  This paper is devoted to the computation of transmission eigenvalues in the
inverse acoustic scattering theory. This problem is first reformulated as a two
by two boundary system of boundary integral equations. Next, utilizing the
Schur complement technique, we develop a Schur complement operator with
regularization to obtain a reduced system of boundary integral equations. The
Nystr\"{o}m discretization is then used to obtain an eigenvalue problem for a
matrix. We employ the recursive integral method for the numerical computation
of the matrix eigenvalue. Numerical results show that the proposed method is
efficient and reduces computational costs.
