mldivide algorithm for an underdetermined system of equations

I'm trying to solve an underdetermined equation (one equation with two unknowns) as a system of the form:

A * x = b

Where

A = [2, 1]
b = 1

If I solve it with the pseudoinverse, I get

x = pinv(A) * b
x = [0.4, 0.2]'

Which is the least squares solution. So far, so good.

However, if I use mldivide of backslash, I get the following solution.

x = A \ b
x = [0.5, 0]'

Which is the sparsest and minimum norm-1 solution. So my question is, how does MATLAB actually calculate this solution?

I've read some other questions, and they just mention that the solution using backslash has at most m nonzero components for an m-by-n matrix A, which is what I'm getting, but they don't say how it's computed.

In older versions (http://matlab.izmiran.ru/help/techdoc/ref/mldivide.html), they say it's calculated with QR decomposition with pivoting, but this yields to:

[Q, R, P] = qr(A)
Q = 1
R = [2, 1]
P = [1, 0;
     0, 1]

Which doesn't help much... if I use this, I get the initial solution of least squares. So I guess it's using a different algorithm.

Answer 1

[Bruno Luong]

The outline of "\" goes as following:

% Test data
A=randi(10,3,2)*randi(10,2,5)

A = 3×5

108 67 38 86 70 137 96 52 122 85 109 84 44 106 65

b=randi(10,3,1)

b = 3×1

5 3 5

[Q, R, P] = qr(A,'vector');
r = rank(A);            % real implementation: r is estimated mainly from diag(R)
x = zeros(size(A,2),1);
y = Q(:,1:r)'*b;
X = R(1:r,1:r);         % X is upper triangular & square matrix with all diagonal terms ~= 0
x(P(1:r)) = inv(X)*y    % real implementation this is a back substitution 

x = 5×1

0.0440 0 0 -0.0097 0

A\b

Warning: Rank deficient, rank = 2, tol = 2.283770e-13.

ans = 5×1

0.0440 0 0 -0.0097 0

TMW calls it "basis/basic" solution.

[Juan Gallego]

Thanks a lot, this is exactly the answer I was looking for!

[Nouasria Hamid]

Thanks Thanks