Extending results of Grossman and Zeitman [1] on an inherently iterative algorithm for the classical Ackermann’s function, the author develops an analogous algorithm for the more logically interesting Grzegorczyk hierarchy [2]. Recall that the diagonal function for the Grzegorczyk hierarchy is yet another deep example of an effectively calculable function that is not primitive recursive. As a bonus, the author’s iterative algorithm results in a space-complexity function and a time-complexity function whose orders of growth improve upon those for the (more obvious) directly implemented algorithm. This note is another interesting contribution to the theory of general recursive functions with super-rapid growth.