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The variationally stable procedure for Nth-order perturbative-transition-matrix elements introduced by the authors [Phys. Rev. Lett. 61, 404 (1988)] is presented here in detail. Its key features are that it is noniterative, involves only two unknown functions regardless of the value of N, is stable even near intermediate-state resonances (due to the presence of energy numerators rather than energy denominators), and uses the inverse of the perturbation operator for N≥3. Explicit formulas are presented for application to high-order multiphoton processes in atomic hydrogen. Numerical results for multiphoton-ionization cross sections (for two, three, and seven photons), for the frequency dependence of the nonlinear susceptibilities for harmonic generation (for the third, fifth, and seventh harmonics), and for harmonic-generation transition rates up to 11th order for λ=1064 nm are presented for the atomic hydrogen ground state and compared with results of others by more standard procedures. We also present detailed analyses of the Z scaling of our results for hydrogenic systems, of the use of the imaginary part of the appropriate 2N-photon nonlinear susceptibility to obtain N-photon-ionization cross sections, and of the relation of the variational principle presented here to variational principles for scattering processes of Nuttall and Cohen [Phys. Rev. 188, 1542 (1969)] and of Schwinger [Phys. Rev. 72, 742 (1947)].