Essays/PolynomialsIntersection
Find the intersection of polynomials
Two polynomials
Given two polynomials, for example:
(a) (b)
how can we find their intersection?
The coefficients of the above polynomials from lowest to highest order are:
a=: 3 1 b=: 1 2 1
The dyadic form of the J primitive p. (Polynomial) evaluates a polynomial of order #x with coefficients x for the argument y. So for polynomial b is
1 2 1 p. _2 1
We can plot these polynomials as follows:
load 'plot' plot _3 3;'3 1 p. y ` 1 2 1 p. y' NB. or plot _3 3;'a p. y ` b p. y'
From the plot we can see that the two polynomials intersect when is _2 or 1. How can we get that result using J?
This is equivalent to finding the roots (the values for where a polynomial is zero) of a polynomial formed by subtracting polynomial a from polynomial b.
Firstly subtract one polynomial from another:
a (-/@,:) b 2 _1 _1
i.e. (c)
Then find the roots of the resulting polynomial c using the monadic form of the J primitive p. (Roots):
p. 2 _1 _1 ┌──┬────┐ │_1│_2 1│ └──┴────┘
The roots are contained in the 2nd box (the first contains the multiplier) so we can put these ideas all together to give a vector of the values where two polynomials intersect:
findIntersect=: 1 {:: [: p. -/@,: a findIntersect b _2 1
Where the roots of the polynomial formed by subtraction are complex, findIntersect will return the complex roots.
6 2 1 findIntersect 1 2 0j2.236068 0j_2.236068
If we only wished to return real roots we could extend findIntersect as follows:
6 2 1 (#~ (= +))@findIntersect 1 2
Two or more polynomials
The examples given above are probably the best way to find the intersection of 2 polynomials. If you want the common intersection of several, here is a less-succinct method.
You can test whether are all equal by forming the sum of and setting it to zero.
ppr =: +//.@(*/) NB. polynomial product pdiff =: -/@,: NB. polynomial difference pps =: ppr~ NB. polynomial square comb=: 4 : 0 k=. i.>:d=.y-x z=. (d$<i.0 0),<i.1 0 for. i.x do. z=. k ,.&.> ,&.>/\. >:&.> z end. ; z ) NB. findIntersectM <list of boxes of coefficients> NB. returns x-coordinates of common intersection points findIntersectM=:3 : 0 a=. >y c=. 2 comb #a ~. (#~ (= +)) 1{:: p. +/ pps@pdiff /"2 c { a )
For example to find the intersection of the polynomials:
(d) (e) (f)
d=: 0 0 1 e=: 2 0 _1 f=: 1
We can plot these as follows.
require 'plot' plot _3 3;'d p. y ` e p. y ` f p. y'
By inspection these polynomials intersect when is 1 or _1.
Given a list of boxed coefficients for each polynomial we can find the coordinates where they all intersect.
]r=: findIntersectM d;e;f 1 _1
We can evaluate each polynomial at those values to show that they all have the same .
d p. r 1 1 e p. r 1 1 f p. r 1 1
Contributions
This essay was compiled by Ric Sherlock from posts in this forum thread by John Randall, Raul Miller and Henry Rich.
See Also
- The Polynomials lab available from the J session (Studio | Labs... | Polynomials).