## Introduction

In recent years, there is more need to deal with large size of control problems, such as multi-purpose control problems that are very difficult to solve analytically. So many researchers have been actively trying to solve these problems numerically. One effective approach is linear matrix inequality (LMI). Many control design specifications can be reformulated as LMIs which are mathematical programming problems. LMI problems can be transformed into Semi-Definite Programming (SDP) problems, which can be solved by numerical optimization algorithms. It is well-known that all numerical values are approximated on the computer. Double precision numbers with finite precision according to IEEE754 are commonly used for the numerical computation. There are rounding errors (round-off errors) between the calculated value and its exact mathematical value. Due to rounding errors, when SDP problems are solved by numerical optimization algorithms, inaccurate or no solutions may be obtained even if the mathematical solutions of SDP problems exist. To overcome these problems, it is necessary to develop a highly accurate numerical software. Due to numerical errors many problems are still unsolved numerically in the control field. We propose a new methodology which verifies whether these problems are really infeasible or not by using highly accurate numerical computation. General-purpose and easy-to-use highly accurate numerical software is indispensable to this approach. Therefore, our group has developed highly accurate SDP solver (SDPJ, Semidefinite Programming solver in Java) by using multiple precision arithmetic.