HP 15c Scientific manual Limiting the Estimation Time

238 Appendix D: A Detailed Look at _

Using the same initial estimates each time, you have found four roots for this equation involving a fourth-degree polynomial. However, the last two roots are quite close to each other and are actually one root (with a multiplicity of 2). That is why the root was not eliminated when you tried deflation once at this root. (Round-off error causes the original function to have small positive and negative values for values of x between 8.4999 and 8.5001; for x = 8.5 the function is exactly zero.)

In general, you will not know in advance the multiplicity of the root you are trying to eliminate. If, after you have attempted to eliminate a root, _ finds that same root again, you can proceed in a number of ways:

∙Use different initial estimates with the deflated function in an attempt to search for a different root.

∙Use deflation again in an attempt to eliminate a multiple root. If you do not know the multiplicity of the root, you may need to repeat this a number of times.

∙Examine the behavior of the deflated function at x-values near the known root. If the function's calculated values cross the x-axis smoothly, either another root or a greater multiplicity is indicated.

∙Analyze the original function and its derivatives algebraically. It may be possible to determine its behavior for x-values near the known root. (A Taylor series representation, for example, may indicate the multiplicity of a root.)

Limiting the Estimation Time

Occasionally, you may desire to limit the time used by _ to find a root. You can use two possible techniques to do this – counting iterations and specifying a tolerance.