By Kenneth E. Iverson
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Additional resources for A Programming Language.
It should not be confused with the special null elenlent D. 8 REDUCTION An operation (such as sumnlation) which is applied to allcornponents of a vector to prod uce a result of a sirTI pier structure is called a reduction. The C-reduction of a vector x is denoted by (~)x and defined as z +- C)lx<¢:>z == ( ... ((Xl x 2) x:~) ... ) 0 X,,), where 0 is any binary operator with a suitable dornain. Thus + Ix is the surn, X Ix is the product, and V Ix is the logical sunl of the components of a vector x.
Clearly, x (\ y == Y (\ x, although x (\ y is not, in general, equal to y (\ x, since the conlponents nlay occur in a different order and may be repeated a differing number of tinles. The vector x (\ y is said to be ordered on x. Thus a is ordered on s. y·oint. The set diJference of y and x is denoted by y ~ x and deflned as follows: y ~ x == €yXjy. Hence y ~ x is obtained fronl y by suppressing those conlponents which belong to x. For exalnple, e/ == (1,0,0,0, I) and s Ll t == (s, d). More over, €/ == (0, 0, 0) and t ~ s == €(O).
A rOH' reduction of a matrix X by an operator () is denoted by Y~ ,:)/X and specifies a vector y of dimension fleX) such that Yi = c) Xi. A COIUJ11n reduction of X is denoted by Z ~ all X and specifies a vector z of dimension veX) such that Zj = c)/Xj • For example, if u= (~ ~ ~) I then +/U = (2,2,3), +//U = (2, L 3, I), A//U = (0,0, 1,0), (0,0,1), =//U = (0, I, I, I), and +/(=//U) = 3. 9 /)U == SELECTION Compression The effective use of structured operands depends not only on generalized operations but also on the ability to specify and select certain elenlents or groups of elements.
A Programming Language. by Kenneth E. Iverson