HP 15c User Manual
Page 103
Section 4: Using Matrix Operations
103
Let
)
(
)
(
)
(
)
(
)
(
)
(
)
(
and
,
)
(
)
(
)
(
)
(
,
1
2
21
1
11
2
1
2
1
x
x
x
x
x
x
x
F
x
x
x
x
f
x
pp
p
p
p
p
p
F
F
F
F
F
F
f
f
f
x
x
x
,
where
)
(
)
(
x
x
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
v
v
versus the alternative that
2
0
2
v
v
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
0
2
2
0
1
v
x
s
or
v
x
s
n
n
,
where
n
i
i
n
i
i
n
y
n
y
and
y
y
s
1
1
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