How do you get a general solution of and underdetermined system?

How can you obtain the general solution of an underdetermined system? What I have now is:

A=[1 2 3;-2 -1 0;0 2 4];
b=[5;-1;6];
x=A\b

but this gives me the answer x =

   NaN
   NaN
   NaN
hope someone can help me

Answer 1

[Star Strider]

I would use the pseudo-inverse pinv function:

x = pinv(A)*b

x =

     166.6667e-003
     666.6667e-003
       1.1667e+000

There may be other options, such as the sparse matrix lsqr funciton, which gives the same result:

x = lsqr(A,b)

[Thomas van de Wiel]

Thank you!

[John D'Errico]

What Star has NOT explained is that this is NOT in fact the general solution!

When you have a singular problem, there will be issues. First of all, there may be no exact solution at all. If we change b slightly, than that would happen.

A=[1 2 3;-2 -1 0;0 2 4];
b=[5;-1;6];

x = pinv(A)*b
x =
    0.16667
    0.66667
     1.1667

A*x
ans =
          5
         -1
          6

bhat = b;
bhat(3) = 5
bhat =
     5
    -1
     5

xhat = pinv(A)*bhat
xhat =
      0.28161
      0.64368
       1.0057

Is the solution now exact? No.

A*xhat
ans =
       4.5862
      -1.2069
       5.3103

In fact, NO exact solution exists to the modified problem, where b was perturbed slightly.

A*xhat = bhat

As we saw in the first case, the solution for b was exact. But is it the unique solution? When A is a singular matrix, the solution will not be unique. Lets see what the true solution is. There will be an undetermined coefficient, that we can vary arbitrarily. I'll call it k here. The general solution for the case of A is given by what I call xtrue, since A is a 3x3 matrix that is of rank 2.

syms k
xtrue = pinv(A)*b + k*null(A)
xtrue =
         (6^(1/2)*k)/6 + 1/6
 2/3 - (2^(1/2)*3^(1/2)*k)/3
         (6^(½)*k)/6 + 7/6

Yes, I know this looks a bit messy, but we can pick a specific, arbitrary value of k, and get this:

vpa(subs(xtrue,k,3))
ans =
  1.3914115380582557157653087040196
 -1.7828230761165114315306174080392
  2.3914115380582557157653087040196

A*ans
ans =
  5.0
 -1.0
  6.0

In fact, we can always pick ANY value for k, and get a different solution, which is equally valid.

simplify(A*xtrue)
ans =
  5
 -1
  6

So when you asked for the general solution, Star misled you, telling you it was obtainable from pinv. In fact, that was NOT the general solution.