HP 15c Scientific manual Finding the Roots of an Equation

Section 13: Finding the Roots of an Equation 191

By making the height 1.5 decimeters, a 5.0×1.0×1.5-decimeter box is specified.

If you ignore the upper limit on the height and use initial estimates of 3 and 4 decimeters (still less than the width), you will obtain a height of 4.2026 decimeters – a root that is physically meaningless. If you use small initial

estimates such as 0 and 1 decimeter,

you will obtain a height of 0.2974 Graph of f(x) decimeter – producing an undesirably

short, flat box.

As an aid for examining the behavior of a function, you can easily evaluate the function at one or more values of x using your subroutine in program memory. To do this, fill the stack with x. Execute the subroutine to calculate the value of the function (press ´ letter label or Glabel.

The values you calculate can be plotted to give you a graph of the function. This procedure is particularly useful for a function whose behavior you do not know. A simple-looking function may have a graph with relatively extreme variations that you might not anticipate. A root that occurs near a localized variation may be hard to find unless you specify initial estimates that are close to the root.

If you have no informed or intuitive concept of the nature of the function or the location of the zero you are seeking, you can search for a solution using trial-and-error. The success of finding a solution depends partially upon the function itself. Trial-and-error is often – but not always – successful.

If you specify two moderately large positive or negative estimates and the function's graph does not have a horizontal asymptote, the routine will seek a zero which might be the most positive or negative (unless the function oscillates many times, as the trigonometric functions do).

If you have already found a zero of the function, you can check for another solution by specifying estimates that are relatively distant from any known zeros.