Question

I am trying to solve the inverse problem for the following function in R.

x + 2 (C1 * y) + C1 * C1 * z = d2

I can currently enter C1 and get d2 but need to enter d2 and get C1. The variables x, y and z are all known and never change.

I already have some known C1 and d2 values to use.

 C1     d2 
 5   0.000316
 0   0.000193
-5   0.000123

Is there an R function which will allow me to enter the function, previous results and a d2 value and for it return the C1 coefficient?

Answer 1

You have a quadratic equation of the form:

(x - d2)*C1^0 + (2*y)*C1^1 + (z)*C1^2 = 0

You can solve quadratics (and in fact any polynomial equation) with the function polyroot() in R:

x <- 1
y <- 2
z <- 3

d <- 0

polyroot(c(x-d, 2*y, z))
[1] -0.3333333+0i -1.0000000+0i

(Which gives two solutions, as you would expect)

To solve for a range of input values, you need to put this into your favourite apply function, in this case sapply():

d <- seq(0, 1, 0.2)

sapply(d, function(dd)polyroot(c(x-dd, 2*y, z)))

              [,1]          [,2]          [,3]          [,4]           [,5]         [,6]
[1,] -0.3333333+0i -0.2450296+0i -0.1722534-0i -0.1088933-0i -0.05203037+0i  0.000000+0i
[2,] -1.0000000+0i -1.0883037+0i -1.1610799+0i -1.2244400+0i -1.28130296+0i -1.333333+0i