Ebook: Symbolic-Numeric Computation

The growing demand of speed, accuracy, and reliability in scientific and engineering computing has been accelerating the merging of symbolic and numeric computations, two types of computation coexisting in mathematics yet separated in traditional research of mathematical computation. This book with 23 chapters presents original research and tutorial survey on the integration and interaction of symbolic and numeric computations. It represents the current state of the art and will serve as a valuable reference on the development of algorithms and software packages for hybrid symbolic-numeric computation.

The book highlights the interaction between symbolic and numeric computations. It contains tutorial surveys by distinguished experts, covers recent advances on key topics, and provides effective methods illustrated with applications.

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The growing demand of speed, accuracy, and reliability in scientific and engineering computing has been accelerating the merging of symbolic and numeric computations, two types of computation coexisting in mathematics yet separated in traditional research of mathematical computation. This book with 23 chapters presents original research and tutorial survey on the integration and interaction of symbolic and numeric computations. It represents the current state of the art and will serve as a valuable reference on the development of algorithms and software packages for hybrid symbolic-numeric computation.

The book highlights the interaction between symbolic and numeric computations. It contains tutorial surveys by distinguished experts, covers recent advances on key topics, and provides effective methods illustrated with applications.

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The growing demand of speed, accuracy, and reliability in scientific and engineering computing has been accelerating the merging of symbolic and numeric computations, two types of computation coexisting in mathematics yet separated in traditional research of mathematical computation. This book with 23 chapters presents original research and tutorial survey on the integration and interaction of symbolic and numeric computations. It represents the current state of the art and will serve as a valuable reference on the development of algorithms and software packages for hybrid symbolic-numeric computation.

The book highlights the interaction between symbolic and numeric computations. It contains tutorial surveys by distinguished experts, covers recent advances on key topics, and provides effective methods illustrated with applications.

Content:
Front Matter....Pages i-viii
On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis....Pages 1-15
Ill-conditioned Properties and Hybrid Computations....Pages 17-45
Rational Interpolation and Its Ill-conditioned Property....Pages 47-53
Computing Approximate GCD of Multivariate Polynomials....Pages 55-68
Structured Low Rank Approximation of a Sylvester Matrix....Pages 69-83
Implementation of Fast Low Rank Approximation of a Sylvester Matrix....Pages 85-97
New Algorithms for Exact and Approximate Polynomial Decomposition....Pages 99-112
Amortized Bound for Root Isolation via Sturm Sequences....Pages 113-129
An Algorithm of Real Root Isolation for Polynomial Systems with Applications to the Construction of Limit Cycles....Pages 131-147
An Algebraic Method for Separating Close-Root Clusters and the Minimum Root Separation....Pages 149-166
On the Location of Zeros of an Interval Polynomial....Pages 167-184
Root-Finding with Eigen-Solving....Pages 185-210
Galois Theory via Eigenvalue Methods....Pages 211-222
Pythagore’s Dilemma, Symbolic-Numeric Computation, and the Border Basis Method....Pages 223-243
Proposal for the Algorithmic Use of the BKK-Number in the Algebraic Reduction of a O-dimensional Polynomial System....Pages 245-267
Evaluation of Jacobian Matrices for Newton’s Method with Deflation to Approximate Isolated Singular Solutions of Polynomial Systems....Pages 269-278
On Approximate Linearized Triangular Decompositions....Pages 279-298
On the Extended Iterative Proportional Scaling Algorithm....Pages 299-313
Preprocessing for Finite Element Discretizations of Geometric Problems....Pages 315-333
Symbolic Computation Sequences and Numerical Analytic Geometry Applied to Multibody Dynamical Systems....Pages 335-347
A Symbolic-Numeric Approach to an Electric Field Problem....Pages 349-359
Financial Applications of Symbolically Generated Compact Finite Difference Formulae....Pages 361-374
Symbolic Analyzer for Large Lumped and Distributed Networks....Pages 375-394
Back Matter....Pages 395-395

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The growing demand of speed, accuracy, and reliability in scientific and engineering computing has been accelerating the merging of symbolic and numeric computations, two types of computation coexisting in mathematics yet separated in traditional research of mathematical computation. This book with 23 chapters presents original research and tutorial survey on the integration and interaction of symbolic and numeric computations. It represents the current state of the art and will serve as a valuable reference on the development of algorithms and software packages for hybrid symbolic-numeric computation.

The book highlights the interaction between symbolic and numeric computations. It contains tutorial surveys by distinguished experts, covers recent advances on key topics, and provides effective methods illustrated with applications.

Content:
Front Matter....Pages i-viii
On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis....Pages 1-15
Ill-conditioned Properties and Hybrid Computations....Pages 17-45
Rational Interpolation and Its Ill-conditioned Property....Pages 47-53
Computing Approximate GCD of Multivariate Polynomials....Pages 55-68
Structured Low Rank Approximation of a Sylvester Matrix....Pages 69-83
Implementation of Fast Low Rank Approximation of a Sylvester Matrix....Pages 85-97
New Algorithms for Exact and Approximate Polynomial Decomposition....Pages 99-112
Amortized Bound for Root Isolation via Sturm Sequences....Pages 113-129
An Algorithm of Real Root Isolation for Polynomial Systems with Applications to the Construction of Limit Cycles....Pages 131-147
An Algebraic Method for Separating Close-Root Clusters and the Minimum Root Separation....Pages 149-166
On the Location of Zeros of an Interval Polynomial....Pages 167-184
Root-Finding with Eigen-Solving....Pages 185-210
Galois Theory via Eigenvalue Methods....Pages 211-222
Pythagore’s Dilemma, Symbolic-Numeric Computation, and the Border Basis Method....Pages 223-243
Proposal for the Algorithmic Use of the BKK-Number in the Algebraic Reduction of a O-dimensional Polynomial System....Pages 245-267
Evaluation of Jacobian Matrices for Newton’s Method with Deflation to Approximate Isolated Singular Solutions of Polynomial Systems....Pages 269-278
On Approximate Linearized Triangular Decompositions....Pages 279-298
On the Extended Iterative Proportional Scaling Algorithm....Pages 299-313
Preprocessing for Finite Element Discretizations of Geometric Problems....Pages 315-333
Symbolic Computation Sequences and Numerical Analytic Geometry Applied to Multibody Dynamical Systems....Pages 335-347
A Symbolic-Numeric Approach to an Electric Field Problem....Pages 349-359
Financial Applications of Symbolically Generated Compact Finite Difference Formulae....Pages 361-374
Symbolic Analyzer for Large Lumped and Distributed Networks....Pages 375-394
Back Matter....Pages 395-395
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