The algorithm for proven and young animals (APY) implements the inverse of the genomic relationship matrix by recursion on a subset of animals. If the subset is small, storage and computations are approximately linear with the number of genotyped individuals, allowing for processing of practically an unlimited number of animals. The APY algorithm was tested with many subsets, including proven bulls, bulls and cows, cows only and random subsets. GEBVs calculated with APY were accurate when the number of animals in the subset was ≥10k, with little difference between different subsets. Best convergence rates when solving equations with APY were obtained with subsets composed of randomly chosen animals. The properties of the APY algorithm can be explained using the concept of a finite number of independent chromosome segments. Assume that each segment has a fixed value, that a fraction of each segment has a value proportional to the length of that segment (infinitesimal model), and that a genome of an individual is composed of a fraction of each segment. Subsequently, in the absence of confounding, n animals allow for identification of a population with n segments. Assuming some errors, lowest estimation errors are with heterogeneous animals. For traits where genes are distributed unequally across the genome, a conceptual division of segments into smaller segments with quasi-equal distribution would result in a larger subset required for the same accuracy of GEBV. If genome sequencing allows for identification of all m QTN, each QTN may be treated as one segment and, assuming a purely additive model, a recursion on m sufficiently heterogeneous animals will capture all variability in the genome. Computation in APY assume ability to compute parts of such a genomic relationship matrix that reflects the genetic architecture for each trait. The APY algorithm may allow routine genomic evaluation for any number of genotyped animals with any model at a cost not much above BLUP.