Solving an Equation without solve command

Hello Everyone,

I'm trying to solve the following equation for h

My problem is that the equation can't have an exact solution. However, there is a tolerance of 0.05. Therefore, I can't use the solve( command. I tried to obtain a solution but I couldn't. Do you guys have a suggestion?. Thank you in advance. This is my code;

load cal.mat

% Parameters
g = 9.81; % Gravity acceletration

% Inputs
Q = 40; % (m^3/sec)
So = 0.02 % slope of the channel
W = 4 % width of the channel
H = 3 % Depth of water
hin = H % depth of the water
hv = hin;% height of the canopy

% Obtain initial data
A = W*H % Cross-sectional Area of the Channel
P = W + hin + hin
R = A/P

% Setting Tolerance
Tol = 0.05 %meter
dif = 1
ht1 = hin + 0.01
ht = hin

for i=1:100
    U = Q/A;

      while dif > Tol
          % for trees
          ht = (((U)^2)*((g*((cal(i,2))^2)/(ht^(1/3))+0.5*((cal(i,1))*(cal(i,4))*ht*(cal(i,3))))))/(g*(1-0.25*(cal(i,4)*pi*((cal(i,3))^2))))
          dif = ht1-ht
          ht = ht + 0.01
      end
  end

Answer 1

[Luke Halberstadt]

Why are you not permitted to find an exact solution? The equation you gave can be solved directly for h since if you look carefully, it is in the form h = a/h^(1/3) + b*h, which can be converted to h = (a/(1-b))^(3/4).