Levin type methods for highly oscillatory integrals

The quadrature methods (Filon, Levin, Asymptotic) to compute highly oscillatory integrals of the form?? ()().bigxaIffxedx ??? (A)are investigated, where ? is given positive number, ()fx ve ()gx are sufficiently smooth functions on the interval [,]ab. It is known that for small ? the methods based on asymptotic expansions don?t converge to the integral value of the form ()A. Filon or Filon-type methods, on the other hand, require the computation of moments []kkIx ?? analytically, but this is not always possible. Unlike Filon-type methods, Levin or Levin-type methods do not require the computation of the moments.In this study, having studied Levin and Levin- type methods which are applied to high oscilatory integrals, we specified why Levin and Levin -type methos are used. Among the methods applied to high oscillatory integrals, Levin-type methods is known to be more common and be applied more easily. But when this method is applied to integrals of the form (A), an algebraic linear system (e.g, Acf ? ) occurs. The condition number of coefficent matrix of this system is sometimes too great and therefore, the solution of the system gets very difficult. For the solution of this system, truncated singular value decomposition was used and obtained results were compared.Key Words: Levin-type methods, TSVD method.