User Manual

Section 4: Using Matrix Operations 93
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





































0
0
109
10
10
with,
999980.1999
999980.1999
999980.1999
999980.1999
000080.2000
6
5
7
AXX
.
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 calculations. The typical least-squares
problem involves an n × p matrix X of observed data and a vector y of n observations from
which you must find a vector b with p coefficients that minimizes



n
i
i
F
r
1
2
2
r
Where r = y – Xb is the residual vector.
Normal Equations
From the expression above,
XbXbyXbyyXbyXbyr
TTTTTT
F
 2)()(
2
.
Solving the least-squares problem is equivalent to finding a solution b to the normal
equations.
X
T
Xb = X
T
y.
However, the normal equations are very sensitive to rounding errors. (Orthogonal
factorization, discussed on page 95, is relatively insensitive to rounding errors.)
The weighted least-squares problem is a generalization of the ordinary least-squares
problem. In it you seek to minimize