By Kenneth E. Iverson
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Extra info for A Programming Language.
Lb) = a n b. * For the purposes of describing algorithms, this notation is superior to the classical "disjoint cycles" notation for permutations [cf. Birkhoff and MacLane, (1941)] because (1) the direction of the transformation (from a to c) is unequivocally indicated, and (2) the notation directly indicates a straightforward and efficient method for actual execu tion, namely, indirect addressing. 17 MapjJing and jJermutations 33 Consequently, if p, q, ... , t are vectors, each contained in b, then each can be represented jointly by the vector b and a mapping vector.
Then (3€) 10 II == (1,2,0, 1,2,0, ... ), and consequently the vector u specified by step 1 is of the form u == (0, 1, I, 0, 1. 1, 0, ... ). Mesh operations on matrices are defined analogously, row mesh and column mesh being denoted by single and double reverse virgules, respec tively. The catenation of vectors x, y, ... , z is denoted by x y z and is defined by the relation x y EB ... EB z == (Xl' x 2, ... , XJ'(X)' YI' Y2' ... , ZI'(Z))' Catenation is clearly associative and for two vectors x and y it is a special case of the mesh \x, u, y\ in which u is a suffix vector.
The conditions on the branches of a properly defined program must be disjoint and exhaustive. 3 illustrates the use of a branch point. Statement ~5 is a comparison which determines the branch to statements /1 I, () I, or }' I, according as z > n, z == n, or z < n. The program represents a crude but effective process for determining x == n'2:\ for any positive cube n. 4 shows the preceding program reorganized into a compact linear array and introduces two further conventions on the labeling of branch points.
A Programming Language. by Kenneth E. Iverson