OutlineComputer simulations are required in broad areas in industry and science from nanolevel to cosmic scales 
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.
Themes

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
