Conversion between RGB and RYB color spaces

Here is a category on UIColor that does the same thing, returning elements between RGB, RYB, and CMYK. Further, you can mix any number of colors in the respective color space (they mix differently, of course, depending).

https://github.com/ddelruss/UIColor-Mixing

I found this JavaScript implementation of RYB->RGB conversion based on cubic splines. Here is my Lua port (all values lie in the interval 0-1):

local ryb2rgb = function( R, Y, B ) 
  local R, Y, B = R*R*(3-R-R), Y*Y*(3-Y-Y), B*B*(3-B-B)
  return 1.0 + B * ( R * (0.337 + Y * -0.137) + (-0.837 + Y * -0.163) ),
    1.0 + B * ( -0.627 + Y * 0.287) + R * (-1.0 + Y * (0.5 + B * -0.693) - B * (-0.627) ),
    1.0 + B * (-0.4 + Y * 0.6) - Y + R * ( -1.0 + B * (0.9 + Y * -1.1) + Y )
end

I managed to solve it in the end.

Take the equations for a trilinear interpolation: wikipedia Edit: Wikipedia revision at the time

Substitute the first equations into the last, the expand and collect the coefficients for: Xd, Yd, Zd, XdYd, XdZd, YdZd, ZdYdZd and the constant.

Then find the partial differentiation of the equation in each of the 3 dimensions each in respect to Xd, Yd and Zd. Use these new equations to populate the (3x3) Jacobian matrix and then use Newton's method to solve in software.

Newton-Raphson Method