HP 50g User Manual page 352

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
with
A
The solution using LSQ is shown next:
2x
1
x – 3x
1
2x
1
2 3 5
⎡ ⎤
⎢ ⎥
1 3 8
⎢ ⎥
⎢ ⎥
2 2 4
⎣ ⎦
+ 3x –5x = 13,
2 3
+ 8x = -13,
2 3
– 2x + 4x = -6,
2 3
⎡ ⎤
x
1
⎢ ⎥
, x ,
x
⎢ ⎥
2
⎢ ⎥
x
⎣ ⎦
3
13
⎡
⎢
b 13
and
⎢
⎢
6
⎣
⎤
⎥
.
⎥
⎥
⎦
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