A variety of block Krylov subspace methods have been successfully developed for linear systems and matrix equations. The application of block Krylov methods to compute matrix functions is, however, less established, despite the growing prevalence of matrix functions in scientific computing. Of particular importance is the evaluation of a matrix function on not just one but multiple vectors. The main contribution of this paper is a class of efficient block Krylov subspace methods tailored precisely to this task. With the full orthogonalization method (FOM) for linear systems forming the backbone of our theory, the resulting methods are referred to as B(FOM)2: block FOM for functions of matrices. Many other important results are obtained in the process of developing these new methods. Matrix-valued inner products are used to construct a general framework for block Krylov subspaces that encompasses already established results in the literature. Convergence bounds for B(FOM)2 are proven for Stieltjes functions applied to a class of matrices which are self-adjoint and positive definite with respect to the matrix-valued inner product. A detailed algorithm for B(FOM)2 with restarts is developed, whose efficiency is based on a recursive expression for the error, which is also used to update the solution. Numerical experiments demonstrate the power and versatility of this new class of methods for a variety of matrix-valued inner products, functions, and matrices.

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Block Krylov subspace methods for functions of matrices

In this paper, we analyze all possible situations of rank deficiency that cause breakdown in block conjugate gradient (BCG) solvers. A simple solution, breakdown-free block conjugate gradient (BFBCG), is designed to address the rank deficiency problem. The rationale of the BFBCG algorithm is to derive new forms of parameter matrices based on the potentially reduced search subspace to handle rank deficiency. Orthogonality properties and convergence of BFBCG in case of rank deficiency are justified accordingly with mathematical rigor. BFBCG yields faster convergence than restarting BCG when breakdown occurs. Numerical examples suffering from rank deficiency are provided to demonstrate the robustness of BFBCG.

A breakdown-free block conjugate gradient method

A new family of global methods for linear systems with multiple right-hand sides

We analyze an expansion of the generalized block Krylov subspace framework of [Electron. Trans. Numer. Anal., 47 (2017), pp. 100--126]. This expansion allows the use of low-rank modifications of th...

Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

The block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides

Generalized global conjugate gradient squared algorithm

Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems

For solving a linear system with a large-scale coefficient matrix which is stored in a factorised form, we present a new interlaced randomised iterative method. The new method can take advantage of the factored form and avoid performing matrix multiplications. Furthermore, its convergence property is studied and theoretical analysis reveals that it is linearly convergent. Finally, some numerical results are given to confirm the effectiveness of the proposed method.

Jianhua Zhang

Papers

A new family of global methods for linear systems with multiple right-hand sides

Generalized global conjugate gradient squared algorithm

Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems

A randomised iterative method for solving factorised linear systems

On maximum residual nonlinear Kaczmarz-type algorithms for large nonlinear systems of equations