Using arrayfun to solve a equation system (with vpasolve) MatLab - matlab

Question

I am struggling with solving a problem as efficient as possible.
I have two equations and want to solve them together. Furthermore, I want to solve them for four different cases. I can solve them independently altering the code for each case. I can also solve them by accessing a vector that contains the desired value for one of them (e.g., the first value of A) by using arrayfun. However I can't manage to do all of them together.
My code:
clc
clear all
close all
% scalar parameters
g = 9.807; % Gravity constant, m/s2
d = 1.225; % Density of air, kg/m3
x = 1000; % Height, m
% vector parameters
%t = 0:2:10; % time, seconds
A = [1.2; 1.7; 1.8; 0.3]; % Area, m^2
m = [82; 84; 90; 25]; % Mass, kg
Cd = [0.3; 1.14; 0.29; 0.045]; % Drag coefficient, -
syms v t
eqn1 = 2*m./(A.*Cd*d).* log(abs(cosh(t.*sqrt(A.*Cd*d*g./(2*m))))) == x;
eqn2 = (2*g*m./(d*A.*Cd)).*tanh(t.* sqrt((g*d*Cd.*A)./(2*m))) - v == 0;
eqns=[eqn1 eqn2];
variables = [v t];
result = arrayfun(#vpasolve, eqns, 'uniform', 0)
%disp('v='),disp(eval(v));
%disp('t='),disp(eval(t));
I get a result for t (which is weirdly negative and I don't know why), but I only get a {1×1 struct} for v, which I don't want. I know I can solve this also by using a for loop, but I wanted to make it more efficient.
I tried the code written above and solved it in various forms, however not as desired.

Answer 1

The way you have it implemented you're effectively trying to solve four equations with two unknowns, which isn't supported by vpasolve except in the case of polynomial systems.
It looks like you're trying to solve the same system for four different combinations of values for the parameters A, m, and Cd. The easiest way to do that is to just break this out as a for loop (there is simply no benefit to using arrayfun here – speed or otherwise). You'll need to create some temporary symbolic variable for the parameter (there are a few ways this could be done). Here is example code to do just that:
% scalar parameters
g = 9.807; % Gravity constant, m/s2
d = 1.225; % Density of air, kg/m3
x = 1000; % Height, m
% vector parameters
%t = 0:2:10; % time, seconds
A = [1.2; 1.7; 1.8; 0.3]; % Area, m^2
m = [82; 84; 90; 25]; % Mass, kg
Cd = [0.3; 1.14; 0.29; 0.045]; % Drag coefficient, -
syms v t real
syms A_sym m_sym Cd_sym real % Temporary symbolic variables
eqn1 = 2*m_sym./(A_sym.*Cd_sym*d).* log(abs(cosh(t.*sqrt(A_sym.*Cd_sym*d*g./(2*m_sym))))) == x;
eqn2 = (2*g*m_sym./(d*A_sym.*Cd_sym)).*tanh(t.* sqrt((g*d*Cd_sym.*A_sym)./(2*m_sym))) - v == 0;
eqns=[eqn1 eqn2];
variables = [v t];
n = length(A); % Number of prameter sets
result = cell(n,1); % Preallocate resultant cell array
for i = 1:n
% Substitute in numeric values for i-th parameter set
eqns_sub = subs(eqns,{A_sym,m_sym,Cd_sym},{A(i),m(i),Cd(i)});
% Solve and store in cell array
result{i} = vpasolve(eqns_sub,variables);
end
The output, result, is a 4-by-1 cell array of structs with your two variables, v and t, as VPA-valued fields for each.
You could replace the last few lines with the following if what you want is a numeric array as the output (this assumes that only one solution is found for each parameter set):
n = length(A); % Number of prameter sets
result = zeros(n,length(variables)); % Preallocate resultant array
for i = 1:n
% Substitute in numeric values for i-th parameter set
eqns_sub = subs(eqns,{A_sym,m_sym,Cd_sym},{A(i),m(i),Cd(i)});
% Solve, convert to double precision, and store in array
out = vpasolve(eqns_sub,variables);
result(i,:) = double([out.v out.t]);
end