Computer Algebra in Scientific Computing

Name: Computer Algebra in Scientific Computing | 19th International Workshop, CASC 2017, Beijing, China, September 18-22, 2017, Proceedings
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19th International Workshop, CASC 2017, Beijing, China, September 18-22, 2017, Proceedings

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

Springer (Publisher)

Published on 7. September 2017

XIII, 407 pages

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978-3-319-66320-3 (ISBN)

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Content

Intro
Preface
Organization
Contents
Linear Differential Systems with Infinite Power Series Coefficients (Invited Talk)
1 Introduction
2 Algorithmic Representation
2.1 Computable Infinite Power Series in the Role of Coefficients of Linear Differential Systems
2.2 Procedures for Constructing Local Solutions
3 Approximate (Truncated) Representation
3.1 Strongly Non-singular Matrices
3.2 When only a Truncated System Is Known
4 The Width
References
On the Asymptotic Stability of a Satellite with a Gravitational Stabilizer
1 Introduction
2 Description and Construction of a Symbolical Model
3 Formulation of the Problem
4 Regions of System's Instability
5 Parametric Analysis of Asymptotic Stability Conditions
5.1 Stabilization in the ``Pitch'' Subsystem
5.2 Stabilization in the ``Yaw-and-Roll'' Subsystem
6 Conclusion
References
Sparse Interpolation, the FFT Algorithm and FIR Filters
1 Sparse Interpolation
2 Divide and Conquer Approach
3 The FFT Algorithm
4 An Analog Version of the Splitting Technique
5 Connection to FIR Filters
6 Conclusion
References
On New Integrals of the Algaba-Gamero-Garcia System
1 Introduction
2 Problem Statement
3 Necessary Conditions of Local Integrability
4 Sufficient Conditions of Integrability
5 Case b2=2/3, Subcase 3 a0 - 2 b0 = b(3 a1 - 2 b1)
6 Analytical Properties of the Integrals
7 Conclusions
References
Full Rank Representation of Real Algebraic Sets and Applications
1 Introduction
2 Full Rank Representation of Real Algebraic Sets
3 Compute Full Rank Representation
4 Applications on Plotting Singular Plane and Space Curves
5 Experimentation
6 Conclusion and Future Work
References
Certifying Simple Zeros of Over-Determined Polynomial Systems
1 Introduction
2 Preliminaries
3 Transforming Over-Determined Polynomial Systems into Square Ones
4 Certifying Simple Zeros of Over-Determined Systems
References
Decomposing Polynomial Sets Simultaneously into Gröbner Bases and Normal Triangular Sets
1 Introduction
2 Preliminaries
2.1 Triangular Set and Triangular Decomposition
2.2 Gröbner Basis and W-Characteristic Set
2.3 (Strong) Characteristic Decomposition and Characterizable Gröbner Basis
3 Algorithm for (Strong) Characteristic Decomposition
3.1 Algorithm to Handle the Variable Ordering Condition
3.2 Algorithms for Characteristic Decomposition
3.3 Algorithm for Strong Characteristic Decomposition
3.4 An Illustrative Example
4 Implementation and Experimental Results
References
Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks
1 Introduction
2 The Systems for the Case Studies
2.1 Biomod-26
2.2 Biomod-28
3 Graph-Theoretical Symbolic Preprocessing
4 Determination of Multiple Steady States
4.1 Numerical Approach
4.2 Symbolic Approach
4.3 Comparison
4.4 Going Further
5 Conclusion and Future Work
References
The Polymake Interface in Singular and Its Applications
1 Introduction
2 An Interface to Polymake
3 User Defined Types in Singular: Polyhedral Divisors
4 Quasihomogeneous Isolated Hypersurface Singularities
