#826 prod for calculate product similarly to sum

[Shigeharu TAKENO]

Attached patch is an implement of "prod" which calculates product of numbers similarly to sum.

[Ethan Merritt]

I can see that this implementation should be sufficient for simple cases.

This is not my area of expertise, so I do not feel confident of judging whether it is sufficient or optimal as the number of terms becomes very large or the magnitude of the individual terms becomes large.

I worry that sequential multiplication can lead to overflow more often than sequential addition. There also may be concerns about loss of precision. For example, I wonder if there is a good way to decide when it would be better to evaluate a sum of logarithms instead?

Product(start, end, <expression>)
    = exp( Sum(start, end, log(<expression>)) )

Does anyone know what other programs do? R? Mathematica? Octave?

[Shigeharu TAKENO]

I think the implementation by exp and log is not sufficient because each term may has minus sign.
For example: Lagrange interpolation:

array a[5] = [1.0, 2.0, 4.0, 8.0, 9.0]
array b[5] = [2.0, 5.0, 1.0, 3.0, 2.0]
f(x,a,b,n) = sum [j=1:n] (prod [i=1:j-1] (x-a[i])/(a[j]-a[i]))*(prod [i=j+1:n] (x-a[i])/(a[j]-a[i]))*b[j]
set xrange [0:10]
set xtics 1
set grid
plot f(x,a,b,5)

[Ethan Merritt]

Good point about handling terms that are less than zero.

Another question:
What should happen if the end is less than the start?
For example
A = prod [ i = 5 : 4 ] i

The code in the patch sets A to 1.0, but this is non-intuitive.
I would expect it to set A to 5.0 or to report an error and set A to NaN or <undefined>.</undefined>

Alternatively, the code could rearrange the terms so that the sequence of multiplications runs from min(start,end) to max(start,end). But this would be different from what "sum" currently does.

[Shigeharu TAKENO]

I think it is reasonable that "prod [i=5:4] i" returns 1, since it is reasonable that "sum [i=5:4] i" returns 0.

This behavior is also used in the Lagrange interpolation I showed, and it may be useful.

[Ethan Merritt]

OK.
Added to the git master by commit 2b27736ab8
I added a description in the documentation.