion
References
On Characteristic Decomposition and Quasi-characteristic Decomposition
1 Introduction
2 Preliminaries
2.1 Gröbner Basis, Triangular Set and Triangular Decomposition
2.2 W-Characteristic Set, Characteristic Pair and Characteristic Decomposition
3 A New Algorithm for Characteristic Decomposition
4 Decomposition into Quasi-characteristic Pairs
5 Two Illustrative Examples
5.1 Characteristic Decomposition
5.2 Quasi-characteristic Decomposition
6 Implementation and Experimental Results
References
About Integrability of the Degenerate System*-12pt
1 Introduction
2 The Model System
3 Transferring the System to Non-degenerate Form
4 About the Normal Form and the Condition A
5 Necessary Condition of Local Integrability
6 Sufficient Conditions of Integrability
7 Case b2=2/3
7.1 Subcase 3 a0 - 2 b0 = b(3 a1 - 2 b1)
7.2 Subcase 3 a0 - 2 b0 =b(3 a1 - 2 b1)
8 Conclusions
References
Construction of a New Implicit Difference Scheme for 2D Boussinesq Paradigm Equation
1 Introduction
2 Generation of a Difference Scheme for BPE
3 Consistency and Conservativity
4 Modified Equation
5 Conclusion
References
Symbolic Investigation of the Dynamics of a System of Two Connected Bodies Moving Along a Circular Orbit*-10pt
1 Introduction
2 Equations of Motion
3 Equilibrium Orientations of Satellite-Stabilizer System
4 Investigation of Equilibria
5 Conclusion
References
Parametric Standard Bases and Their Applications
1 Introduction
2 Standard Bases Computation
3 Comprehensive Standard Systems
4 Applications
4.1 Comprehensive Milnor Systems
4.2 Comprehensive Determinacy Systems
4.3 Automated Geometric Theorem Discovery
5 Conclusions
References
An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and Its Application to the Construction of Exponential Integrators
1 Introduction
2 Coefficients of Words in Expressions Involving Exponentials
2.1 A Family of Homomorphisms
2.2 Maple Implementation of the Algorithm
3 Order Conditions for Exponential Integrators
3.1 Grading of Words and Homogeneous Lie Elements
3.2 Leading Error Term
3.3 Symmetry
3.4 Lyndon Words and Lyndon Bases
3.5 Order Conditions
3.6 Example
4 Magnus-Type Integrators
4.1 Legendre Expansions
4.2 Order Conditions for Magnus-Type Integrators
4.3 8th Order Commutator-Free Magnus-Type Integrators
References
Revisit Sparse Polynomial Interpolation Based on Randomized Kronecker Substitution
1 Introduction
1.1 Comparing with Other Reduction Based Methods
1.2 Comparing with Interpolation Algorithms over Finite Fields
2 Reduction Based on Randomized Kronecker Substitution
2.1 Find an ``ok" Random Kronecker Substitution
2.2 Recover Non-colliding Terms
2.3 Algorithms
3 Sparse Interpolation over Finite Fields
3.1 The Ben-Or and Tiwari Sparse Interpolation Algorithm
3.2 Univariate Ben-Or and Tiwari Algorithm over Finite Field
3.3 Multivariate Polynomial Interpolation over Finite Fields
4 Experimental Results
References
Root-Finding with Implicit Deflation
1 Introduction
2 Some Efficient Functional Iterations for Root-Finding
3 The Problem of Taming Wild Roots
4 Taming Wild Roots by Means of Deflation
5 Taming Wild Roots by Means of Mapping the Variable with Linear Maps and Reversion
6 Taming Wild Roots by Means of Squaring the Variable
7 Two Remarks
8 Efficiency of Root-Finding Iterations
References
On Linear Invariant Manifolds in the Generalized Problem of Motion of a Top in a Magnetic Field
1 Introduction
2 On Linear IMs of the Dissipative System
2.1 Formulation of the Problem
2.2 Finding the Linear IMs
2.3 On Linear IMs of 2nd-Level and Higher
2.4 Finding Stationary Solutions and IMs
2.5 On Stability of the Stationary Solutions and IMs
3 On Linear IMs of the Conservative System
3.1 Finding Stationary Solutions and IMs
3.2 On Stability of the Stationary Solutions and IMs
4 Conclusion
References
Robust Schur Stability of a Polynomial Matrix Family
1 Introduction
2 Algebraic Preliminaries
2.1 Newton Sums of a Polynomial
2.2 Positivity of a Multivariate Polynomial
2.3 Distance from a Point to an Algebraic Manifold
3 Stability Domain in the Parameter Space
3.1 Structure of the Boundary
3.2 The Algorithm
4 Numerical Examples
5 Conclusions
References
Symbolic and Numerical Methods for Searching Symmetries of Ordinary Differential Equations with a Small Parameter and Reducing Its Order
1 Introduction
2 Theoretical Section
2.1 Point Symmetries of Differential Equations
2.2 Invariant Representation of Differential Equations
2.3 Order Reduction for Differential Equations
2.4 Symmetries, Invariants, and OID for Equations with a Small Parameter
2.5 Symmetries of Equations with Fractional Derivatives
3 Algorithms and Computer Algebra Modules
3.1 Determining Lie Symmetries
