Solving system of equations using matlab

Question

So I have the following system of equations

 x1 - x2 =  20
 x2 - x3 =  30
 x3 - x4 =  75
 x4 - x5 = -49
-x1 + x5 = -20

how would I solve the system using Matlab? I I'm a little stuck.

There's a good chance there's no solution but if someone could let me know how to do it that would be great!

Answer 1

First, convert this equation into matrix notation:

A = [ 1 -1  0  0  0
      0  1 -1  0  0
      0  0  1 -1  0
      0  0  0  1 -1
     -1  0  0  0  1];

b = [ 20
      30
      75
     -49
     -20];

You are trying to find x giving Ax = b. You can not take the inverse of A since it is singular. To see this check its rank; rank(A) == 4. It would be 5 if A were non-singular.

So, you should find best x approximating b when multiplied by A from left. This is an optimization problem: you want to minimize the error between Ax and b. Usually, people use least squares method. That is, you minimize the sum of squares of the residuals. This can be done by pseudo inverse as follows:

x = pinv(A) * b

gives

x =

   31.8000
   23.0000
    4.2000
  -59.6000
    0.6000

Best approximation is found by

b2 = A*x


b2 =

    8.8000
   18.8000
   63.8000
  -60.2000
  -31.2000

The least squares error is found to be

e = norm(b-b2)

e =

   25.0440

If you want to try other methods alternative to least squares to minimize Ax-b, you can google l1-minimization, sparse encoding, etc.