More About Solving; How Solve Finds A Root - HP 32SII Owner's Manual

More about Solving

This appendix provides information about the SOLVE operation beyond that
given in chapter 7.

How SOLVE Finds a Root

SOLVE is an iterative operation; that is, it repetitively executes the specified
equation. The value returned by the equation is a function f(x) of the unknown
variable x. (f(x) is mathematical shorthand for a function defined in terms of
the unknown variable x.) SOLVE starts with an estimate for the unknown
variable, x, and refines that estimate with each successive execution of the
function, f(x).
If any two successive estimates of the function f(x) have opposite signs, then
SOLVE presumes that the function f(x) crosses the x–axis in at least one place
between the two estimates. This interval is systematically narrowed until a root
is found.
For SOLVE to find a root, the root has to exist within the range of numbers of
the calculator, and the function must be mathematically defined where the
iterative search occurs. SOLVE always finds a root, provided one exists
(within the overflow bounds), if one or more of these conditions are met:
Two estimates yield f(x) values with opposite signs, and the function's
graph crosses the x–axis in at least one place between those estimates
(figure a, below).
f(x) always increases or always decreases as x increases (figure b,
below).
The graph of f(x) is either concave everywhere or convex everywhere
(figure c, below).
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24 Size : 17 .7 x 25.2 cm
More about Solving
C
C–1