HP 33s - Manual (Page 325)

More about Solving D–1
D
More about Solving
This appendix provides information about the SOLVE operation beyond that given
in chapter 7.
How SOLVE Finds a Root
SOLVE first attempts to solve the equation directly for the unknown variable. If the
attempt fails, SOLVE changes to an iterative(repetitive) procedure. The iterative
operation is to execute repetitively 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 of f(x) (figure d, below).