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Krylov Subspace Methods for Linear Systems: Principles of Algorithms

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This book focuses on Krylov subspace methods for solving linear systems, which are known as one of the top 10 algorithms in the twentieth century, such as Fast Fourier Transform and Quick Sort (SIAM News, 2000). Theoretical aspects of Krylov subspace methods developed in the twentieth century are explained and derived in a concise and unified way. Furthermore, some Krylov subspace methods in the twenty-first century are described in detail, such as the COCR method for complex symmetric linear systems, the BiCR method, and the IDR(s) method for non-Hermitian linear systems.

The strength of the book is not only in describing principles of Krylov subspace methods but in providing a variety of applications: shifted linear systems and matrix functions from the theoretical point of view, as well as partial differential equations, computational physics, computational particle physics, optimizations, and machine learning from a practical point of view.

The book is self-contained in that basic necessary concepts of numerical linear algebra are explained, making it suitable for senior undergraduates, postgraduates, and researchers in mathematics, engineering, and computational science. Readers will find it a useful resource for understanding the principles and properties of Krylov subspace methods and correctly using those methods for solving problems in the future.

Author(s): Tomohiro Sogabe
Series: Springer Series in Computational Mathematics, 60
Publisher: Springer
Year: 2023

Language: English
Pages: 232
City: Singapore

Preface
Acknowledgements
Contents
1 Introduction to Numerical Methods for Solving Linear Systems
1.1 Linear Systems
1.1.1 Vector Norm
1.1.2 Matrix Norm
1.2 Condition Number
1.3 Direct Methods
1.3.1 LU Decomposition
1.3.2 LU Decomposition with Pivoting
1.3.3 Iterative Refinement
1.4 Direct Methods for Symmetric Linear Systems
1.4.1 Cholesky Decomposition
1.4.2 LDL Decomposition
1.5 Direct Methods for Large and Sparse Linear Systems
1.6 Stationary Iterative Methods
1.6.1 The Jacobi Method
1.6.2 The Gauss–Seidel Method
1.6.3 The SOR Method
1.6.4 Convergence of the Stationary Iterative Methods
1.7 Multigrid Methods
1.8 Krylov Subspace Methods
1.9 Orthogonalization Methods for Krylov Subspaces
1.9.1 The Arnoldi Process
1.9.2 The Bi-Lanczos Process
1.9.3 The Complex Symmetric Lanczos Process
1.9.4 The Lanczos Process
2 Some Applications to Computational Science and Data Science
2.1 Partial Differential Equations
2.1.1 Finite Difference Methods
2.1.2 The Finite Element Method
2.1.3 Weak Form
2.1.4 Derivation of Linear Systems
2.1.5 Example
2.2 Computational Physics
2.2.1 Large-Scale Electronic Structure Calculation
2.2.2 Lattice Quantum Chromodynamics
2.3 Machine Learning
2.3.1 Least-squares Regression
2.3.2 Least-squares Classification
2.4 Matrix Equations
2.5 Optimization
2.5.1 Tensor Notations
2.5.2 Newton's Method on Euclidean Space
2.5.3 Newton's Method on Riemannian Manifold
3 Classification and Theory of Krylov Subspace Methods
3.1 Hermitian Linear Systems
3.1.1 The Conjugate Gradient (CG) Method
3.1.2 The Conjugate Residual (CR) Method
3.1.3 The Minimal Residual (MINRES) Method
3.2 Complex Symmetric Linear Systems
3.2.1 The Conjugate Orthogonal Conjugate Gradient (COCG) Method
3.2.2 The Conjugate Orthogonal Conjugate Residual (COCR) Method
3.2.3 The Quasi-Minimal Residual (QMR_SYM) Method
3.3 Non-Hermitian Linear Systems
3.3.1 The Bi-Conjugate Gradient (BiCG) Method
3.3.2 The Composite Step Bi-Conjugate Gradient (CSBiCG) Method
3.3.3 The Bi-Conjugate Residual (BiCR) Method
3.3.4 The Quasi-Minimal Residual (QMR) Method
3.3.5 The Generalized Minimal Residual (GMRES) Method
3.3.6 The Generalized Conjugate Residual (GCR) Method
3.3.7 The Full Orthogonalization Method (FOM)
3.3.8 Product-Type Krylov Subspace Methods
3.3.9 Induced Dimension Reduction (IDR(s)) Method
3.3.10 Block Induced Dimension Reduction (Block IDR(s)) Method
3.4 Other Krylov Subspace Methods
3.4.1 Krylov Subspace Methods for Normal Equations
3.5 Preconditioning Techniques
3.5.1 Incomplete Matrix Decomposition Preconditioners
3.5.2 Approximate Inverse Preconditioners
3.5.3 Matrix Polynomial Preconditioners
3.5.4 Preconditioners Based on Stationary Iterative Methods
3.5.5 Reorderings for Preconditioners
4 Applications to Shifted Linear Systems
4.1 Shifted Linear Systems
4.2 Shifted Hermitian Linear Systems
4.2.1 The Shifted CG Method
4.2.2 The Shifted CR Method
4.2.3 The Shifted MINRES Method
4.3 Shifted Complex Symmetric Linear Systems
4.3.1 The Shifted COCG Method
4.3.2 The Shifted COCR Method
4.3.3 The Shifted QMR_SYM Method
4.4 Shifted Non-Hermitian Linear Systems
4.4.1 The Shifted BiCG Method
4.4.2 The Shifted BiCGSTAB Method
4.4.3 The Shifted GMRES Method
4.4.4 The Shifted IDR(s) Method
5 Applications to Matrix Functions
5.1 Jordan Canonical Form
5.2 Definition and Properties of Matrix Functions
5.3 Matrix Square Root and Matrix pth Root
5.3.1 Matrix Square Root
5.3.2 Matrix pth Root
5.4 Matrix Exponential Function
5.4.1 Numerical Algorithms for Matrix Exponential Functions
5.4.2 Multiplication of a Matrix Exponential Function and a Vector
5.5 Matrix Trigonometric Functions
5.6 Matrix Logarithm
5.7 Matrix Fractional Power
Appendix Software
Appendix References
Index