Difference between mldivide and lsqr

Hi, I am curious about the difference between mldivide and lsqr.

Both seem to be used for calculating x where A*x=b.

Thank you in advance.

Answer 1

[Bruno Luong]

The difference manifests obviously when you try to solve underdetermined linear system A*x = b.

A=rand(3,5)

A = 3×5

0.1391 0.7822 0.5816 0.3031 0.3500 0.4841 0.6309 0.9509 0.7327 0.0768 0.7137 0.8771 0.4724 0.4694 0.4100

b=rand(3,1)

b = 3×1

0.9778 0.2986 0.9737

The backslash returns a solution with most 0s and lsqr returns solution with minimum l2-norm.

xbackslash=A\b

xbackslash = 5×1

-0.4389 1.8328 -0.6786 0 0

xlsqr=lsqr(A,b)

lsqr converged at iteration 3 to a solution with relative residual 8.3e-14.

xlsqr = 5×1

-0.1969 1.1213 -0.1120 -0.3746 0.8767

xminnorm=lsqminnorm(A,b)

xminnorm = 5×1

-0.1969 1.1213 -0.1120 -0.3746 0.8767

Both minimize norm(A*x-b,2), in case not overdetermined system A*x match b (as withe example here)

A*xbackslash

ans = 3×1

0.9778 0.2986 0.9737

A*xlsqr

ans = 3×1

0.9778 0.2986 0.9737

norm(xbackslash)

ans = 2.0031

norm(xlsqr) % smaller even xlsqr has no O

ans = 1.4891

On the runtime the backslash is usualy faster, at least for dense matrix.