User Manual

Section 4: Using Matrix Operations 103
Let













































)()(
)()(
)()(
)(and,
)(
)(
)(
)(,
1
221
111
2
1
2
1
xx
xx
xx
xF
x
x
x
xfx
ppp
p
p
pp
FF
FF
FF
f
f
f
x
x
x





,
where
)()( xx
i
j
ij
f
x
F



for i, j = 1, 2, …, p.
The system of equations can be expressed as f(x) = 0. Newton's method starts with an initial
guess x
(0)
to a root x of f(x) = 0 and calculates
x
(k + 1)
= x
(k)
− (F(x
(k)
))
−1
f(x
(k)
) for k = 0, 1, 2, …
until x
(k+1)
converges.
The program in the following example performs one iteration of Newton's method. The
computations are performed as
x
(k + 1)
= x
(k)
− d
(k)
,
where d
(k)
is the solution to the p×p linear system
F(x
(k)
)d
(k)
= f(x
(k)
) .
The program displays the Euclidean lengths of f(x
(k)
) and the correction d
(k)
at the end of each
iteration.
Example: For the random variable y having a normal distribution with unknown mean m and
variance v
2
, construct an unbiased test of the hypothesis that
2
0
2
vv 
versus the alternative that
2
0
2
vv 
for a particular value
2
0
v
.
For a random sample of y consisting of y
1
, y
2
, … , y
n
an unbiased test rejects the hypothesis if
2
02
2
01
vxsorvxs
nn

,
where



n
i
i
n
i
in
y
n
yandyys
11
2
1
)(
for some constants x
1
and x
2
.
If the size of the test is a (0 < a < 1), you can find x
l
and x
2
by solving the system of
equations f
1
(x) = f
2
(x) = 0, where
f
1
(x) = (n – 1) ln(x
2
/ x
1
) + x
1
– x
2