Shinya Miyajima

Fast Enclosure for All Eigenvalues and Invariant Subspaces in Generalized Eigenvalue Problems

Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms.

Fast enclosure for solutions in underdetermined systems

Fast algorithms for enclosing minimal 2-norm solutions in underdetermined systems are proposed. For developing these algorithms, theory for computing error bounds for numerical solutions is established. Moreover techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms.

Detecting Incorrect Rules Automatically in Equivalent Transformation Programs

In the equivalent transformation computation model, a program (called an ET program) is a set of rules for meaning-preserving transformation of problems. In this paper we propose a method for automatically detecting incorrect rules in a given ET program.

Fast verified computation for stabilizing solutions of discrete-time algebraic Riccati equations

Fast iterative algorithms for computing interval matrices containing solutions of discrete-time algebraic Riccati equations are proposed. These algorithms involve only cubic complexity per iteration. The stabilizability and uniqueness of the contained solution can moreover be verified by these algorithms. Numerical results show the properties of the algorithms.

Fast verified computation for the matrix principal pth root

Abstract A fast iterative algorithm for numerically computing an interval matrix containing the principal p th root of an n × n matrix A is proposed. This algorithm is based on a numerical spectral decomposition of A , and is applicable when a computed eigenvector matrix of A is not ill-conditioned. Particular emphasis is put on the computational efficiency of the algorithm which has only O ( n 3 + p n ) operations per iteration. The algorithm moreover verifies the uniqueness of the contained p th root. Numerical results show the efficiency of the algorithm.

Fast verified computation for solutions of algebraic Riccati equations arising in transport theory

Summary A fast algorithm for enclosing the solution of the nonsymmetric algebraic Riccati equation arising in transport theory is proposed. The equation has a special structure, which is taken into account to reduce the complexity. By exploiting the structure, the enclosing process involves only quadratic complexity under a reasonable assumption. The algorithm moreover verifies the uniqueness and minimal positiveness of the enclosed solution. Numerical results show the efficiency of the algorithm.

Verified solutions of delay eigenvalue problems

A numerical algorithm for enclosing solutions of nonlinear eigenvalue problems arising in delay differential equations is proposed. This algorithm verifies the uniqueness of the enclosed eigenvalue when it is simple, and is applicable even for multiple eigenvalues. Numerical results show properties of the algorithm.

Fast enclosure for the minimum norm least squares solution of the matrix equation AXB = C

Summary Fast algorithms for enclosing the minimum norm least squares solution of the matrix equation AXB = C are proposed. To develop these algorithms, theories for obtaining error bounds of numerical solutions are established. The error bounds obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. Copyright

Verified computation for the Hermitian positive definite solution of the conjugate discrete-time algebraic Riccati equation

Abstract A fast iterative algorithm is proposed for numerically computing an interval matrix containing the Hermitian positive definite solution to the conjugate discrete-time algebraic Riccati equation. This algorithm requires only cubic complexity per iteration, and verifies the uniqueness of the contained solution. Numerical results show effectiveness and robustness of the algorithm.

Verified computation of the matrix exponential

Two numerical algorithms for computing interval matrices containing the matrix exponential are proposed. The first algorithm is based on a numerical spectral decomposition and requires only cubic complexity under some assumptions. The second algorithm is based on a numerical Jordan decomposition and applicable even for defective matrices. Numerical results show the effectiveness and robustness of the algorithms.