Solving singular integral equations using Gaussian quadrature and overdetermined system

Gauss-Chebyshev quadrature and collocation at the zeros of the Chebyshev polynomial of the first kind T_n(x), and second kind U_n(x) leads to an overdetermined system of linear algebraic equations. The size of the coefficient matrix for the overdetermined system depends on the degrees of Chebyshev polynomials used. We show that we can get more accurate solution using T_4n+4(x), than other T_n(x). The regularization method using Generalized Singular Value Decomposition is described and compared to Gauss-Newton method for solving the overdetermined system of equations. Computational tests show that GSVD with an appropriate choice of regularization parameter gives better solution in solving singular integral equations.