4.1 Finding Quasihomogeneous Isolated Hypersurface Singularities
4.2 Reconstruction of QHIS from the Milnor Algebra
References
Computation of Some Integer Sequences in Maple
1 Introduction
1.1 Definitions of Numbers
2 Stirling Partition Numbers
2.1 Sequence Calculation
3 Stirling Cycle Numbers
3.1 Singleton Computation
3.2 A Finite Sum
3.3 Sequence Calculation
4 Associated Stirling Numbers
4.1 Singleton Stirling 2-Partition and 2-Cycle
4.2 Sequence Calculation of 2-Partition and 2-Cycle Numbers
4.3 Singleton Stirling r-Partition and r-Cycle Numbers
4.4 Sequence Calculation of r-Partition and r-Cycle Numbers
4.5 Implementation in Maple
5 A Multiple Threads Approach to Sequence Calculations
6 Implementation of Eulerian Numbers
6.1 Timings for Eulerian Number Calculations
References
Symbolic-Numerical Algorithm for Generating Interpolation Multivariate Hermite Polynomials of High-Accuracy Finite Element Method
1 Introduction
2 Setting of the Problem
3 FEM Calculation Scheme
3.1 Lagrange Interpolation Polynomials
3.2 Algorithm for Calculating the Basis of Hermite Interpolating Polynomials
3.3 Example: HIP for d=2
3.4 Piecewise Polynomial Functions
4 Results and Discussion
5 Conclusion
References
Symbolic-Numerical Algorithms for Solving the Parametric Self-adjoint 2D Elliptic Boundary-Value Problem Using High-Accuracy Finite Element Method
1 Introduction
2 FEM Algorithm for Solving the Parametric 2D BVP
3 Fully Symmetric High-Order Gaussian Quadratures
4 The Algorithm for Calculating the Parametric Derivatives of Eigenfunctions and Effective Potentials
5 Benchmark Calculations of Helium Atom Ground State
6 Conclusion
References
A Symbolic Study of the Satellite Dynamics Subject to Damping Torques
1 Introduction
2 Equations of Motion
3 Equilibrium Orientations of Satellite
4 Conditions for the Existence of Equilibrium Orientations of the Satellite
5 Necessary and Sufficient Conditions of Asymptotic Stability of the Equilibrium Orientations of Satellite
6 Conclusion
References
Characteristic Set Method for Laurent Differential Polynomial Systems
1 Introduction
2 Laurent Polynomial Systems
2.1 Laurent Regular Chain
2.2 Characteristic Set Method
2.3 Laurent Gröbner Basis and Minimal Decomposition
3 Differential Polynomial Systems
3.1 Laurent Regular Differential Chains
3.2 Decision of Univariate Laurent Regular Differential Polynomial
References
Sparse Polynomial Interpolation with Finitely Many Values for the Coefficients
1 Introduction
2 Univariate Polynomial Interpolation
2.1 Sparse Interpolation with Finitely Many Coefficients
2.2 The Sparse Interpolation Algorithm
2.3 The Rational Number Coefficients Case
3 Multivariate Polynomial Sparse Interpolation with Modified Kronecker Substitution
3.1 Find a Good Prime
3.2 A Deterministic Algorithm
3.3 Probabilistic Algorithm
4 Experimental Results
5 Conclusion
References
On Stationary Motions of the Generalized Kowalewski Gyrostat and Their Stability
1 Introduction
2 Formulation of the Problem
3 Finding the Stationary Solutions
3.1 Permanent Rotations
3.2 Equilibria
4 On Invariant Manifolds of Codimension 2
5 On Stability of the Stationary Solutions
5.1 On Stability of the Permanent Rotations
5.2 On Stability of the Equilibria
6 Conclusion
7 Appendix
References
Computing the Integer Points of a Polyhedron, I: Algorithm
1 Introduction
2 Polyhedral Sets
3 Integer Solutions of Linear Equation Systems
4 Integer Solutions of Linear Inequality Systems
4.1 Normalization of Linear Inequality Systems
4.2 Representing the Integer Points
4.3 The IntegerSolve Procedure: Specifications
4.4 The DarkShadow Procedure
4.5 The GreyShadow Procedure
