Vector vol. 23 Nos. 1&2 Contents Editorial Stephen Taylor 2 Sustaining Members News
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- Navigate this page:
- From Foundation to Structure
- Take and drop
- Mirrors and Transpositions
- Back to Primary School
GeneralisationIn truth, the expression we just wrote is an example of an algorithm for “changing the frame of reference”. Not to panic, the name is recondite, but the concept is simple: a list of area numbers (the initial set) is translated into a list of discount rates (the final set). Let us imagine the initial set to be an alphabet composed of lower case and upper case letters, and the final set to be composed of only upper case letters: Almin
Almaj
Fable ← 'Le petit Chaperon-Rouge a bouffé le Loup' The expression below converts from lower to upper case. Almaj[Almin ⍳ Fable]
As one might expect, the characters - and é, which are absent from the initial alphabetic set have been replaced by the * of the final set, but the conversion is acceptable. This solution can easily be improved. Once more, the rational steps to be taken to resolve a computing problem are entirely different from the ways traditionally taught, and the programmer has thereby gained a much more extensive insight.
It is quite otherwise with APL, which offers many tools for working with the shape of the data. We shall only look at a few of them. Take and dropThe functions Take (↑) and Drop (↓) serve, as their name suggests, to take or drop part of a variable. Here we shall show only examples based on vectors, but all the other shapes of data can be treated in a similar way. Recalling that Vec has values 15 40 63 18 27 40 33 29 40 88 4 ↑ Vec ⍝ take the first 4 items
5 ↓ Vec ⍝ drop the first 5 items 40 33 29 40 88 If the left argument is negative, these same functions count from the end of the vector. ¯3 ↑ Vec ⍝ take the last 3 items
¯7 ↓ Vec ⍝ drop the last 7 items 15 40 63 If one drops the last 7 items, it leaves only the first 3, which we could have accomplished with 3↑Vec. To have two functions seems unnecessary. What purpose does this serve? Let us imagine a business with a turnover which has grown over 12 years. The variable Tome is turnover in Millions of Euros. Tome
We want to calculate the difference between each year and the next; how to do it? 1↓Tome
¯1↓Tome
We see that all that remains is to subtract item from item: (1↓Tome)-(¯1↓Tome)
Without a program or loops; all very simple! In place of a subtraction, one division calculates the rates of growth instead of the differences, with some obvious adjustments: 100 × ((1↓Tome)÷(¯1↓Tome)) – 1 5.35 13.56 ¯4.48 ¯6.25 1.67 11.47 7.35 6.85 ¯3.85 8 3.70 Mirrors and TranspositionsAPL is also well endowed with functions which pivot data about any axis, as suggested by the shape of the symbol used. It applies both to numeric and text data; as we are going to show by applying these functions to the variable Towns met above.
The symbols used ⌽ ⊖ ⍉ are self-describing, no effort is required to remember any of them. They also have dyadic uses, with different but always interesting results. Back to Primary SchoolRemember when we learned our multiplication tables. In that practically palaeolithic era, to make sure that we knew all our tables, my instructor made us calculate the multiplication table for the integers 1 to 9:
You see, I haven’t forgotten! Probably you have done all this just like me. And then we quickly forgot the imposition: thus sidestepping a very powerful tool, one which APL provides, under the name outer product. The task consists of taking pairs of items of two vectors, (the column and row headings) and making them the left and right arguments of the function at the top left. Then we shall go on to see what we get if we change the values a little:
This operator is written thus in APL: 5 4 10 3 ∘.× 8 5 15 9 11 40 40 25 75 45 55 200 32 20 60 36 44 160 80 50 150 90 110 400 24 15 45 27 33 120 Now imagine replacing the symbol for multiplication by any of a number of other functions, or programs which you could have defined yourself, and you will understand, as for reduction already encountered, that outer product is an operator of amazing power.
And go on to practical applications. Directory: issues issues -> Protecting the rights of the child in the context of migration issues -> Submission for the Office of the High Commissioner for Human Rights (ohchr) report to the General Assembly on the protection of migrants (res 68/179) June 2014 issues -> Human rights and access to water issues -> October/November 2015 Teacher's Guide Table of Contents issues -> Suhakam’s input for the office of the high commissioner for human rights (ohchr)’s study on children’s right to health – human rights council resolution 19/37 issues -> Office of the United Nations High Commissioner issues -> The right of persons with disabilities to social protection issues -> Human rights of persons with disabilities issues -> Study related to discrimination against women in law and in practice in political and public life, including during times of political transitions issues -> Super bowl boosts tv set sales millennials most likely to buy Download 0.53 Mb. Share with your friends:
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