In science and engineering, mathematical models are constructed to understand phenomena, and by solving these models, it becomes possible to predict unknown phenomena and design new engineering products. In most cases, these models are difficult to solve analytically, and thus are often solved by numerical computations using computers. On the other hand, numerical computations performed on computers are not exact. The results of arithmetic operations are approximated to a finite number of digits at each step, and all infinite processes involving limits are approximated by finite operations. As a consequence, there are cases in which even the sign of the computed result differs from the true value.

To draw correct conclusions from computed results, it is necessary to determine how many digits of those results are reliable. In our laboratory, we develop methods that provide not only the computed results but also information on how many digits of those results are guaranteed to be correct. In particular, we are interested in matrix problems—such as matrix equations, matrix functions, eigenvalue problems, computation of singular values and singular vectors, and linear least squares problems—and we work on developing methods that are both faster and capable of guaranteeing a larger number of correct digits. To construct such methods, it is necessary to prove theorems that quantitatively determine bounds on the true solution using only the values obtained within a computer.