AEF.NSEq Numerical Estimates for Solving Equations
Beginning and intermediate algebra spend a considerable amount of time solving
equations exactly without any reference to functions.
Very little time is spent in those courses visualizing equations or making estimates
for solutions- though some time is spent on estimating roots or
discussing the fact that technology often gives only estimates for numbers.
more advance courses before calculus, functions and estimation become
more important but still little is done to make sense visually of the
solution of equations beyond the connection to intersections of curves with
other curves or with the x-axis.
Estimates are visualized in almost
haphazard fashion, along with discussion of limit concepts.
Estimating a solution to an equation (or the value of a function) does have some general numerical
techniques. These are still evolving as technology improves to make them more effective and efficient. [See Root-finding algorithm, Wikipedia]
In this section we provide mapping diagrams that illustrate some important yet simple numerical
algorithms for estimating function values and equation solutions.
Visualizing Numerical Estimation:
Basic approaches: In most estimation techniques the key idea is to
find a procedure that is applied to some initial estimate to arrive at a
second estimate. Following a general principle that what's good for one
is good for all, the procedure is then repeated on each result.
Iteration of a procedure with some initial estimating information is
possible as long as the initial conditions needed for the procedure
continue to be satisfied.
Each of the following three techniques (bisection, false position, and
Newton secant) has some positive and negative attributes, which are
discussed in the comparison, where the issues of effectiveness and
efficiency are illustrated with mapping diagrams.