how to find unknown within equation having unknown

I have a equation which includes 2 matrix and 1 unknown I want to calculate the unknown for each row and save the results as another matrix for each term my equation is R/X=( 0.8*log(L*X/10e-06)+33 )

R is 132x1 matrix (known)

L is 132x1 matrix (known)

I want to find X matrix

Answer 1

[Star Strider]

I have no idea if this actually has a robust (and mathematically correct) solution.

It is always appropriate to experiment, so I am doing that here —

RLX_fcn = @(X,R,L) R./X - ( 0.8*log(L.*X/10e-06)+33 )

RLX_fcn = function_handle with value:
    @(X,R,L)R./X-(0.8*log(L.*X/10e-06)+33)

R = randn(10,1);
L = randn(10,1);
X0 = rand(size(R))

X0 = 10×1

0.1263 0.0376 0.5791 0.9550 0.3068 0.1994 0.0835 0.2704 0.8998 0.1432

[Xv,fv] = fsolve(@(X)RLX_fcn(X,R,L), X0)

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

Xv = 
   0.0112 - 0.0007i
   0.0033 + 0.0002i
  -0.0156 + 0.0010i
  -0.0161 + 0.0000i
   0.0074 - 0.0005i
  -0.0114 + 0.0000i
  -0.0041 + 0.0000i
   0.0328 - 0.0021i
  -0.0150 + 0.0010i
   0.0236 - 0.0000i

fv = 
   1.0e-14 *

   0.0000 + 0.0000i
   0.0000 + 0.0000i
   0.0000 - 0.0444i
   0.7105 + 0.0098i
   0.0000 + 0.0000i
  -0.7105 + 0.0018i
   0.0000 + 0.0000i
   0.0000 + 0.0000i
   0.0000 + 0.0444i
   0.0000 + 0.0000i

Experiment further with the correct vectors.

.

[John D'Errico]

Actually, no that is not correct, and it has an analytical solution, in terms of the wrightOmega function. It is sort of a cousin to the Lambert W, and probably about as well known.

Answer 2

[John D'Errico]

R and L are known, with X being an unknown.

syms X L R
Xsol = solve(R/X==( 0.8*log(L*X/10e-06)+33),X)

Xsol = 

The script w there is the wrightOmega function in MATLAB. Most people don't seem to know it even exists, but it does, as a special function, essentially to solve this exact class of problem..

help wrightOmega

--- help for double/wrightOmega ---

 WRIGHTOMEGA Wright omega function.
    W = WRIGHTOMEGA(X) is a solution of the equation Y + log(Y) = X.
 
    Reference:
        [1] Corless, R. M. and Jeffrey, D. J. "The Wright omega Function."
        In Artificial Intelligence, Automated Reasoning, and Symbolic
        Computation (Ed. J. Calmet, B. Benhamou, O. Caprotti, L. Henocque
        and V. Sorge). Berlin: Springer-Verlag, pp. 76-89, 2002.

    Documentation for double/wrightOmega
       doc double/wrightOmega

    Other functions named wrightOmega

       single/wrightOmega    sym/wrightOmega

So now you can form the solution simply enough, as...

Xfun = matlabFunction(Xsol)

Xfun = function_handle with value:
    @(L,R)(R.*(5.0./4.0))./wrightOmega(-log(1.0./(L.*R.*1.25e+5))+1.65e+2./4.0)

So Xfun is a function of two variables, L and R. We can use this as follows:

Xfun(1,3)

ans = 0.0747

If L and R are vectors, it will still work.

Xfun([1 2 3],[4 5 6])

ans = 1×3

0.0991 0.1217 0.1444