Nothing new is being introduced in this section, but
instead, we will review the five tools which we
will use from here on. These are:
- Linear algebra,
- Taylor series, and
Note that we already used iteration to solve a system of linear
equations, we used linear algebra to find interpolating polynomials,
and, from Calculus, you will recall that Taylor were found using
limits of interpolating polynomials
to determine that a function may be described by the derivatives evaluated at
All numerical techniques
taught in this class after this point are based on one or more these tools.
- Almost all algorithms rely on iteration, that is,
taking one approximation and finding a better approximation,
- As well as using linear systems to find interpolating polynomials,
any time we are solving problems with n variables, we invariably
need to solve a system of linear equations.
- In finding an approximation, bracketing techniques
are the worst performers and should only be used when
no other tools are available,
- In engineering, data is sampled from the
system, and therefore we may use interpolation,
- In many cases, we will have models (usually in
the form of differential equations) which describe
systems, in which case, we may use Taylor series to
help find solutions.
As a general rule, we will take non-linear systems and approximate
them by polynomials (often linear polynomials) and then solve the
corresponding polynomial problem.