A vector of polynomials each defined as a function

Question

I'm trying to get a vector of polynomials, but within the vector have each polynomial defined by a function in Pari.

For example, I want to be able to output a vector of this form: [f(x) = x-1 , f(x) = x^2 - 1, f(x) = x^3 - 1, f(x) = x^4 - 1, f(x) = x^5 - 1]

A simple vector construction of vector( 5, n, f(x) = x^n-1) doesn't work, outputting [(x)->my(i=1);x^i-1, (x)->my(i=2);x^i-1, (x)->my(i=3);x^i-1, (x)->my(i=4);x^i-1, (x)->my(i=5);x^i-1].

Is there a way of doing this quite neatly?

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Update:

I have a function which takes a polynomial in two variables (say x and y), replaces one of those variables (say y) with exp(I*t), and then integrates this between t=0 and t=1, giving a single variable polynomial in x: int(T)=intnum(t=0,1,T(x,exp(I*t)))

Because of the way this is defined, I have to explicitly define a polynomial T(x,y)=..., and then calculate int(T). Simply putting in a polynomial, say int(x*y)-1, returns:

*** at top-level: int(x*y-1) *** ^---------- *** in function int: intnum(t=0,1,T(x,exp(I*t))) *** ^-------------- *** not a function in function call *** Break loop: type 'break' to go back to GP prompt

I want to be able to do this for many polynomials, without having to manually type T(x,y)=... for every single one. My plan is to try and do this using the apply feature (so, putting all the polynomials in a vector - for a simple example, vector(5, n, x^n*y-1)). However, because of the way I've defined int, I would need to have each entry in the vector defined as T(x,y)=..., which is where my original question spawned from.

Defining T(x,y)=vector(5, n, x^n*y-1) doesn't seem to help with what I want to calculate. And because of how int is defined, I can't think of any other way to go about trying to tackle this.

Any ideas?

Answer 1

The PARI inbuilt intnum function takes as its third argument an expression rather than a function. This expression can make use of the variable t. (Several inbuilt functions behave like this - they are not real functions).

Your int function can be defined as follows:

int(p)=intnum(t=0, 1, subst(p, y, exp(I*t)))

It takes as an argument a polynomial p and then it substitutes for y when required to do so.

You can then use int(x*y) which returns (0.84147098480789650665250232163029899962 + 0.45969769413186028259906339255702339627*I)*x'.

Similarly you can use apply with a vector of polynomials. For example:

apply(int, vector(5, n, x^n*y-1))

Coming back to your original proposal - it's not technically wrong and will work. I just wouldn't recommend it over the subst method, but perhaps if you are were wanting to perform numerical integration over a class of functions that were not representable as polynomials. Let's suppose int is defined as:

int(T)=intnum(t=0,1,T(x,exp(I*t)))

You can invoke it using the syntax int((x,y) -> x*y). The arrow is the PARI syntax for creating an anonymous function. (This is the difference between an expression and a function - you cannot create your own functions that work like PARI inbuilt functions)

You may even use it with a vector of functions:

apply(int, vector(5, n, (x,y)->x^n*y-1))

I am using the syntax (x,y)->x^n*y-1 here which is preferable to the f(x,y)=x^n*y-1 you had in your question, but they are essentially the same. (the latter form also defines f as a side effect which is not wanted so it is better to use anonymous functions.