Topic 3. Non-Linear Equations and Taylor's Theorem

• Examples of non-linear equations are equations that relate observed quantities to unknown parameters using functions of sine, cosine, tangent and that are raised to second and higher order powers.
• A standard step employed to solve non-linear equations is by transforming them such that linear equations are created, more popularly known as “linearization”.
• These linear equations can then be solved but the values of the

  unknown parameters are not exact, but only an approximation.
• Linearization of non-linear equation can be done using 1st order Taylor Series Approximation through the application of Taylor’s Theorem.
• x0 and y0 = the approximations for x and y.
• f(x0, y0) are the nonlinear function evaluated at these approximations.
• R = remainder
• dx and dy = corrections to the approximations
• Increasing the value of n results to a more exact approximation.
• As the order of each successive term increases, its significance in the overall expression decreases.
• The linear equation can then be solved using an iterative procedure.