This primer includes example reductions.
An array is filled with random values and the locations of the
maximum and minimum are found. The Euclidean norm of the array’s
columns is computed using
+ reductions over slices of
&& reduction is used to compute whether all values in
greater than 0.25 and the results of the computations are printed.
For an example usage of the standard module Random, see its primer.
use Random; // For random number generation config const seed = 31415; // Random generation seed config const size = 10; // The size of each side of the array var A: [1..size, 1..size] real; // The 2D work array
Fill an array with random values¶
A with random real values between 0 and 1.
Uses the NPB random number generator for historical reasons.
fillRandom(A, seed, algorithm=RNG.NPB); writeln("A is: "); writeln(A); writeln();
Find the average value of an array¶
We find the average value of the array by summing over its elements and dividing by the number of elements it contains.
var eltAvg = (+ reduce A) / size**2; writeln("The average element of A has the value ", eltAvg); writeln();
Find the 1-norm of an array¶
We can find the 1-norm of
A by summing over the absolute value of the
abs(A) creates a new matrix which contains in each of its
elements the absolute value of the corresponding element in
+ reduce clause just sums these up.
var oneNorm = + reduce abs(A); writeln("The 1-norm of A is ", oneNorm); writeln();
The Frobenius norm¶
The Frobenius norm is the square root of the sum over all elements of their respective squares.
The expression below can be broken down thus:
The Frobenius norm is the square root of some quantity.
The quantity is the sum over all elements of a matrix.
That matrix is the promotion of
**2. That is, each of its elements is the square of the corresponding element of
var frobNorm = sqrt(+ reduce A**2); writeln("The Frobenius norm of A is ", frobNorm); writeln();
maxloc reductions. We capture the results into
minloc reductions expect a 2-tuple argument
that can be iterated over using zippered iteration. They produce
a 2-tuple result. The first component of the result is the
maximum (or minimum) over the first component of the argument.
The second component of the result indicates its location, i.e.
the corresponding element of the second component of the argument.
var (maxVal, maxLoc) = maxloc reduce zip(A, A.domain); var (minVal, minLoc) = minloc reduce zip(A, A.domain); writeln("The maximum value in A is: A", maxLoc, " = ", maxVal); writeln("The minimum value in A is: A", minLoc, " = ", minVal); writeln("The difference is: ", maxVal - minVal); writeln();
The Euclidean norm¶
Compute Euclidean norms for each column using
Breaking down the statement below:
vecNormsis a 1-D array containing
sizeelements (indexed by
The elements of
vecNormsare the square-root of some quantity.
The quantity is the sum over all of the elements of some vector.
The vector consists of the promotion of the j-th column of
**2. That is, each element of that column vector is squared.
var vecNorms = [j in 1..size] sqrt(+ reduce A(1..size, j)**2); writeln("The Euclidean norm of each column is: ", vecNorms); writeln();
&& reduction to compute if all values in
A are greater
The parenthesized value is the promotion of
> 0.25. This yields
an array of the same size as
A, containing boolean values that are true
if the corresponding element in
A exceeds 0.25 and false otherwise.
&& reduce computes the result of applying the Boolean AND
operator between all elements of the promoted array.
var onlyBigValues = && reduce (A > 0.25); if onlyBigValues then writeln("The values in A are all greater than 0.25"); else writeln("Some values in A are less than 0.25"); writeln();