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By Kenneth E. Iverson

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Congruency is transitive and reflexive and is denoted by In == n (nlod b). In classical treatments, such as Wright (1939), only the O-residue is considered. The use of I-origin indexing (cf. Sec. 5) accounts for the interest of the I-residue . , in a base ten or a base two nUlnber system) Inust, in practice, employ only a finite nunlber of digits. It is therefore often desirable to approxilnate a nunlber or by an integer. For this purpose two functions are defined: I. r, 2. r. Thus f3. 00J == -3.

Upon register addresses or other numeric synlbols) in the automatic execution of programs, the mapping is frequently performed in three successive phases, determining in turn the following quantities: 1. the index i = b l h, 2. the index k such that ale 3. the element ale. I2a. The ranking is performed (steps 1-3) by scanning the set b in order and comparing each element with the argument b. The second phase is a permutation of the integers 1,2, ... , v(b), which may be described by a permutation vector j, such that Ii = k.

Compression by the full matrix E (defined by E == 0) produces either a r()\\, list (EI A) or a colurnn list (Ell A) of the matrix A. Moreover, a nunleri­ cal nlatrix X can be represented jointly by the logical nlatrix V and the row list VIX (or the column list VIIX), where V == (X 0). , the components are predominantly zero), this provides a conlpact representation which may reduce the computer storage req uired for X. The compression operations controlled by matrices also generate a group of corresponding mesh and mask operations as shown in Sec.

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