HP HP-15C - Changing Signs

122 Section 4: Using Matrix Operations
Solving a System of Nonlinear Equations
Consider a system ofp nonlinear equations inp unknowns:
fi(xi, x2,..., xp) = 0 for / = 1, 2, ...,p
for which the solution x^, x2,..., xp is sought.
Let
,andF(x) =
xl
X2
xp
,f(x) =
A(x)
/2(X)
/p(x)
... F(x)
where
J?..{f} — f-i^f} fnri 7— 19 n
rij\)~ o //Ax/1 ion,7 — i, z, ...,p.
3*;
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
(F(x(*)))-
for k = 0, 1, 2, ...
until
+ ^ converges.
The program in the following example performs one iteration of
Newton's method. The computations are performed as
where d^ ' is the solution to the/? Xp linear system
The program displays the Euclidean lengths of f(x(/2)) and the
correction d(/e) at the end of each iteration.
Example: For the random variable y having a normal distribution
with unknown mean m and variance u2, construct an unbiased test
of the hypothesis that v2 — VQ versus the alternative that u2 ¥= v2 for
a particular value VQ.
For a random sample of y consisting of y1; y2, ..., yn, an unbiased
test rejects the hypothesis if