HP -15C Advanced Functions Handbook page 163

Section 4: Using Matrix Operations 1 61
Once the direction is determined from the gradient, the program
looks for the optimum distance to move from x, in the direction
indicated by s
;
—the distance that gives the greatest improvement
in /(x) toward a minimum or maximum.
To do this, the program finds the optimum value tj by calculating
the slope of the function
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 tj within an interval. When the program
finds a change of sign, it then reduces the interval by halving it
7 + 1 times to find the best t value near t = 0. This procedure is
called "interval reduction"—it yields more accurate values for tj as
KJ converges toward the desired solution. (These two processes are
collectively called "line search.") The new value of x is then
X • j_ — X • + t -S •
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 HI for the bounding search. The
first value of t tried is
This corresponds to a distance of
which shows that d and the iteration number define how close
to the last x value the program starts the bounding search.
Determines the values u
2
> u^, ... of subsequent steps in the
bounding search. These values of t are defined by