  A new symbolic algorithmic implementation of the functional-discrete (FD-)
method is developed and justified for the solution of fourth order
Sturm--Liouville problem on a finite interval in the Hilbert space. The
eigenvalue problem for the fourth order ordinary differential equation with
polynomial coefficients is investigated. The sufficient conditions of an
exponential convergence rate of the proposed approach are received. The
obtained estimates of the absolute errors of FD-method significantly improve
the accuracy of the estimates obtained earlier by I.P~Gavrilyuk, V.L.~Makarov
and A.M.~Popov in 2010. Our algorithm is symbolic and operates with the
decomposition coefficients of the eigenfunction corrections in some basis. The
number of summands in these decompositions depends on the degree of the
potential coefficients and the correction number. Our method uses only the
algebraic operations and basic operations on $(2\times 1)$ column vectors and
$(2\times 2)$ matrices. The proposed approach does not require solving any
boundary value problems and computations of any integrals, unlike the previous
variants of FD-method by I.P.~Gavrilyuk, V.L.~Makarov, A.M.~Popov and
N.M.~Romaniuk in 2010 and 2017. The corrections to eigenpairs are computed
exactly as analytical expressions, and there are no rounding errors. The
numerical examples illustrate the theoretical results. The numerical results
obtained with the FD-method are compared with the numerical test results
obtained with other existing numerical techniques.
