HP 15c - Manual (Page 136)

136 Section 4: Using Matrix Operations
136




 

maximum. a finding for ) (
minimum a finding for ) (
x
x
s
f
f
Once the direction is determined from the gradient, the program looks for the optimum
distance to move from x
j
in the direction indicated by s
j
—the distance that gives the greatest
improvement in f(x) toward a minimum or maximum
To do this, the program finds the optimum value t
j
by calculating the slope of the function
g
j
(t) = f(x
j
+ ts
j
)
at increasing values of t until the slope changes sign. This procedure is called "bounding
search" since the program tries to bound the desired value t
j
within an interval. When the
program finds a change of sign, it then reduces the interval by halving it j + 1 times to find
the best t value near t=0. This procedure is called "interval reduction"—it yields more
accurate values for t
j
as x
j
converges toward the desired solution. (These two processes are
collectively called "line search.") The new value of x is then
x
j+1
= x
j
+ t
j
s
j
.
The program uses four parameters that define how it proceeds toward the desired solution.
Although no method of line search can guarantee success for finding an optimum value of t,
the first two parameters give you considerable flexibility in specifying how the program
samples t.
d Determines the initial step u
1
for the bounding search. The first value of t tried is
F
j
j
d
u
s ) 1 (
1 

.
This corresponds to a distance of
 
1
1 
  
j
d
u
F
j j j
x s x ,
which shows that d and the iteration number define how close to the last x value the
program starts the bounding search.
a Determines the values u
2
, u
3
, … of subsequent steps in the bounding search. These
values of t are defined by
u
i+1
= au
i
Essentially, a is an expansion factor that is normally greater than 1, producing an
increasing sequence of values of t.
e Determines the acceptable tolerance on the size of the gradient. The iterative process
stops when
||

f(x
j
)||
F

e.