Parallel Algorithms for Numerical Linear Algebra

Front Cover

Parallel Algorithms for Numerical Linear Algebra

Copyright Page

Table of Contents

Preface

Part 1: Systolic array algorithms

Chapter 1. A quadratically convergent parallel Jacobi process for diagonally dominant matrices with distinct eigenvalues

1. Introduction

2. Parallel annihilators; the first step

3. The effect of a complete sweep

4. Numerical examples

5. Conclusions

References

Chapter 2. A Jacobi-like algorithm for computing the generalized Schur form of a regular pencil

1. Introduction

2. Normal pencils

3. Description of the method

4. Global convergence

5. Ultimate convergence

6. Numerical tests

7. Conclusion

References

Chapter 3. Canonical correlations and generalized SVD: applications and new algorithms

1. Introduction

2. Applications

3. SVD of products of three matrices

4. New algorithms

5. Final remarks

Acknowledgements

References

Chapter 4. From Bareiss' algorithm to the stable computation of partial correlations

1. Introduction

2. The Generalized Bareiss algorithm

3. Cybenko's algorithm

4. The Hyperbolic Cholesky algorithm

5. Application to the computation of certain sample partial correlations

6. Computation of arbitrary partial correlations

7. Conclusions

Acknowledgement

References

Part 2: Message-passing systems

Chapter 5. A recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors

1. Introduction

2. The LU decomposition algorithm

3. Solution of tridiagonal systems using prefix algorithms

4. Parallel prefix algorithms on hypercube multiprocessors

5. Estimated speedup and efficiency

6. Experimental results and conclusions

References

Chapter 6. Least squares modifications with inverse factorizations: parallel implications

1. Introduction

2. Updating R–1

3. Downdating R–1

4. Summary and parallel implications

Acknowledgements

References

Chapter 7. Solution of sparse positive definite systems on a hypercube

1. Introduction

2. Solution of sparse symmetric positive definite systems

3. Parallel Cholesky factorization

4. Symbolic factorization

5. Sparse triangular solution

6. Ordering

7. Some experiments and concluding remarks

References

Chapter 8. Some aspects of parallel implementation of the finite-element method on message passing architectures

1. Introduction

2. The model problem and finite-element discretization

3. Overview of computations

4. Cost analysis

5. Numerical experiments

6. Conclusions

Appendix

References

Part 3: Algorithms for parallel shared-memory systems

Chapter 9. An overview of parallel algorithms for the singular value and symmetric eigenvalue problems

1. Introduction

2. Jacobi methods

3. Reduction to tridiagonal form and multisectioning

4. Performance of eigensolvers

5. Singular value decomposition

6. Performance of SVD algorithms

7. Conclusions

Acknowledgements

References

Chapter 10. Block reduction of matrices to condensed forms for eigenvalue computations

1. Introduction

2. The algorithm: reduction to tridiagonal form

3. Reduction to Hessenberg form

4. Reduction to bidiagonal form

5. Relationship to the WY-factorization

6. Pipelining reduction to condensed form with determination of eigenvalues

7. Operations counts and storage

8. Experimental results

References

Chapter 11. Multiprocessing a sparse matrix code on the Alliant FX/8

1. Introduction

2. Alliant FX/8

3. Multifrontal codes and elimination trees

4. Data management issues

5. Task spawning and granularity

6. Management of work queue and performance of code

7. Use of the SCHEDULE package

8. Conclusions

Acknowledgements

References

Chapter 12. Vector and parallel methods for the direct solution of Poisson's equation

1. Introduction

2. Parallel communication algorithms

3. The FFT

4. Tridiagonal solvers

5. Direct methods for solving Poisson's equation

References

Part 4: Design of fast algorithms and implementations for vector supercomputers

Chapter 13. Factoring with the quadratic sieve on large vector computers

1. Introduction

2. The multiple polynomial quadratic sieve

3. Implementation of the MPQS-algorithm on the CYBER 205 and the NEC SX-2

4. Results

5. Conclusions

Note added in proof

Acknowledgements

References

Chapter 14. Efficient vectorizable PDE solvers

1. Introduction

2. Families of difference and error formulas

3. The error-equation and its consequences

4. Linear solver and vectorization

5. The FIDISOL program package

6. Examples

7. Concluding remarks

Acknowledgements

References

Chapter 15. Vectorizable preconditioners for elliptic difference equations in three space dimensions

1. Introduction

2. Factorization of matrices partitioned in block tridiagonal form

3. Limit matrix analysis of preconditioning matrices

4. Plane block factorizations

5. Numerical results

6. Conclusions

Acknowledgement

References

Chapter 16. Solving 3D block bidiagonal linear systems on vector computers

1. A sketch of the problem

2. Vectorization techniques

3. Implementations of the hyperplane ordering for the CYBER 205

4. Preconditioned linear systems and Eisenstat's trick

5. Examples

References