Least-Squares Calculations; Normal Equations - HP -15C Advanced Functions Handbook

110 Section 4: Using Matrix Operations
Note that r
T
E was scaled by 10
7
so that each row of E and A has
roughly the same norm as every other. Using this new system, the
HP-15C calculates the solution
2000.000080
1999.999980
1999.999980
1999.999980
1999.999980
, with AX =
10
7
-10"
5
-9 X 10~
6
0
0
This solution differs from the earlier solution and is correct to 10
digits.
Sometimes the elements of a nearly singular matrix E are
calculated using a formula to which roundoff contributes so much
error that the calculated inverse E"
1
must be wrong even when it is
calculated using exact arithmetic. Preconditioning is valuable in
this case only if it is applied to the formula in such a way that the
modified row of A is calculated accurately. In other words, you
must change the formula exactly into a new and better formula by
the preconditioning process if you are to gain any benefit.
Least-Squares Calculations
Matrix operations are frequently used in least-squares calcula-
tions. The typical least-squares problem involves an n X p matrix
X of observed data and a vector y of n observations from which you
must find a vector b wiihp coefficients that minimizes
ill-
where r = y — Xb is the residual vector.
Normal Equations
From the expression above,
||r||| - (y - Xb)
T
(y - Xb) = y
T
y -
Solving the least-squares problem is equivalent to finding a
solution b to the normal equations