Real solutions to polynomials

When I do this code:

%Find all real solutions of the equations

syms x

A = x^5 - 12*x^4 + 55*x^3 - 120*x^2 + 124*x == 48;

a = solve(A,x,'Real',true)

B = x^6 - x^4/5 + 14*x^2 - 2*x - 10 == 0;

b = solve(B,x,'Real',true)

C = x^5 - 4*x^3 + 4*x^2 - (4*x)/3 - 10/3 == 0;

c = solve(C,x,'Real',true)

D = x^5 - 13*x^4 + 64*x^3 - 152*x^2 + 176*x == 80;

d = solve(D,x,'Real',true)

I get:

>> Matlab412

a =

1

2

2

3

4

b =

root(z^6 - z^4/5 + 14*z^2 - 2*z - 10, z, 1)

root(z^6 - z^4/5 + 14*z^2 - 2*z - 10, z, 2)

c =

root(z^5 - 4*z^3 + 4*z^2 - (4*z)/3 - 10/3, z, 1)

root(z^5 - 4*z^3 + 4*z^2 - (4*z)/3 - 10/3, z, 4)

root(z^5 - 4*z^3 + 4*z^2 - (4*z)/3 - 10/3, z, 5)

d =

2

2

2

2

5

In other words; I just get an expression for the real roots for b) and c).

Does anyone have a clue why?

[Robert Bag]

I mean, I do not ger the numbers.

Answer 1

[Alan Stevens]

Try

B = x^6 - x^4/5 + 14*x^2 - 2*x - 10 == 0;
b = vpasolve(B,x)
C = x^5 - 4*x^3 + 4*x^2 - (4*x)/3 - 10/3 == 0;
c = vpasolve(C,x)

then delete the complex values.

Answer 2

[David Hill]

Why not just use roots()?

a=roots([5 -12 55 -120 124 -48]);
r=[];
for k=1:length(a)
  if isreal(a(k))
     r=[r,a(k)];
  end
end

Answer 3

[John D'Errico]

When you use solve, remember that first of all, it will be impossible to find algebraic solutions to a general polynomial of degree 5 or higher. This was proved a zillion years ago. Read about Abel-Ruffini. So all that solve can do in some cases is to use a numerical solver.

This one worked, because the polynomial is in fact a really easy one to solve:

syms x
A = x^5 - 12*x^4 + 55*x^3 - 120*x^2 + 124*x == 48;
a = solve(A,x,'Real',true)

a = 

But this one is more difficult:

B = x^6 - x^4/5 + 14*x^2 - 2*x - 10 == 0;
b = solve(B,x,'Real',true)

b = 

So MATLAB gets hung up. It returns a result that says, well, I know the result ins in there somewhere, but I could not find it.

We can force MATLAB to turn that into numerical values using either double or vpa.

format long g
double(b)

ans = 2×1

-0.770857012663665 0.902280599422831

vpa(b)

ans = 

Which one you use depends on whether you want a double as a result, or a symbolic floating point number.

[Robert Bag]

You are an angel.