  The calculable $R$-matrix theory has been formulated successfully for regular
boundary conditions with vanishing radial wave functions at the coordinate
origins [P. Descouvemont and D. Baye, Rept. Prog. Phys. 73, 036301 (2010)]. We
generalize the calculable $R$-matrix theory to the incoming wave boundary
condition (IWBC), which is widely used in theoretical studies of low-energy
heavy-ion fusion reactions to simulate the strong absorption of incoming flux
inside the Coulomb barriers. The generalized calculable $R$-matrix theory also
provides a natural starting point to extend eigenvector continuation (EC) [D.
Frame et al., Phys. Rev. Lett. 121, 032501 (2018)] to fusion observables. The
${}^{14}\text{N}+{}^{12}\text{C}$ fusion reaction is taken as an example to
validate these new theoretical tools. Both local and nonlocal potentials are
considered in numerical calculations. Our generalizations of the calculable
$R$-matrix theory and EC are found to work well for IWBC.
