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By Kenneth E. Iverson

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Example text

For example, if A is an arbitrary 3 x 4 matrix, u === (0, 1,0, I) and v === (I, 0, 1); the n ul A === AI) (A/ A A: 2 2 A 23 A 43 2 (A A A A1) A~3 A A A> ' 41 (A ) A === A:3 A 1 , 2 vilA === 1 3 3 1 3 3 2 1 and ulvll A === vllul A 3 4 It is clear that rOlf compression suppresses columns corresponding to zeros of the logical vector and that column compression suppresses rOlfS. This illustrates the type of confusion in nomenclature which is avoided by the convention adopted in Sec. 3: an operation is called a rOH' operation if the underlying operation from which it is generalized is applied to the row vectors of the matrix, and a column operation if it is applied to columns.

The interred rector lJ(n) is defined as the vector of integers beginning \vith j. Thus l O( 4) == (0, 1, 2, 3), II ( 4) == (1, 2, 3, 4) , and l-' ( 5) == ( - 7, - 6, - 5, -4, -3). Four types of logical vectors are defined as follo\vs. The jth lInit rector €J(n) has a one in the )th position, that is, (€J(n))k == (k == j). Thejul! rector €(n) consists of all ones. The vector consisting of all zeros is denoted both by and by "€(n). The prefix rector (~llt'eight j is denoted by a/(n) and possesses ones in the fIrst k positions, where k is the lesser ofj and n.

The importance of the latter characteristic in developing clear and comprehensible programs is not sufficiently appre­ ciated. 2 The language y ~O 2 k ~ 3 x ~y 4 5 1 x y z~yxx > z . 4) which includes one or more iterated subprocesses is said to be iteratire. 2 PrograJJlS 5 defined in the usual way as v(A) C/ = L Akl x B/" k=l where the dimension of an fn columns) is denoted by p(X) X n x ri = 1,2, ,p,(A), lj = 1,2, , v(B), rectangular nlatrix X (of 111 rows and n veX). 5. Steps 1-3 initialize the indices, and the loop 5-7 continues to add successive products to the partial sum until k reaches zero.

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