Interpolation

bezier(x, y, xguides, yguides)

An application of Hermitic polynomials to draw Bezier curves between points.

Notes

Each argument should be a one-to-one mapping of points, (xᵢ, yᵢ) and (xᵢ₊₁, yᵢ₊₁) and their respective guide points, (xᵢ⁺, yᵢ⁺) and (xᵢ₊₁⁻, yᵢ₊₁⁻).

clamped(x, f, fp)

The bookend polynomials will have the same slope entering and exiting the interval as the derivative at the respective endpoint.

lagrange(x, f[; n=nothing])

Given a domain, x and range, f, construct the nth Lagrangian polynomial.

Notes

If n=nothing, then method will utilize entire dataset. Polynomials will quickly oscillate for larger datasets.

linearinterpolation(x0, y0, x1, y1, x)

$y = y₀ + (x - x₀)*(y₁ - y₀)/(x₁ - x₀)$

linearinterpolation(x, y, p)

Calls linearinterpolation with the first index in x less than and greater than p. If for any p ∈ x, then the first occurrence in y is returned.

linearleastsquares(x, f, n::Int64)

Construct a polynomial of degree, n while minimizing the least squares error.

Notes

Least squares error := $E = \sum_{i=1}^{m}[y_{i} - P_{n}(x_{i})]^{2}$

Constructed polynomial of the form: $P(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + \dots + a_{1}x + a_{0}$

linearleastsquares(x, f, type::Symbol)

Given a domain and range, yield the coefficients for an equation and the equation of the form $y = ax^{b}$.

natural(x, f)

The bookend polynomials do not assume the slope entering and exiting the interval as the derivative at the respective endpoint.

newtondifference(x, f, α[; dir::Symbol=:auto])

Given a domain, x and range, f, construct some polynomial by Newton's Divided Difference centered around α. :forward or :backward construction.

Notes

Direction will be chosen if not specified. Polynomials best made with even spacing in x; although, this is not completely necessary.