how to use solve() without a 'z' variable solution

Hi, i'm trying to code a 'hn' and 'hc' solver, it uses those equations and data, but

close all
clear all
clc
%DATA
alfa = degtorad(30);, b = 0.14;, L = b;, m = L*sin(alfa);,k = L*sin(alfa);, pendiente = 0.001;,q = 4;, n = 0.011;
%FUNCTIONS
syms h
d = @(h) b+h.*(m+k);
a = @(h) b.*h+(h.^2)*(m+k)./2;
pm = @(h) b+h.*(sqrt(1+m)+sqrt(1+k));
eta = @(h) h.*(b+h*(m+k)+2*b)./(3*(b+h*(m+k)+b));
fr = @(h) (((q.^2).*d(h))./9.8.*a(h).^3).^0.5;
man = @(h) q.*n./(pendiente.^0.5) == (a(h).^(5/3))./(pm(h).^(2/3));
%SOLUTIONS
rug = eta(h);
hn = solve(man(h),h)    %%%%HERE IS MY PROBLEM
hc = solve(fr(h).^2==1,h,)  %%%%%HERE IS MY PROBLEM
Ec = hc  + (q.^2)/((a(hc).^2)*2*9.8)
En = hn  + (q.^2)/((a(hn).^2)*2*9.8) 
 

And when I run it, matlab show me this:

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve. 
> In sym/solve (line 304)
  In calculos (line 15) 
 
hn =
 
5.992149502432489941281916778964
 
 
hc =
 
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6)
 root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7)
 
Warning: Solution is not unique because the system is rank-deficient. 
> In symengine
  In sym/privBinaryOp (line 1030)
  In  /  (line 373)
  In calculos (line 17) 
 
Ec =
 
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 2)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 3)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 4)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 5)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 6)]
[ root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7) + 400000/(2401*(root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1)^2 + 2*root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 1))^2), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7), root(z^7 + 7*z^6 + 18*z^5 + 20*z^4 + 8*z^3 - 625000/49, z, 7)]
 
 
En =
 
6.0647894215630416283421290892246
 
>> 
 

so what I want is to get a real number 'hc' without a 'z' variable on them, I think it could be because there are no real solutions but I'm not sure, don't know how to solve that problem, don't even know what it means

Answer 1

[Alan Stevens]

The following gives real results.

%DATA
alfa = deg2rad(30); b = 0.14; L = b; m = L*sin(alfa); k = L*sin(alfa); pendiente = 0.001; q = 4; n = 0.011;
%FUNCTIONS
%syms h
d = @(h) b+h.*(m+k);
a = @(h) b.*h+(h.^2)*(m+k)./2;
pm = @(h) b+h.*(sqrt(1+m)+sqrt(1+k));
eta = @(h) h.*(b+h*(m+k)+2*b)./(3*(b+h*(m+k)+b));
fr = @(h) (((q.^2).*d(h))./9.8.*a(h).^3).^0.5;
fr2 = @(h) (((q.^2).*d(h))./9.8.*a(h).^3) - 1;
man = @(h) q.*n./(pendiente.^0.5) - (a(h).^(5/3))./(pm(h).^(2/3));
%SOLUTIONS
hn = fzero(man, [0 10])
hc = fzero(fr2, [0 10])

rug = eta(hn)  % not sure if this is what is wanted for rug
Ec = hc  + (q.^2)/((a(hc).^2)*2*9.8)
En = hn  + (q.^2)/((a(hn).^2)*2*9.8)   

[Alexis]

Thanks, it is a huge help, but doesn't work with 'fr' function because I mispelled it (sorry), is that one:

fr = @(h) (((q.^2).*d(h))./(9.8.*a(h).^3).^0.5;

so if we equal to zero, it should be:

fr = @(h) (((q.^2).*d(h))./(9.8.*a(h).^3)-1;

and that equation doesn't work with fzero function, matlab returns an error

[Alan Stevens]

That is exactly the function I called fr2 and it works with fzero! It returns a value of hc = 2.9695.

[Alan Stevens]

I suggest you plot your functions to see where the real roots are likely to lie. I used a range for h of 0 to 10, and the quick plotting I did seemed to suggest that was the range within which the roots lie. However, I didn't do a more detailed look.

[Alexis]

Isn't the same

fr = @(h) (((q.^2).*d(h))./9.8.*a(h).^3).^0.5;     %%wrong function
fr = @(h) (((q.^2).*d(h))./(9.8.*a(h).^3).^0.5;    %%correct function

that parenthesis in 9.8 changes the function, that's why your code don't give me the correct solution and when I change it to the right way, the fzeros function don't work

thanks for all your help, going to plot it rn

[Alan Stevens]

You are right! But then you only need to change the search range slightly:

%DATA
alfa = deg2rad(30); b = 0.14; L = b; m = L*sin(alfa); k = L*sin(alfa); pendiente = 0.001; q = 4; n = 0.011;
%FUNCTIONS
%syms h
d = @(h) b+h.*(m+k);
a = @(h) b.*h+(h.^2)*(m+k)./2;
pm = @(h) b+h.*(sqrt(1+m)+sqrt(1+k));
eta = @(h) h.*(b+h*(m+k)+2*b)./(3*(b+h*(m+k)+b));
fr = @(h) ((q.^2).*d(h))./(9.8.*a(h).^3).^0.5;
fr2 = @(h) ((q.^2).*d(h))./(9.8.*a(h).^3) - 1;
man = @(h) q.*n./(pendiente.^0.5) - (a(h).^(5/3))./(pm(h).^(2/3));
%SOLUTIONS
hn = fzero(man, [0 10])
hc = fzero(fr2, [1 10])

rug = eta(hn)  % not sure if this is what is wanted for rug
Ec = hc  + (q.^2)/((a(hc).^2)*2*9.8)
En = hn  + (q.^2)/((a(hn).^2)*2*9.8)   

Answer 2

[Walter Roberson]

R = @(v) sym(v);

%DATA
alfa = R(deg2rad(30));
b = R(0.14);
L = b;
m = L*sin(alfa);
k = L*sin(alfa);
pendiente = R(0.001);
q = R(4);
n = R(0.011);
%FUNCTIONS
syms h
d = b+h.*(m+k);
g = R(9.81);
a = b.*h+(h.^R(2))*(m+k)./R(2);
pm = b+h.*(sqrt(1+m)+sqrt(1+k));
eta = h.*(b+h*(m+k)+R(2)*b)./(R(3)*(b+h*(m+k)+b));
fr = sqrt(((q.^R(2)).*d)./g.*a.^R(3));
man = q.*n./sqrt(pendiente) == (a.^(R(5)/R(3)))./(pm.^(R(2)/R(3)));
%SOLUTIONS
rug = eta;
hn = solve(man,h);
display(hn)

hc = solve(fr.^R(2)==R(1), h);
display(hc)

fprintf('hc has %d total solutions\n', length(hc));
hcr = hc;
hcr(imag(hcr)~=0) = [];
fprintf('hc has %d real-valued solutions\n', length(hcr));
if isempty(hcr)
    fprintf('No real solutions for hc. Giving up\n');
else
  Ec = hcr  + (q.^R(2))/((subs(a,h,hcr).^R(2))*R(2)*g);
  display(Ec)
  En = hn  + (q.^R(2))/((subs(a,h,hn).^R(2))*R(2)*g);
  display(En)
end