# Essays/PolynomialsIntersection

## Contents

## 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 c;d;e 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, RaulMiller and Henry Rich.

## See Also

- The Polynomials lab available from the J session (
`Studio | Labs... | Polynomials`).