| Introduction | p. 1 |
| Parallel Computers | p. 1 |
| Measurements for Algorithms' Performance | p. 2 |
| Vector Computers | p. 3 |
| Summary | p. 10 |
| Exercises | p. 11 |
| References | p. 11 |
| Storage Schemes for the Coefficient Stiffness Matrix | p. 13 |
| Introduction | p. 13 |
| Full Matrix | p. 14 |
| Symmetrical Matrix | p. 14 |
| Banded Matrix | p. 14 |
| Variable Banded Matrix | p. 14 |
| Skyline Matrix | p. 15 |
| Sparse Matrix | p. 16 |
| Detailed Procedures For Determining The Mapping Between 2-D Array and 1-D Array in Skyline Storage Scheme | p. 17 |
| Determination of the Column Height (ICOLH) of a Finite Element Model | p. 19 |
| Computer Implementation For Determining Column Heights | p. 23 |
| Summary | p. 25 |
| Exercises | p. 26 |
| References | p. 26 |
| Parallel Algorithms for Generation and Assembly of Finite Element Matrices | p. 27 |
| Introduction | p. 27 |
| Conventional Algorithm to Generate and Assemble Element Matrices | p. 27 |
| Node-by-Node Parallel Generation and Assembly Algorithms | p. 29 |
| Additional Comments on Baddourah-Nguyen's (Node-by-Node) Parallel Generation and Assembly (GandA) Algorithm | p. 37 |
| Application of Baddourah-Nguyen's Parallel GandA Algorithm | p. 38 |
| Qin-Nguyen's GandA Algorithm | p. 41 |
| Applications of Qin-Nguyen's Parallel GandA Algorithm | p. 46 |
| Summary | p. 48 |
| Exercises | p. 49 |
| References | p. 50 |
| Parallel-Vector Skyline Equation Solver on Shared Memory Computers | p. 51 |
| Introduction | p. 51 |
| Choleski-based Solution Strategies | p. 51 |
| Factorization | p. 52 |
| Basic sequential skyline Choleski factorization: computer code (version 1) | p. 55 |
| Improved basic sequential skyline Choleski factorization: computer code (version 2) | p. 59 |
| Parallel-vector Choleski factorization (version 3) | p. 60 |
| Parallel-vector (with "few" synchronization checks) Choleski factorization (version 4) | p. 64 |
| Parallel-vector enhancement (vector unrolling) Choleski factorization (version 5) | p. 66 |
| Parallel-vector (unrolling) skyline Choleski factorization (version 6) | p. 69 |
| Solution of Triangular Systems | p. 72 |
| Forward solution | p. 72 |
| Backward solution | p. 78 |
| Force: A Portable, Parallel FORTRAN Language | p. 81 |
| Evaluation of Methods on Example Problems | p. 82 |
| Skyline Equation Solver Computer Program | p. 86 |
| Summary | p. 86 |
| Exercises | p. 87 |
| References | p. 88 |
| Parallel-Vector Variable Bandwidth Equation Solver on Shared Memory Computers | p. 91 |
| Introduction | p. 91 |
| Data Storage Schemes | p. 91 |
| Basic Sequential Variable Bandwidth Choleski Method | p. 96 |
| Vectorized Choleski Code with Loop Unrolling | p. 101 |
| More on Force: A Portable, Parallel FORTRAN Language | p. 103 |
| Parallel-Vector Choleski Factorization | p. 103 |
| Solution of Triangular Systems | p. 108 |
| Forward solution | p. 109 |
| Backward solution | p. 112 |
| Relations Amongst the Choleski, Gauss and LDL[superscript T] Factorizations | p. 115 |
| Choleski (U[superscript T]U) factorization | p. 115 |
| Gauss (with diagonal terms L[subscript ii]=1) LU factorization | p. 117 |
| Gauss (LU) factorization with diagonal terms U[subscript ii]=1 | p. 118 |
| LDL[superscript T] factorization with diagonal term L[subscript ii]=1 | p. 120 |
| Similarities of Choleski and Gauss methods | p. 122 |
| Factorization Based Upon "Look Backward" Versus "Look Forward" Strategies | p. 123 |
| Evaluation of Methods For Structural Analyses | p. 129 |
| High speed research aircraft | p. 130 |
| Space shuttle solid rocket booster (SRB) | p. 131 |
| Descriptions of Parallel-Vector Subroutine PVS | p. 134 |
| Parallel-Vector Equation Solver Subroutine PVS | p. 136 |
| Summary | p. 137 |
| Exercises | p. 138 |
| References | p. 139 |
| Parallel-Vector Variable Bandwidth Out-of-Core Equation Solver | p. 141 |
| Introduction | p. 141 |
| Out-of-Core Parallel/Vector Equation Solver (version 1) | p. 141 |
| Memory usage and record length | p. 142 |
| A synchronous input/output on Cray computers | p. 144 |
| Brief summary for parallel-vector incore equation solver on the Cray Y-MP | p. 145 |
| Parallel-vector out-of-core equation solver on the Cray Y-MP | p. 146 |
| Out-of-Core Vector Equation Solver (version 2) | p. 149 |
| Memory usage | p. 149 |
| Vector out-of-core equation solver on the Cray Y-MP | p. 149 |
| Out-of-Core Vector Equation Solver (version 3) | p. 155 |
| Application | p. 157 |
