An efficient shooting algorithm is derived for the Sturm-Liouville eigenvalue problem (SLP) on \([a,b]\) and for the rational Sturm-Liouville problem (RSLP), where the potential is a rational function of the eigenvalue. In both problems an approximate solution is computed starting with an approximation \(\mu\) of the eigenvalue \(\lambda\) by solving an initial value problem from the left \(x=b\) and stepwise to a central point \(x=c\) with stepsize \(h\). For the SLP case let \(F(\mu,h)\) be the miss distance at \(x=c\). Choosing \(t(h)\), called a discretization parameter, to reduce stepsize \(\lim_{h\to 0} t(h)/h=0\) it is possible to compute \[ F^{(k)}(\mu,t^i(h))= (\delta/\delta\mu)^kF(\mu,t^i(h)),\quad k=0,1,2,\dots,r+1,\;i= 1,2,\dots \] Expanding \(F(\mu+\delta,t^i(h))\) in Taylor series \(\delta\) for \(r+1\) terms leads to a polynomial equation for \(\delta\) such that \(\mu+\delta\simeq\lambda\). Varying \(h\) and \(i\) the authors obtain a set of linear equations for \(\delta\) which is employed to modify \(\mu\) in the shooting algorithm. A similar but more complex method in employed for the RSLP problem. Results of numerical computations are presented for both SLP and RSLP problems using \(t=h^2+dh^3\) and \(t=h^3+dh^4\), \(d\) constant.