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In many fields of scientific computing and Data Science, we frequently face the problem of solving large and sparse linear systems of the form Ax = b, which is one of the most time-consuming parts of all computations. From this fact, many researchers have devoted themselves to developing efficient numerical algorithms for solving the linear systems, and Krylov subspace methods are nowadays popular numerical algorithms and are known as one of the top ten algorithms of the twentieth century, others including fast Fourier transform and Quick Sort. 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. Linear systems arise in the field of Machine Learning. In this section, through two examples: least-squares regression and least-squares classification, we will see the importance of linear systems.
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.
Introduction to Numerical Methods for Solving Linear Systems.
Some Applications to Computational Science and Data Science.
Classification and Theory of Krylov Subspace Methods.
Applications to Shifted Linear Systems.
Applications to Matrix Functions