| Version 1 performance | p. 157 |
| Version 2 performance | p. 159 |
| Version 3 performance | p. 160 |
| Summary | p. 162 |
| Exercises | p. 163 |
| References | p. 163 |
| Parallel-Vector Skyline Equation Solver for Distributed Memory Computers | p. 165 |
| Introduction | p. 165 |
| Parallel-Vector Symmetrical Equation Solver | p. 165 |
| Basic symmetrical equation solver | p. 165 |
| Parallel-vector performance improvement in decomposition | p. 166 |
| Communication performance improvement in factorization | p. 176 |
| Forward/backward elimination | p. 177 |
| Numerical Results and Discussions | p. 181 |
| FORTRAN Call Statement to Subroutine Node | p. 185 |
| Summary | p. 187 |
| Exercises | p. 188 |
| References | p. 188 |
| Parallel-Vector Unsymmetrical Equation Solver | p. 191 |
| Introduction | p. 191 |
| Parallel-Vector Unsymmetrical Equation Solution Algorithms | p. 191 |
| Basic unsymmetric equation solver | p. 191 |
| Detailed derivation for the [L] and [U] matrices | p. 193 |
| Basic algorithm for decomposition of "full" bandwidth/column heights unsymmetrical matrix | p. 194 |
| Basic algorithm for decomposition of "variable" bandwidths/column heights unsymmetrical matrix | p. 198 |
| Algorithms for decomposition of "variable" bandwidths/column heights unsymmetrical matrix with unrolling strategies | p. 199 |
| Parallel-vector algorithm for factorization | p. 200 |
| Forward solution phase [L]{y}={b} | p. 202 |
| Backward solution phase [U] {x{ = {y{ | p. 204 |
| Numerical Evaluations | p. 206 |
| A Few Remarks On Pivoting Strategies | p. 211 |
| A FORTRAN Call Statement to Subroutine UNSOLVER | p. 212 |
| Summary | p. 214 |
| Exercises | p. 214 |
| References | p. 216 |
| A Tridiagonal Solver for Massively Parallel Computers | p. 217 |
| Introduction | p. 217 |
| Basic Sequential Solution Procedures for Tridiagonal Equations | p. 217 |
| Cyclic Reduction Algorithm | p. 221 |
| Parallel Tridiagonal Solver by Using Divided and Conquered Strategies | p. 226 |
| Parallel Factorization Algorithm for Tridiagonal System of Equations Using Separators | p. 229 |
| Forward and Backward Solution Phases | p. 236 |
| Forward solution phase: [L] {z} = {y} | p. 236 |
| Backward solution phase: [U] {x} = {z} | p. 238 |
| Comparisons between Different Algorithms | p. 239 |
| Numerical Results | p. 240 |
| A FORTRAN Call Statement To Subroutine Tridiag | p. 241 |
| Summary | p. 244 |
| Exercises | p. 244 |
| References | p. 245 |
| Sparse Equation Solver with Unrolling Strategies | p. 247 |
| Introduction | p. 247 |
| Basic Equation Solution Algorithms | p. 248 |
| Choleski algorithm | p. 248 |
| LDL[superscript T] algorithm | p. 249 |
| Storage Schemes for the Coefficient Stiffness Matrix | p. 252 |
| Reordering Algorithms | p. 254 |
| Sparse Symbolic Factorization | p. 255 |
| Sparse Numerical Factorization | p. 271 |
| Forward and Backward Solutions | p. 278 |
| Forward substitution phase | p. 279 |
| Backward substitution phase | p. 279 |
| Sparse Solver with Improved Strategies | p. 280 |
| Finding master (or super) degree-of-freedom (dof) | p. 280 |
| Sparse matrix (with unrolling strategies) times vector | p. 281 |
| Modifications for the chained list array ICHAINL (-) | p. 288 |
| Sparse numerical factorization with unrolling strategies | p. 289 |
| Out-of-core sparse equation solver with unrolling strategies | p. 299 |
| Numerical Performance of the Developed Sparse Equation Solver | p. 301 |
| FORTRAN Call Statement to SPARSE Equation Solver | p. 306 |
| Summary | p. 308 |
| Exercises | p. 308 |
| References | p. 309 |
| Algorithms for Sparse-Symmetrical-Indefinite and Sparse-Unsymmetrical System of Equations | p. 311 |
| Introduction | p. 311 |
| Basic Formulation for Indefinite System of Linear Equations | p. 311 |
| Rotation Matrix [R] Strategies | p. 318 |
| Natural 2 x 2 Pivoting | p. 323 |
| Switching Row(s) and Column(s) During Factorization | p. 325 |
| Simultaneously Performing Symbolic and Numerical Factorization | p. 329 |
| Restart Memory Managements | p. 329 |
| Major Step-by-Step Procedures for Mixed Look Forward/Backward, Sparse LDL[superscript T] Factorization, Forward and Backward Solution With 2x2 Pivoting Strategies | p. 331 |
| Numerical Evaluations | p. 332 |
| Some Remarks on Unsymmetrical-Sparse System of Linear Equations | p. 334 |
| Summary | p. 338 |
| Exercises | p. 338 |
| References | p. 338 |
| Index | p. 341 |
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