Computer Algebra in Scientific Computing

Name: Computer Algebra in Scientific Computing | 17th International Workshop, CASC 2015, Aachen, Germany, September 14-18, 2015, Proceedings
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17th International Workshop, CASC 2015, Aachen, Germany, September 14-18, 2015, Proceedings

Vladimir P. Gerdt Wolfram Koepf Werner M. Seiler Evgenii V. Vorozhtsov(Editor)

Springer (Publisher)

Published on 10. September 2015

XIII, 494 pages

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978-3-319-24021-3 (ISBN)

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Content

Intro
Preface
Organization
Contents
Hypergeometric Solutions of First-Order Linear Difference Systems with Rational-Function Coefficients
1 Introduction
2 The Problem
3 The Reasoning Behind the Algorithm
3.1 The Resolving Equation and Matrix
3.2 The Minimal Subspace Containing All Solutions with yi=0
3.3 The Use of RNF
3.4 The Space of Solutions with yi=0
3.5 When k=m
3.6 Selection of yi
4 The Algorithm
5 On the Resolving Procedure
6 Implementation and Experiments
6.1 Implementation
6.2 Some Experiments
6.3 Comparison with the Cyclic Vector Approach
Janet Bases and Resolutions in CoCoALib
1 Introduction
2 Involutive Bases and Free Resolutions
3 Free Resolutions with Janet Bases
4 Benchmarks
Regular Chains under Linear Changesof Coordinates and Applications
1 Introduction
2 Preliminaries
3 Algorithm for Linear Change of Coordinates
4 Noether Normalization and Regular Chains
5 Applications of Random Linear Changes of Coordinates
6 On the Computation of lim(W(T)) and sat(T)
7 Conclusion
A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve
1 Introduction
2 Preliminaries
2.1 Tangent Cone of a Space Curve
2.2 Regular Chains
3 Computing Intersection Multiplicities in Higher Dimension
4 Computing Tangent Lines as Limits of Secants
4.1 An Algorithmic Principle
4.2 Algorithm
4.3 Equations of Tangent Cones
4.4 Examples
5 Conclusion
Research on the Stability of Relative Equilibria of Oblate Axisymmetric Gyrostat by Meansof Symbolic-Numerical Modelling
1 Introduction
2 Relative Equilibria
3 Construction of Symbolical Model and Parametrization of a Problem
4 Stability of the Third Class Equilibria
4.1 Necessary Conditions of Stability
4.2 Stable and Unstable Equilibria
4.3 Gyroscopic Stabilization: Symbolic-Numerical Modelling
5 Parametrical Analysis of the Conditions of Gyroscopic Stabilization for Second Class Equilibria
6 Conclusion
A New Approach for Computing Regular Solutions of Linear Difference Systems
1 Factorial Series
1.1 Definition
1.2 Ring Structure
1.3 Translation z z +
2 The Functions n
2.1 Definition
2.2 Properties
3 Regular Solutions of Linear Difference Systems
4 Appendix
4.1 Proof of Proposition 1
4.2 Proof of Theorem 1
Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation
1 Introduction
2 Related Work and Alternative Approaches
3 Design and Implementation
4 Solving by Polynomial Homotopy Continuation
4.1 Running a Blackbox Solver
4.2 The Scripting Interface phcpy
5 Pattern Matching with a Database
5.1 The Classification Problem
5.2 The Graph Isomorphism Problem
5.3 Computing Canonical Graph Labelings With Nauty
5.4 Benchmarking the Canonization
5.5 Storing Labelings in a Database
6 Conclusions
Finding First Integrals Using Normal Forms Modulo Differential Regular Chains
1 Introduction
2 Basis of Linear Dependences of Rational Functions
2.1 Preliminary Results
2.2 Algorithm findKernelBasis
2.3 A Variant of findKernelBasis
3 Incremental Computation of Linear Dependences
3.1 Algorithm incrementalFindDependence
3.2 Improvement Using a LU-decomposition
3.3 Finding the First Linear Dependence
3.4 Complexity of the Linear Algebra
4 Application to Finding First Integrals
4.1 Basic Differential Algebra
4.2 Normal Form Modulo a Differential Regular Chain
4.3 Normal Form Modulo a Decomposition
4.4 First Integrals in Differential Algebra
4.5 Algorithm findAllFirstIntegrals
4.6 Complexity
Simplification of Cylindrical Algebraic Formulas
1 Introduction
2 Preliminary
3 Motivating Examples
4 Algorithm
5 Experimentation
6 Conclusions
Quasi-Steady State - Intuition, Perturbation Theory and Algorithmic Algebra
1 Introduction
2 Transferring Scientific to Mathematical Notions
3 Preliminaries and Notation
3.1 Lie Derivatives and Invariance Criteria
3.2 Singular Perturbations
4 The Ad Hoc Approach
5 Reduction in the SPT Setting
