polynomial

Polynomials are objects that represent expressions like \(x^2+6x+9\).
Where the expression its a series of coefficients using the "standard" power series as its basis.

Polynomial objects in numluau are a conveniance class that allows you to perform algebraic, differential and integral operations on that series.

Each coefficient is sorted in ascending power, so a polynomial like \(1x^2+2x+3\) is implemented as:

const poly = numluau.polynomial({3,2,1})

evaluation¶

we can treat polynomials as functions, where we can subsituite x for a given value.

const f = numluau.polynomial({9,6,1})
const A = f(5) -- 64

const B = f(1,2,3,4,5) -- array([16 25 36 49 64])
const C = f(numluau.linspace(-2,2,10)) -- array([1 2.1 3.6 5.4 7.7 10 13 17 21 25])

operations¶

differentiation & integration¶

Unlike numluau.gradient() or numluau.cumsum() which preform integration and differentiation using numerical methods. We can do these same operations on polynomials algebraically.

local f = numluau.polynomial({0,0,1}) -- f(x) = x²
local x = numluau.linspace(-2,2,10)

local df = f:deriv() -- f'(x) = 2x
local dy = df(x) -- array([-4 -3.1 -2.2 -1.3 -0.44 0.441.3 2.2 3.1 4])

local x = numluau.linspace(-2,2,10)
local df = numluau.gradient(x ^ 2) -- array([-1.6 -1.4 -0.99 -0.59 -0.2 0.2 0.59 0.99 1.4 1.6])

While doing it numerically looks simpler, polynomial differentiation allows us to evaluate the derivative at a single point. numluau.gradient cant do this as it relies on there being multiple values.

local f = numluau.polynomial({0,0,1}) -- f(x) = x²
local df = f:deriv()

df(5)   -- 10
df(512) -- 1024
df(-20) -- -40

polynomial:integ also works the same way, instead if finds the anti derivative of the polynomial.

local f = numluau.polynomial({0,0,2}) -- f(x) = x²
local x = numluau.linspace(-2,2,10)

local F = f:integ() -- F(x) = x^3 / 3
local Y = F(x)

arithmetic¶

We can also do polynomial arithmetic using the standard +-/* etc. operations

local polyA : numluau.Polynomial = numluau.polynomial({9,6,1})
local polyB : numluau.Polynomial = numluau.polynomial({3,1})

print(polyA + polyB)  -- 12 + 7x + 1x^2
print(polyA - polyB)  -- 6 + 5x + 1x^2
print(polyA * polyB)  -- 27 + 27x + 9x^2 + 1x^3
print(polyA // polyB) -- 3 + 1x

print(polyA * 2) -- 18 + 12x + 2x^2
print(polyA / 2) -- 4.5 + 3x + 0.5x^2

interpolation¶

In data science we might have a collections of points, we may want to fit a function to those points. This allows us to notice trends and interpolate/extrapolate missing data.

In numluau this is done using numluau.polynomial.fit().

local x = numluau.array({1,2,3,4,5,6})
local y = numluau.array({2,4,6,8,10,12})

local fitted = numluau.polynomial.fit(y,x,1) -- 1.3 + 2.72x

we can raise the degree to get a tighter fit

local x = numluau.array({-4,-3,-2,-1,0,1,2,3,4,5})
local y = x * 2

local fitted = numluau.polynomial.fit(x,y,2) -- -0.02 + 1.73x - 0.02x^2

solving polynomial equations¶

If we had a equation like \(x^2-6x+9=0\), we can simply just use the quadratic formula to find its roots.

local a = 1
local b = -6
local c = 9

local x0 = (-b + math.sqrt(b^2 - 4 * a * c)) / (2 * a) -- 3
local x1 = (-b - math.sqrt(b^2 - 4 * a * c)) / (2 * a) -- 3

but for an equation like \(x^3+9x^2+27x+27=0\) or even \(x^4-10x^3+35x^2-50x+24=0\), you'd need more complicated methods for solving these equations.
numluau solves this by using polynomial:roots().

local poly_A = numluau.polynomial({27,27,9,1})
local result_A = poly_A:roots() -- array([-3 -3 -3])

local poly_B = numluau.polynomial({24,-50,35,-10,1})
local result_B = poly_B:roots() -- array([4 3 2 1])