The primitive p. is defined
(x p. y) = +/x*y^i.#x
for all open x .
Its inverse, according to the interpreter, is undefined:
6 5 4 p. 2
32
6 5 4 p.^:_1: 32
|domain error
| 6 5 4 p.^:_1:32
What are the major reasons for this? That is, given:
z =: x p. y
why can't we have :
y = x p.^:_1 z
?
I'm no mathematician, but it seems to me that producing y from x and z is
just a matter of solving a polynomial, and we already have a polynomial solver.
In broad strokes, we could do something like this:
x =: 6 5 4
y =: 2
] z =: x p. y
32
p. (({.x)-z),}.x
+-+-------+
|4|_3.25 2|
+-+-------+
pInv =: ((-~ {.) p.@, [EMAIL PROTECTED])~
x pInv z
+-+-------+
|4|_3.25 2|
+-+-------+
y e. > {: x pInv z
1
Now, if p.'s inverse has been left undefined because it's multivalued, then
we have a few options:
(A) Give all the values, as pInv does. But this loses the value of
things like (-: ]&.(x&p.)) .
(B) Select one input. This has the drawback of biasing results, but
(i) That's not so bad. For example, %: is multivalued but only one
value is selected.
(ii) Users could always substitute the user-written pInv for p.^:_1
if they always wanted a result.
To (B) , if we had to select a single output, I'd say pick the "normalest"
one. Prefer real to complex,
positive to negative, whole to fractional, etc.
Of course, there are still problems with pInv :
12 pInv 32
|domain error: pInv
| 12 pInv 32
and probably more that I havent noticed, which may be the reason it wasnt
implemented in the first place.
-Dan
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