As a result of its high numerical accuracy and versatility to include complex tool configurations and arbitrary spatial distributions of material properties, the Monte Carlo method is the foremost numerical technique used to simulate borehole nuclear measurements. Although recent advances in computer technology have considerably reduced the computer time required by Monte Carlo simulations of borehole nuclear measurements, the efficiency of the method is still not sufficient for estimation of layer-by-layer properties or combined quantitative interpretation with other borehole measurements. We develop and successfully test a new linear iterative refinement method to simulate nuclear borehole measurements accurately and rapidly. The approximation stems from Monte Carlo-derived geometric response factors, referred to as flux sensitivity functions (FSFs), for specific density and neutron-tool configurations. Our procedure first invokes the integral representation of Boltzmann's transport equation to describe the detector response from the flux of particles emitted by the radioactive source. Subsequently, we use the Monte Carlo N-particle (MCNP) code to calculate the associated detector response function and the particle flux included in the integral form of Boltzmann's equation. The linear iterative refinement method accounts for variations of the response functions attributable to local perturbations when numerically simulating neutron and density porosity logs. We quantify variations in the FSFs of neutron and density measurements from borehole environmental effects and spatial variations of formation properties. Simulations performed with the new approximations yield errors in the simulated value of density of less than with respect to Monte Carlo-simulated logs. Moreover, for the case of radial geometric factor of density, we observe a maximum shift of at 90% of the total sensitivity as a result of realistic variations of formation density. For radial variation of neutron properties (migration length), the maximum change in the radial length of investigation is . Neutron porosity values simulated with the new approximation differ by less than 10% from Monte Carlo simulations. The approximations enable the simulation of borehole nuclear measurements in seconds of CPU time compared to several hours with MCNP.