5.1 Conditions
5.2 Reduction of Rational Systems
5.3 Algorithmic Aspects
6 Identifying ``Small Parameters
6.1 Definition and Basic Properties
6.2 Structure of the TFPV Set
6.3 Algorithmic Aspects
7 The Ad Hoc Approach Revisited
7.1 Basics and Approximation Properties
7.2 Polynomial Systems and Algorithmic Aspects
8 Conclusion
Polynomial Complexity Recognizing a Tropical Linear Variety
Computing Highest-Order Divisors for a Classof Quasi-Linear Partial Differential Equations
1 Bound on a Degree of a Divisor
2 Algorithm to Find the Algebraic Variety of All the Divisors
Symbolic Algorithm for Generating Irreducible Bases of Point Groups in the Space of SO(3) Group
1 Introduction
2 Rotations
2.1 Geometric Rotation
2.2 Rotation in Functional Spaces
3 Intrinsic Group
4 Generalized Projection Operators
5 Example of Using the Algorithm for the Octahedral Group
5.1 Construction of Elements of the Octahedral Group from Its Generators
5.2 Construction of Irreducible Representations of the Octahedral Group O in the Cartesian Bases
5.3 GPOs Implementation for Standard and Intrinsic Point Groups
6 Conclusion
Symbolic-Numeric Solution of Boundary-Value Problems for the Schrödinger Equation Usingthe Finite Element Method: Scattering Problem and Resonance States
1 Introduction
2 Formulation of Boundary-Value Problems
2.1 Scattering Problem: The Physical Asymptotic Solutions in Longitudinal Coordinates and the Scattering Matrix
3 Generation of Algebraic Problems
3.1 The Calculation Scheme for the Solution Matrix h=h
3.2 The Calculation Scheme for the Solution Matrix h=h
3.3 Algorithm for Calculating the Complex Eigenvalues and Eigenfunctions of Metastable States
4 Benchmark Calculations
5 Conclusion
Application of Computer Algebra Methodsto Investigation of Influence of Constant Torque on Stationary Motions of Satellite
1 Introduction
2 Equations of Motion
3 Equilibrium Orientations
4 Conclusion
Bounds for the Condition Number of Polynomials Systems with Integer Coefficients
1 Introduction
1.1 Our Results
1.2 Notation
2 Condition Number for Univariate Polynomials
3 Condition Number for Polynomial Systems
3.1 Multivariate Aggregate Condition Number
On Invariant Manifolds and Their Stabilityin the Problem of Motion of a Rigid Bodyunder the Influence of Two Force Fields
1 Introduction
2 Formulation of the Problem
3 Finding Invariant Manifolds
3.1 Finding the Invariant Manifolds Embedded in One Another
3.2 Finding the Enveloping Invariant Manifolds
3.3 On Invariant Manifolds under Restrictions on the Constantsof the Problem Integrals
4 On the Stability of Stationary Invariant Manifolds
5 Conclusion
Homotopy Analysis Method for StochasticDifferential Equations with Maxima
1 Introduction
2 The Model
3 Homotopy Analysis Method
4 Homotopy Analysis Method and Stochastic Differential Equation with Third-Order Nonlinearity
5 Concluding Remarks
On the Topology and Visualization of Plane Algebraic Curves
1 Introduction
2 Topology Computation for Plane Algebraic Curves
2.1 Solving the System {f=fy=0}
2.2 Computing the Fibers
2.3 Branch Number Computation and Connection
3 Isotopic Meshing for a Plane Curve
3.1 Tracing Regular Curve Segments
3.2 Error Control for the Meshing of a Plane Curve
4 Experiments
5 Conclusions
Piecewise-Quadratics and Reparameterizations for Interpolating Reduced Data
1 Introduction
2 Problem Formulation and Motivation
3 Main Result
4 Experiments
5 Conclusions
Parametric Solvable Polynomial Rings and Applications
1 Introduction
1.1 Related Work
1.2 Outline
2 Solvable Polynomial Rings
2.1 Parametric Solvable Polynomial Rings
2.2 Solvable Polynomial Coefficient Rings
2.3 Recursive Solvable Polynomial Rings
2.4 Solvable Quotient and Residue Rings
2.5 Solvable Quotient Rings as Coefficient Rings
3 Implementation of Solvable Polynomial Rings
3.1 Polynomial Rings
3.2 Solvable Polynomial Rings
3.3 Recursive Solvable Polynomial Rings
3.4 Solvable Quotient and Residue Rings
3.5 Solvable Polynomial Rings with Non-commutative Coefficients
4 Applications
4.1 Comprehensive Gröbner Bases
4.2 Gröbner Bases and Applications
4.3 Examples
4.4 Extension to Free Non-commutative Coefficients
5 Conclusions
Triangular Decomposition of Matrices in a Domain
1 Introduction
2 Preliminary. Triangular Decomposition in Domain
2.1 LDU Algorithm for the Matrix with Nonzero Diagonal Minors Up to the Rank
3 Triangular Matrix Decomposition
