We first look at computational problems programmed using loops over numbers and arrays. A loop is a command for repeating a sequence of operations. A number is an integer or a real number with an internal representation in a computer. An array is an arrangement of computer memory elements in one or more dimensions. Problems involving arithmetic on numbers were the first problems for which computing devices were built. Problems involving operations on arrays were at the center of many subsequent larger computer applications. Because nontrivial computations involve performing operations iteratively, loops are a most commonly used, most important construct in programming solutions to problems.

Clear and straightforward problem solutions tend to have expensive computations in loops, where the values that these computations depend on are updated slightly in each iteration. To improve efficiency, the results of these computations can be stored and incrementally maintained with respect to updates to the values that they depend on. The transformation of programs to achieve this is called incrementalization.

We will use a small example and several variations to explain the basic ideas of incrementalization. We then describe two larger examples: one in hardware design, to show additional loop optimizations enabled by incrementalization, and the other in image processing, to show handling of nested loops and arrays in incrementalization. We discuss the need for higher-level languages at the end.