Solving an equation with two vectors

I am trying to solve for an unknown using an equation with two vectors. I have tried using solve but the output is a 0x1 vector. I am trying to solve for p in the equation below and want a 37411x1 vector output with all the solutions. I considered using linsolve, but it looks like that only works for equations with one vector.

TotalDensity=WaterContent{:,2};%37411x1 vector	
Drydensity=WaterContent{:,5};%37411x1 vector

syms p
eqn = TotalDensity==p*.99819+(1-p)*Drydensity;
S=solve(eqn,p);

[Torsten]

TotalDensity=WaterContent{:,2};%37411x1 vector	
Drydensity=WaterContent{:,5};%37411x1 vector

TotalDensity = cell2mat(TotalDensity);
DryDensity = cell2mat(DryDensity);
p = (TotalDensity - DryDensity)./(.99819 - DryDensity)

Answer 1

[Walter Roberson]

You have 37411 equations in one single variable. There is no value, p, such that all of the equations are satisfied simultaneously.

solve() is for simultaneous equations. For example,

syms x y
solve([x + y == 5, 3*x + 4*y == 5])

ans = struct with fields:
    x: 15
    y: -10

Notice that this does not attempt to solve for ((x + y == 5) OR (3*x + 4*y == 5)) -- it solves for ((x + y == 5) AND (3*x + 4*y == 5))

There are two strategies:

1. For linear equations and low-degree polynomials, solve a template and then subs() in the actual numeric values to get the vector of results; OR
2. for nonlinear equations, loop doing one equation at a time, perhaps using arrayfun()

syms TD DD p
eqn = TD == p*.99819+(1-p)*DD;
S = solve(eqn, p)

S = 

%some numeric values for testing
TotalDensity = rand(4,1), Drydensity = rand(4,1).^2

TotalDensity = 4×1

0.9501 0.5393 0.9953 0.9124

Drydensity = 4×1

0.4446 0.5514 0.2766 0.2357

%get numeric solutions
SN = double(subs(S, {TD, DD}, {TotalDensity, Drydensity}))

SN = 4×1

0.9131 -0.0271 0.9961 0.8875