RogerBW's Blog

The Weekly Challenge 170: Kronecker's Primorial 23 June 2022

I’ve been doing the Weekly Challenges. The latest involved primes and an unusual form of matrix multiplication. (Note that this is open until 26 June 2022.)

Task 1: Primorial Numbers

Write a script to generate the first 10 Primorial Numbers.

In other words a sort of "prime factorial": 2, 2×3, 2×3×5, etc. (though by convention the list starts at 1). This was touched on in

155.1, though then I at least didn't split out their generation

into a separate function.

I reuse my existing code to generate a list of primes. To keep the list to a manageable size, I also bring back nthprimelimit that I originally wrote for #146.1, i.e. a function that reasonably cheaply returns an upper bound for the nth prime number. (So if I'm looking at the first ten primorial numbers, I need the first nine primes; the function gives me a limit of 27, so I generate only the primes up to that.)

Raku:

sub nthprimelimit($n) {
    my $m=15;
    if ($n >= 6) {
        $m=floor(1+$n*log($n*log($n)));
    }
    return $m;
}

Then the main body of the work becomes relatively simple.

sub primorial($ct) {

Initialise the output list. (1 not being prime or the product of primes.)

    my @o=(1);

For each of the primes…

    for genprimes(nthprimelimit($ct)) -> $p {

Multiply the last value in the list by the new prime, and push it on.

        push @o,@o[*-1] * $p;

The nth prime limit function isn't exact, so it's possible that I'll get more primes than I actually need. So bail out if I've got enough.

        if (@o.elems >= $ct) {
            last;
        }
    }
    return @o;
}

This looks pretty much the same in all the other languages, except Ruby – for I already have the prime library which gives me an infinite sequence of primes reasonably cheaply. And in Perl, I'm finally giving up on my home-grown primality code and moving over to Math::Prime::Util with its next_prime function:

sub primorial($ct) {
  my @o = (1);
  my $lp = 1;
  while (scalar @o < $ct) {
    $lp = next_prime($lp);
    push @o,$o[-1] * $lp;
  }
  return \@o;
}

Task 2: Kronecker Product

You are given 2 matrices.

Write a script to implement Kronecker Product on the given 2 matrices.

This turns out to be a sort of combinatorial multiplication: the second matrix is multiplied linearly by each element in the first, and the results arranged matching those elements. Which gives a way of working out how to do it: if the two input matrices have size (ax, ay) and (bx, by), then for each set of output coordinates (x, y), the result will be the product of the element of a at (x / bx, y / by) and the element of b at (x % bx, y % by). (Assume integer division of course. And this is one of the few cases where Lua's 1-based indices make a difference; they aren't as annoying in general as I thought they'd be, but they certainly are here.)

I use actual variables for ax etc. for clarity, and assume that all rows are of equal length in the input. Rust is one of the majority of languages I'm using here that lets me say "this variable is an integer, dammit, and everything involving it will also be an integer unless I tell you otherwise", so has no need for a special integer-division operator.

fn kronecker(a: Vec<Vec<usize>>, b: Vec<Vec<usize>>) -> Vec<Vec<usize>> {
    let mut o = Vec::new();
    let ax = a[0].len();
    let ay = a.len();
    let bx = b[0].len();
    let by = b.len();
    for y in 0..ay * by {
        let byi = y % by;
        let ayi = y / by;
        let mut row = Vec::new();
        for x in 0..ax * bx {
            let bxi = x % bx;
            let axi = x / bx;
            row.push(a[ayi][axi] * b[byi][bxi]);
        }
        o.push(row);
    }
    o
}

Full code on github.

See also:
The Weekly Challenge 146: Curious Prime
The Weekly Challenge 155: Pisano's Fortune

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