Tomohiro Sogabe

Yuki Satake, Takeshi Fukaya, Tomohiro Sogabe et al.

Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments.

Motohiro Otsuka, Fuminori Tatsuoka, Tomohiro Sogabe et al.

This study considers quadrature-based algorithms to compute $A^α\boldsymbol{b}$, the action of a real power of a Hermitian positive-definite matrix $A$ on a vector $ \boldsymbol{b}$. In these algorithms, the computation of an integral representation of $A^α \boldsymbol{b}$ is reduced to solving several tens or hundreds of shifted linear systems. Current approaches usually analyze the quadrature discretization error, but rarely take into account the additional error introduced by solving these shifted linear systems with iterative solvers. Here, we bound this error with the residual of the approximated solution of these linear systems. This allows the derivation of a stopping criterion for iterative solvers to keep the error of $A^α\boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^α\boldsymbol{b}$ within prescribed tolerance limits.

Fuminori Tatsuoka, Yuto Miyatake, Tomohiro Sogabe

Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of $\boldsymbol{A}$, which enables the computation of $\mathrm{e}^{\boldsymbol{A}}\boldsymbol{b}$ with a prescribed accuracy. However, when the input matrix has the structure $\boldsymbol{A} = τ\boldsymbol{M}^{-1} \boldsymbol{K}$, this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where $\boldsymbol{M}$ is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of $\boldsymbol{A}$. The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of $\boldsymbol{K}$. Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.