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8 The Five Tools

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:

  1. Iteration,
  2. Linear algebra,
  3. Interpolation,
  4. Taylor series, and
  5. Bracketing.

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 a point.

All numerical techniques taught in this class after this point are based on one or more these tools.

  1. Almost all algorithms rely on iteration, that is, taking one approximation and finding a better approximation,
  2. 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.
  3. In finding an approximation, bracketing techniques are the worst performers and should only be used when no other tools are available,
  4. In engineering, data is sampled from the system, and therefore we may use interpolation,
  5. 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.