Appendix D: A Detailed Look at _ 233
| ( | (
1.0000 -20
Restore the stack.
´ _ .0
Error 8
−
1.1250 -20
Another x-value.
)
1.5626 -16
Previous value.
)
2.0000
Same function value.
In each of the three cases, _ initially
searched for a root in a direction
suggested by the graph around the initial
estimate. Using 10 as the initial estimate,
_ found the horizontal asymptote
(value of 1.0000). Using 1 as the initial
estimate, a minimum of 0.3788 at
x = 2.1213 was found. Using 10
−20
as the
initial estimate, the function was
essentially constant (at a value of 2.0000)
for the small range of x that was sampled.
Finding Several Roots
Many equations that you encounter have more than one root. For this
reason, you will find it helpful to understand some techniques for finding
several roots of an equation.
The simplest method for finding several roots is to direct the root search
in different ranges of x where roots may exist. Your initial estimates
specify the range that is initially searched. This method was used for all
examples in Section 13. You can often find the roots of an equation in this
manner.
Another method is known as deflation. Deflation is a method by which
roots are “eliminated” from an equation. This involves modifying the
equation so that the first roots found are no longer roots, but the rest of
the roots remain roots.
If a function f(x) has a value of zero at x = a, then the new function
f (x)/(x − a) will not approach zero in this region (if a is a simple root of
f (x) = 0). You can use this information to eliminate a known root. Simply
add a few program lines at the end of your function subroutine. These
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