Abstract: We constructed a polynomial error approximant of the error function en(x) of the Lanczos Tau method for ordinary differential equations, based on the error of the Lanczos economization process. In the present research, we modify this approximant for boundary value problems in ordinary differential equations by perturbing some of the homogenous condition of en(x) and show that the new approximant, thus obtained, yields a more accurate estimate of the maximum error. Numerical results further confirm that the order of the Tau approximant is also accurately estimated.