#Solving (Cubic Function)
iron warrenBOT
narrow root
n+1 points define a unique polynomial of degree <=n
slim yew
Yeah
that would help
but how is it unique if the value of t in that image changes the equation?
narrow root
prove that the polynomial of degree <=n with n+1 solutions is 0: it's of a form (x-x1)(x-x2)...(x-xn)(x-xn+1)*quotient(x) so it's degree n+1 which can only be a 0, then say there are 2 different polynomials of degree <=n f(x) and g(x) which pass through n+1 same points, then f(x)-g(x) has n+1 solutions and is of degree <=n so by above is a 0
so there is no two different polynomials of degree <=n passing through n+1 same points
slim yew
prove that the polynomial of degree <=n with n+1 solutions is 0: it's of a form ...
Wait, here n refers to the degree of the polynomial?
narrow root
yup
slim yew
ohh, so for a polynomial of degree n you need n+1 points for a unique solution?
narrow root
yup
narrow root
don't forget to .close if there are no more questions
slim yew
.close
iron warrenBOT
Solved
strong lodge
don't forget to .close if there are no more questions
Is there a formular too calculate a,b,c and d easily?
narrow root
Is there a formular too calculate a,b,c and d easily?
wut?
craggy pebble
wut?
I'm pretty sure they meant is there a formula to calculate the coefficients $a, b, & c$ easily
upbeat roostBOT
saddayyy_
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narrow root
given the points?
radiant creek
given the points?
Yes, more or less
narrow root
well, use lagrange interpolation and get the polynomial from the points
radiant creek
The system of three equations you get is something like:
$ax_i^2 + bx_i + c = 0$
You get a "classic" matrix to invert to find a, b and c using the vandermonde determinant
upbeat roostBOT
Daddy_314
strong lodge
wut?
Just something like:
a= y0*y1/x1 or somthing.
A formular with the coordinates as the input and a,b,c or d as the output.
narrow root
its simple fractions lol
strong lodge
its simple fractions lol
But i´ve seen stuff where the use a matrix, loops and all of that fancy stuff.
How does calculating those unknowns, fractions and this fancy stuff connect?
narrow root
strong lodge
https://en.m.wikipedia.org/wiki/Lagrange_polynomial
But how do i bring a computer to calculate that for me?
radiant creek
Compute 'interpolate [(77,2),(71,3),(84,40)]' with the Wolfram|Alpha website (https://www.wolframalpha.com/input/?i=interpolate+[(77%2C2)%2C(71%2C3)%2C(84%2C40)]) or mobile app (wolframalpha:///?i=interpolate+%5B%2877%2C2%29%2C%2871%2C3%29%2C%2884%2C40%29%5D).
radiant creek
But how do i bring a computer to calculate that for me?
Like this
You ask it to interpolate the $n+1$ points into a degree $n$ polynomial
upbeat roostBOT
Daddy_314
strong lodge
This is just hello kitty
But it seams to interpolate the same, and also has a similar formular
But it still doesn´t show how it´s calculating the unknowns a,b,c and d
radiant creek
I already said above
How the coefficients can be explicitly obtained (by inverting the Vandermonde matrix)
strong lodge
What if i already got the derivitaves?
strong lodge
But you are changing the problem
I´m sorry for the confusion.
Here i got one point as an example:
location is x, Value is y and the derivitave is the derivitave
"location": -1.02,
"derivative": 0,
"value": -0.2222
strong lodge
What is the question
I´m trying to turn a bunch of points like this one and use it to modify a noise texture
"location": -1.02,
"derivative": 0,
"value": -0.2222
radiant creek
The derivative of what is zero
strong lodge
The derivative of what is zero
The derivitave of the point
radiant creek
A point doesnt have a derivative
radiant creek
Looks a bit like kernel densities
strong lodge
Looks a bit like kernel densities
It´s a list of spline points which edit a noise texture.