The stability of the Gauss-Chebyshev method for Cauchy singular integral equations

Abstract

AbstractWe show that the infinity condition number of the Gauss-Chebyshev method, for the complete Cauchy singular integral equation (CSIE) of the first kind, is bounded above and below by the condition number of the dominant equation times a constant. The condition number of the dominant equation is asymptotically equal to {(0.718 + 0.344 ln n)n − 0.344}. This implies that the Gauss-Chebyshev method is stable for very large n's provided that the multiplying constants are not too large. The magnitude of the constants depends on the eigenvalues of the CSIE