# Numerical computation

## Outline

Computer simulations are required in broad areas in industry and science from nano-level to cosmic scales , such as the development of new materials, functional analysis of protein and DNA, efficient design and development of products for vehicles, accurate weather prediction, supernova explosion, and so on. In such simulations, linear systems of equations and eigenvalue problems have become significant in recent years. Therefore, it is important to solve such problems efficiently. This computational group studies high-performance algorithms and develops computer programs, collaborating with researchers such as in life science.

Solutions of linear systems of equations Solving a linear system of equations is to find x of Ax=b. The equations arise in numerical simulations such as thermal fluid analysis, structural analysis, DNA analysis, etc. The solution dominates the computational time. The methodologies for the solutions include direct and iterative methods.

Direct methods include the Gaussian elimination and LU decomposition. These approaches can solve linear systems in a finite number of operations but require considerable computational costs and memory.
On the other hand, iterative methods include Krylov subspace methods and require less computational costs and memory but may require many iterations. This numerical computation group mainly studies Krylov subspace iterative methods. Themes

• BSAIC preconditioner
• Linear systems of equations from weather simulations
• Block IDR(s)
Solutions of eigenvalue problems An eigenvalue problem is to find eigenvalues and eigenvectors of a matrix. Eigenvalue problems include the standard eigenvalue problem Ax=λx, generalized eigenvalue problem Ax=λBx, nonlinear eigenvalue problems of quadratic order, with square root, more complicated terms, etc. This numerical computation group mainly studies eigensolvers using contour integrals (SS methods). The SS methods can deal with a variety kind of eigenvalue problems. Moreover, they are highly parallelizable and are efficient in parallel computers. The eigensolvers involve linear systems to solve.

Themes

• Fault-tolerance of SS methods
• Nonlinear eigenvalue problems arising from delay differential equation
• Fifth-order eigenvalue problems arising in quantum dot
• Estimation of eigenvalue distributions of nonlinear eigenvalue problems