HP 15c Scientific manual Finding Several Roots, Error 0000 1250 5626

Appendix D: A Detailed Look at _ 233

Keystrokes	Display
´ _.0	Error 8
−	1.0000	–20
)	1.1250	–20
)	2.0000
((	1.0000	–20
´ _ .0	Error 8
−	1.1250	–20
)	1.5626	–16
)	2.0000

Best x-value. Previous value. Function value. Restore the stack.

Another x-value. Previous value. 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 10–20as 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