3.2 Computing the Operator of Invariant Differentiation in the Specified Form
4 Conclusion
References
An Algorithm for Computing Invariant Projectors in Representations of Wreath Products
1 Introduction
2 Irreducible Invariant Projectors of Wreath Product
2.1 Centralizer Ring of Wreath Product
2.2 Complete Set of Irreducible Invariant Projectors
3 Calculation Example: S4(octahedron)A5(icosahedron)
3.1 Space Group
3.2 Local Group
3.3 Applying C Program
4 Application Remarks
A Computing Invariant Projectors for A5(icosahedron)A5(icosahedron)
References
PAF Reconstruction with the Orbits Method
1 Introduction
2 Combinatorial Concepts and Properties
3 Technical Details
3.1 The Orbits Method
3.2 PSD Test
3.3 Ranking/Unranking Technique
4 Algorithm and Example
5 Experiments: 0=Dto.1880-Dokovic's D-optimal Solutions
5.1 n=73
5.2 n=79
5.3 n=93
6 Conclusion
References
Analytic Complexity of Hypergeometric Functions Satisfying Systems with Holonomic Rank Two
1 Introduction
2 Notation and Definitions
3 Hypergeometric Systems of Holonomic Rank Two
4 Analytic Complexity of Solutions to Systems of Holonomic Rank Two
References
On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds
1 Introduction
2 Autonomous Systems on Manifolds
3 Numerical Integration and Conservation Laws
4 Explicit and Implicit Difference Schemes
5 Explicit Conservative Difference Schemes
6 Algebraic Background
7 Difference Scheme as Algebraic Correspondence
8 Autonomous System on a Curve
9 Conclusion
References
On Berlekamp-Massey and Berlekamp-Massey-Sakata Algorithms
1 Introduction
2 Preliminaries
3 BM and BMS Algorithms in a Unified Flow Chart
4 Comparing the BM and BMS Algorithms Applied to a Sequence
5 Updates in the BMS Algorithm
6 Concluding Remarks
References
An Algorithm for Solving a Quartic Diophantine Equation Satisfying Runge's Condition
1 Introduction
2 Solving Algorithm
2.1 The Equation z2=P(x)
2.2 The Main Case
3 Estimates for Integer Solutions
4 Concluding Remarks
References
Old and New Nearly Optimal Polynomial Root-Finders
1 Introduction
2 Four Fundamental Computational Problems
3 Boolean Complexity: Lower Estimates and Nearly Optimal Upper Bounds
4 Ehrlich's Iterations and Deflation
4.1 Ehrlich's Iterations and Their Super-Linear Convergence
4.2 Precision Management
4.3 Ehrlich's Iterations with Deflation
4.4 Deflation Algorithms for Ehrlich's Iterations
4.5 Extension to Other Functional Iterations
4.6 Boolean Complexity of Problems 1-4 with and Without MPSolve
5 Approximation of a Factor of a Polynomial
5.1 Isolation of a Domain and Its Boundary
5.2 Approximation of a Factor with Root Set in an Isolated Disc
5.3 DLS Deflation: Outline and Complexity
5.4 Deflation in the Case of an Isolated Cluster of a Small Number of Roots
6 Subdivision Iterations with Deflation
7 Root-Finding on a Line Segment with Deflation
References
Symbolic-Numeric Implementation of the Four Potential Method for Calculating Normal Modes: An Example of Square Electromagnetic Waveguide with Rectangular Insert
1 Introduction
2 Preliminary Information: Four Potential Method
3 Setting a Computational Problem
4 Symbolic-Numeric Algorithm for the Approximate Calculation of Normal Modes
4.1 Introductory Notes
4.2 Symbolic-Numeric Algorithm
5 Numerical Calculation of Normal Modes in a Square Waveguide with a Rectangular Insert
6 Numerical Analysis of Convergence of Eigenvalues and Eigenfunctions
7 Verification Method (Comparison with a Hollow Waveguide)
8 Conclusion
References
A Divergence-Free Method for Solving the Incompressible Navier-Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation
1 Introduction
2 Description of the CLS Method for Numerical Solution of the 2D Navier-Stokes Equations on Non-uniform Grids
3 Two-Parameter Preconditioner for the CLS Method
4 Convergence Acceleration Algorithm Based on Krylov's Subspaces
5 Convergence Acceleration with the Aid of the Multigrid Algorithm
6 Results of Numerical Experiments
6.1 Testing
6.2 Lid-Driven Cavity Flow
7 Conclusion
References
Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling
1 Introduction
2 Winding Number Calculation via Sampling
3 Correctness of the Winding Number Algorithm
References
Determining the Heilbronn Configuration of Seven Points in Triangles via Symbolic Computation
1 Introduction
2 A Proof to Heilbronn Number for Seven Points in Triangles
3 Three Unsolved Non-linear Programming Problems Related to Heilbronn Configurations
4 Two Generalizations of the Heilbronn Triangle Problem
References
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Computer Algebra in Scientific Computing

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