I'm trying to create a Lagrange polynomial 'P(x)' for a set of points, but without the x-coordinate of the point I'm trying to find. For those who don't know, a Lagrange polynomial is a method of interpolating a set of data, such that it creates a polynomial that goes through every point in the set of data. The polynomial is formed from each of the x and y values of the data points, as well as the variable 'x,' which represents the x-coordinate of a point you are trying to find the y-value for.
However, in my situation, I have the y-value (y = 43) of a point I am trying to find an x-value for (which will later be done with a root-finding method). I initially thought to use symbolic calculations, as can be seen below:
y0 = 42.997791034585575 ;
y1 = 42.999274955274394 ;
y2 = 43.000758746700800 ;
y3 = 43.002242408852400 ;
L0 = ((x - x1)/(x0 - x1))*((x - x2)/(x0 - x2))*((x - x3)/(x0 - x3)) ;
L1 = ((x - x0)/(x1 - x0))*((x - x2)/(x1 - x2))*((x - x3)/(x1 - x3)) ;
L2 = ((x - x0)/(x2 - x0))*((x - x1)/(x2 - x1))*((x - x3)/(x2 - x3)) ;
L3 = ((x - x0)/(x3 - x0))*((x - x1)/(x3 - x1))*((x - x2)/(x3 - x2)) ;
P1 = y0 * L0 + y1 * L1 + y2 * L2 + y3 * L3
P2 = sym2poly(P1)
P2 =
-2.0651 14.1836 -10.8771 -1.8562
From this, I was given a polynomial (rounded to two decimal places) of 
(This is slightly different to what the above code will give as I assigned values to each of the x and y-coordinates). I need to then 'lower' the polynomial so that the x-coordinate of the point I am trying to find lies on the x-axis, which I did by converting the symbolic polynomial to a vector expression (a 1x4 matrix in the form of [-2.06 14.15 -10.82 -1.88] and subtracting 43 (the y-coordinate of the point) from the fourth term. This worked perfectly and gave me exactly what I needed, however I later realised we weren't allowed to use symbolic calculations in this process. I really don't know where to go from here. I tried assigning each of L0, L1, L2 and L3 to function handles:
L0 = @(x) (((x) - x1)/(x0 - x1))*(((x) - x2)/(x0 - x2))*(((x) - x3)/(x0 - x3)) ;
L1 = @(x) (((x) - x0)/(x1 - x0))*(((x) - x2)/(x1 - x2))*(((x) - x3)/(x1 - x3)) ;
L2 = @(x) (((x) - x0)/(x2 - x0))*(((x) - x1)/(x2 - x1))*(((x) - x3)/(x2 - x3)) ;
L3 = @(x) (((x) - x0)/(x3 - x0))*(((x) - x1)/(x3 - x1))*(((x) - x2)/(x3 - x2)) ;
Unfortunately this didn't work as when I went to form 'P1' I got the error that operations using '*' could not be performed on function handles. I suppose I COULD hypothetically just skip forming L0, L1, L2 and L3 and then P1 by just putting all together in one line, but this would be very messy and harder for a reader to interpret.
Thank you for being so patient as to read through this really long question (I just wanted to give as many details as I could to be as clear as possible), and any ideas are welcome.