4.6 The IntegerSolve Procedure: Algorithm
References
Computing the Integer Points of a Polyhedron, II: Complexity Estimates
1 Introduction
2 Properties of the Projection of Faces of a Polyhedron
3 Complexity Estimates for Fourier-Motzkin Elimination
4 Proof of Theorem1
5 Experimentation
References
Non-linearity and Non-convexity in Optimal Knots Selection for Sparse Reduced Data
1 Problem Formulation
2 Non-Linearity of JTF and Numerical Difficulties
3 Non-Convexity of JTF
4 Numerical Experiments for Fitting Sparse Reduced Data
5 Conclusions
References
The Convergence Conditions of Interval Newton's Method Based on Point Estimates
1 Introduction
2 Notation and Preliminaries
3 Main Results
4 Example
References
Normalization of Indexed Differentials Based on Function Distance Invariants
1 Introduction
2 Indexed Differential Polynomial Ring
3 Distances Between Indexed Functions
4 Normalization with Respect to Monoterm Symmetries
5 Normalization
References
Symbolic-Numeric Integration of the Dynamical Cosserat Equations
1 Introduction
2 Governing Cosserat Equations and the General Solution of Their Kinematic Part
3 Symbolic-Numeric Integration Method
3.1 Naive Approach: Explicit Numerical Solving
3.2 Advanced Approach Based on Exponential Integration
4 Numerical Comparison with the Generalized alpha-Method
5 Conclusison
A Generalized alpha-Method
References
Algorithms for Zero-Dimensional Ideals Using Linear Recurrent Sequences
1 Introduction
2 Generalities on Sequences and Their Annihilators
3 Computing Annihilators of Sequences
3.1 A First Algorithm
3.2 An Algorithm Under Genericity Assumptions
4 Main Algorithm
4.1 Representing Primary Zero-Dimensional Ideals
4.2 The Algorithm
References
Symbolic-Numerical Analysis of the Relative Equilibria Stability in the Planar Circular Restricted Four-Body Problem
1 Introduction
2 Equilibrium Solutions
3 Stability Analysis in Linear Approximation
4 Normalization of the Hamiltonian
5 Conclusion
References
The Method of Collocations and Least Residuals Combining the Integral Form of Collocation Equations and the Matching Differential Relations at the Solution of PDEs
1 Introduction
2 The ``Differential'' CLR Method
2.1 Description of the Method
2.2 Preconditioners for the CLR Method
2.3 Convergence Acceleration Algorithm Based on Krylov's Subspaces
2.4 Convergence Acceleration by Using the Multigrid Algorithm
3 The Use of the Integral Form of Collocation Equations
4 Results of Numerical Experiments
5 Conclusions
References
A Special Homotopy Continuation Method for a Class of Polynomial Systems
1 Introduction
2 Preliminary
2.1 Algebraic Sets and Genericity
2.2 Trackable Paths
2.3 Witness Set and Degree of an Algebraic Set
2.4 Critical Points
3 Main Idea
4 Algorithm
5 Real Critical Set
5.1 Critical Points on a Real Algebraic Set
5.2 Illustrative Example
6 Experiment Performance
6.1 Dense Examples
6.2 Sparse examples
6.3 RAT/Density
References
Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems
1 Introduction
2 Augmented System
3 Error Control with Rank Assumption
4 Error Control for General Systems
4.1 Degree Index
4.2 Improve Accuracy
5 Examples
5.1 ISSAC 2016 System
5.2 Seiler System
5.3 Larger Examples
6 Conclusions
References
Computing Multiple Zeros of Polynomial Systems: Case of Breadth One (Invited Talk)
1 Introduction
2 Local Dual Space
3 Local Separation Bound and Cluster Location
4 Verified Error Bound
5 Modified Newton Iterations
6 Conclusion
References
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Computer Algebra in Scientific Computing

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