Inverse spectral/pseudospectral methods are effective tools for solving differential equations. In this talk we will present inverse matrices constructed analytically by stepwise integrations and low-rank updates for pseudospectral differential operators. Since these inverse operators are derived analytically, applying these operators effectively suppresses serious roundoff errors induced by those ill-conditioned pseudospectral differentiation matrices in computations. Numerical validations are conducted, and the expected exponential convergence is observed.