As a new type of technique, simplicial methods have yielded extremely important contributions toward solutions of a system of nonlinear equations. Theoretical investigations and numerical tests have shown that the performance of simplicial methods depends critically on the triangulations underlying them. This monograph describes some recent developments in triangulations and simplicial methods. It includes the D1-triangulation and its applications to simplicial methods. As a result, efficiency of simplicial methods has been improved significantly. Thus more effective simplicial methods have been developed.…mehr
As a new type of technique, simplicial methods have yielded extremely important contributions toward solutions of a system of nonlinear equations. Theoretical investigations and numerical tests have shown that the performance of simplicial methods depends critically on the triangulations underlying them. This monograph describes some recent developments in triangulations and simplicial methods. It includes the D1-triangulation and its applications to simplicial methods. As a result, efficiency of simplicial methods has been improved significantly. Thus more effective simplicial methods have been developed.
Produktdetails
Produktdetails
Lecture Notes in Economics and Mathematical Systems 421
1 Introduction.- 2 Preliminaries.- 2.1 Notations.- 2.2 Fixed Point Theorems.- 2.3 Applications.- 3 Existing Triangulations.- 3.1 Existing Triangulations of Sn.- 3.2 Existing Triangulations of Rn.- 3.3 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Sn.- 3.4 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Rn.- 4 The D1-Triangulation of Rn.- 4.1 The D1-Triangulation of Rn.- 4.2 Pivot Rules of the D1-Triangulation.- 4.3 The Number of Simplices of the D1-Triangulation in a Unit Cube.- 4.4 The Diameter of the D1-Triangulation.- 4.5 The Average Directional Density of the D1-Triangulation.- 5 The T1-Triangulation of the Unit Simplex.- 5.1 The T1-Triangulation.- 5.2 Pivot Rules of the T1-Triangulation.- 5.3 Comparison of the Triangulations of the Unit Simplex.- 6 The D1-Triangulation in Variable Dimension Algorithms on the Unit Simplex.- 6.1 The Dv1-Triangulation.- 6.2 Pivot Rules of the Dv1-Triangulation.- 6.3 The (n + 1)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 6.4 The (2n+1 - 2)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 7 The D1-Triangulation in Variable Dimension Algorithms on the Euclidean Space.- 7.1 The Dv2-Triangulation.- 7.2 Pivot Rules of the Dv2-Triangulation.- 7.3 The 2n-Ray Variable Dimension Method Based on the D1-Triangulation.- 7.4 The 2n-Ray Variable Dimension Algorithm Based on the Dv2-Triangulation.- 8 The D3-Triangulation for Simplicial Homotopy Algorithms.- 8.1 Definition of the D3-Triangulation.- 8.2 Construction of the D3-triangulation.- 8.3 Pivot Rules of the D3-Triangulation.- 8.4 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 9 The D2-Triangulation for Simplicial Homotopy Algorithms.- 9.1 Construction of the D2-Triangulation.- 9.2Description of the D2-Triangulation.- 9.3 Pivot Rules of the D2-Triangulation.- 9.4 Description of the D2*-Triangulation.- 9.5 Pivot Rules of the D2*-Triangulation.- 9.6 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 10 Conclusions.
1 Introduction.- 2 Preliminaries.- 2.1 Notations.- 2.2 Fixed Point Theorems.- 2.3 Applications.- 3 Existing Triangulations.- 3.1 Existing Triangulations of Sn.- 3.2 Existing Triangulations of Rn.- 3.3 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Sn.- 3.4 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Rn.- 4 The D1-Triangulation of Rn.- 4.1 The D1-Triangulation of Rn.- 4.2 Pivot Rules of the D1-Triangulation.- 4.3 The Number of Simplices of the D1-Triangulation in a Unit Cube.- 4.4 The Diameter of the D1-Triangulation.- 4.5 The Average Directional Density of the D1-Triangulation.- 5 The T1-Triangulation of the Unit Simplex.- 5.1 The T1-Triangulation.- 5.2 Pivot Rules of the T1-Triangulation.- 5.3 Comparison of the Triangulations of the Unit Simplex.- 6 The D1-Triangulation in Variable Dimension Algorithms on the Unit Simplex.- 6.1 The Dv1-Triangulation.- 6.2 Pivot Rules of the Dv1-Triangulation.- 6.3 The (n + 1)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 6.4 The (2n+1 - 2)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 7 The D1-Triangulation in Variable Dimension Algorithms on the Euclidean Space.- 7.1 The Dv2-Triangulation.- 7.2 Pivot Rules of the Dv2-Triangulation.- 7.3 The 2n-Ray Variable Dimension Method Based on the D1-Triangulation.- 7.4 The 2n-Ray Variable Dimension Algorithm Based on the Dv2-Triangulation.- 8 The D3-Triangulation for Simplicial Homotopy Algorithms.- 8.1 Definition of the D3-Triangulation.- 8.2 Construction of the D3-triangulation.- 8.3 Pivot Rules of the D3-Triangulation.- 8.4 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 9 The D2-Triangulation for Simplicial Homotopy Algorithms.- 9.1 Construction of the D2-Triangulation.- 9.2Description of the D2-Triangulation.- 9.3 Pivot Rules of the D2-Triangulation.- 9.4 Description of the D2*-Triangulation.- 9.5 Pivot Rules of the D2*-Triangulation.- 9.6 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 10 Conclusions.
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