Function-valued Padé-type approximant via E-algorithm and its applications in solving integral equations

The function-valued Padé-type approximant (FPTA) was defined in the inner product space [8]. In this work, we choose the coefficients in the Neumann power series to make the inner product with both sides a function-valued system of equations to yield a scalar system. Then we express an FPTA in the determinant form. To avoid the direct computation of the determinants, we present the E-algorithm for FPTA based on the vector-valued E-algorithm given by Brezinski [4]. The method of FPTA via E-algorithm (FPTAVEA) not only includes all previous methods but overcomes their essential difficulties. The numerical experiment for a typical integral equation [1] illustrates that the method of FPTAVEA is simpler and more effective for obtaining the characteristic values and the characteristic functions than all previous methods. In addition, this method is also applicable to other Fredholm integral equations of the second kind without explicit characteristic values and characteristic functions. A corresponding example [12] is given and the numerical result is the same as that in [12].