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•
If
A
is a square matrix and
A
is non-singular (i.e., it’s inverse matrix
exist, or its determinant is non-zero), LSQ returns the exact solution to
the linear system.
•
If
A
has less than full row rank (underdetermined system of equations),
LSQ returns the solution with the minimum Euclidean length out of an
infinity number of solutions.
•
If
A
has less than full column rank (over-determined system of
equations), LSQ returns the "solution" with the minimum residual value
e
=
A
⋅
x
–
b
. The system of equations may not have a solution,
therefore, the value returned is not a real solution to the system, just
the one with the smallest residual.
Function LSQ takes as input vector
b
and matrix
A
, in that order. Function
LSQ can be found in Function catalog (
‚N
). Next, we use function LSQ
to repeat the solutions found earlier with the numerical solver:
Square system
Consider the system
2x
1
+ 3x
2
–5x
3
= 13,
x
1
– 3x
2
+ 8x
3
= -13,
2x
1
– 2x
2
+ 4x
3
= -6,
with
The solution using LSQ is shown next:
.
6
13
13
,
,
4
2
2
8
3
1
5
3
2
3
2
1
−
−
=
=
−
−
−
=
b
x
A
and
x
x
x