HP 48gII - Manual (Page 347)

Page 11-21
Let’s store the latest result in a variable X, and the matrix into variable A, as
follows:
Press
K~x` to store the solution vector into variable X
Press ƒ ƒ ƒ to clear three levels of the stack
Press K~a` to store the matrix into variable A
Now, let’s verify the solution by using:
@@@A@@@ * @@@X@@@ `, which results in
(press ˜ to see the vector elements): [-9.99999999999 85. ], close enough
to the original vector b = [-10 85].
Try also this,
@@A@@@ * [15,10/3,10] ` ‚ï`, i.e.,
This result indicates that x = [15,10/3,10] is also a solution to the system,
confirming our observation that a system with more unknowns than equations
is not uniquely determined (under-determined).
How does the calculator came up with the solution
x = [15.37… 2.46…
9.62…] shown earlier? Actually, the calculator minimizes the distance from a
point, which will constitute the solution, to each of the planes represented by
the equations in the linear system. The calculator uses a least-square method,
i.e., minimizes the sum of the squares of those distances or errors.
Over-determined system
The system of linear equations
x
1
+ 3x
2
= 15,
2x
1
– 5x
2
= 5,
-x
1
+ x
2
= 22,