4 LDU Algorithm with Permutation Matrices
4.1 Left Upper Block A Is Not Zero Block
4.2 Matrix A Has Full Rank, Matrices C and (or) B Are Zero Matrices
4.3 Matrix A Has No Full Rank, Matrices C and (or) B Are Zero Matrices, A =0
5 Matrix A is Zero Matrix
5.1 Matrix A Is Zero Matrix, C and (or) B Are Nonzero Matrices
5.2 All Cases When Half of the Matrix Is Equal to Zero
6 Example of Triangular Decomposition on Z
6.1 Recursive Step of the First Level
6.2 Recursive Step of the First Level
6.3 Final Result
7 Conclusion
Automated Reasoning in Reduction RingsUsing the Theorema System
1 Introduction
2 Reduction Rings
2.1 Gröbner Bases in Reduction Rings
3 The Theorema System - A Short Overview
4 The ReductionRingProver Special Prover
4.1 Order Relations
4.2 Commutative Rings with Unit
5 Formalized Reduction Rings
5.1 Structure of the Formalization
5.2 Example: Noetherianity of in Polynomial Rings
6 Conclusion and Future Work
On the Partial Analytical Solution
1 Introduction
2 Kirchhoff's Rod Theory
3 Derivation of Explicit Analytical Solution
3.1 Arbitrariness in General Solution
3.2 Reduction to Two Dependent Variables
3.3 Lie Symmetry Analysis
3.4 Parametrized Analytical Solution
4 Numerical Experiments
5 Conclusion
Efficient Computation of Algebraic Local Cohomology Classes and Change of Ordering for Zero-Dimensional Standard Bases
1 Introduction
2 Preliminaries
3 Main Results
3.1 Computing a Basis of the Vector Space HF from a Given Standard Basis of "426830A F "526930B
3.2 Change of Ordering for Standard Bases and Reduced Standard Bases
Polynomial Real Root Isolation by Means of Root Radii Approximation
1 Introduction
2 Some Definitions and Auxiliary Results
3 Root Radii and Their Estimation
4 Isolation and Approximation of Simple and Well-Conditioned Real Roots
5 Working Example
6 The Tests for Real Root-Finding with Algorithm 1
7 Conclusions
Randomized Circulant and GaussianPre-processing
1 Introduction
2 Background
3 Pre-processing with an f-Circulant Multiplier
4 Does Random Circulant Pre-processing Make GENPNumerically Stable?
References
Symbolic Computation and Finite Element Methods
1 Overview
2 Brief Introduction to FEM
3 Sparsity Optimized Basis Functions
4 Symbolic Local Fourier Analysis
Approximate Quantum Fourier Transform and Quantum Algorithm for Phase Estimation
1 Introduction
2 Quantum Fourier Transform
3 Quantum Algorithm for Phase Estimation
3.1 Case of the Full QFT
3.2 Case of the Approximate QFT
4 Simulation
5 Conclusion
Two-Point Boundary Problems with One Mild Singularity and an Application to Graded Kirchhoff Plates
1 Introduction
2 A Simple Example
3 Two-Point Boundary Problems with One Singularity
4 Application to Functionally Graded Kirchhoff Plates
5 Conclusion
Analysis of Reaction Network Systems Using Tropical Geometry
1 Introduction
2 Definitions and Settings
3 Branches of Tropical Equilibrations
4 Algorithm
4.1 Newton Polytope and Edge Filtering
4.2 Computing Branches of Tropical Equilibrations
4.3 Computation of Minimal Branches
5 Results
5.1 Summary
5.2 Running Time
5.3 Minimal Branches and Role of
5.4 Slow-Fast Variables
5.5 Connected Components
6 Discussions
Safety Verification of Hybrid Systems Using Certified Multiple Lyapunov-Like Functions
1 Introduction
2 Hybrid Systems Definition
3 Verification Using Multiple Lyapunov-Like Functions
3.1 Safety Verification of Continuous Systems
3.2 Safety Verification of Hybrid Systems
3.3 SOS Formulation for Safety Verification
4 Computation of Certified Multiple Lyapunov-Like Functions
5 An Illustrating Example
6 More Examples with Discussions
6.1 More Examples
6.2 Discussions
7 Conclusions
A New Polynomial Bound and Its Efficiency
1 Introduction
1.1 The Bound
1.2 Auxiliary Results
1.3 Comparison with the Threshold
1.4 Conclusion
Distance Evaluation Between an Ellipse and an Ellipsoid
1 Introduction
2 Algebraic Preliminaries
3 Intersection Conditions
4 Distance Equation
5 Conclusions
Algebraic General Solutions of First Order Algebraic ODEs
1 Introduction
2 Algebraic General Solutions of AODEs
3 The Associated System of a Parametrizable First Order AODE
4 Explicit Algebraic Solutions of Planar Rational System
5 Algebraic General Solutions with Bounded Degree of Parametrizable First Order AODEs
6 Conclusion
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Computer Algebra in Scientific Computing

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