by Chi-Fai Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 53-54). / Abstract also in Chinese. / Chapter Chapter 1 --- INTRODUCTION --- p.1 / Chapter §1.1 --- Polynomial Interpolation --- p.1 / Chapter §1.2 --- Conjugate Gradient Type Methods --- p.6 / Chapter §1.3 --- Outline of the Thesis --- p.10 / Chapter Chapter 2 --- INTEGRAL EQUATIONS --- p.11 / Chapter §2.1 --- Integral Equations --- p.11 / Chapter §2.2 --- Numerical Treatments of Second Kind Integral Equations --- p.15 / Chapter Chapter 3 --- FAST ALGORITHM FOR SECOND KIND INTEGRAL EQUATIONS --- p.20 / Chapter §3.1 --- Introduction --- p.20 / Chapter §3.2 --- The Approximation --- p.24 / Chapter §3.3 --- Error Analysis --- p.35 / Chapter §3.4 --- Numerical Examples --- p.40 / Chapter §3.5 --- Concluding Remarks --- p.51 / References --- p.53

Graduation date: 1991

John N. Tsitsiklis, Bernard C. Levy. / Bibliography: leaves 18-19. / "December, 1981." / "National Science Foundation ... Grant ECS-80-12668"

Indiana University-Purdue University Indianapolis (IUPUI) / We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_)$ for all values of the real parameter $\gamma$.