More About Solving; How Solve Finds A Root - HP 33s 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).
If f(x) has one or more local minima or minima, each occurs singly between
adjacent roots off f(x) (figure d, below).
File name 33s-E-Manual-1008-Publication(1st).doc
Printed Date : 2003/10/8
Page : 386
Size : 13.7 x 21.2 cm
More about Solving
D
D–1