07 Aug 2007 KKKQ 3013 PENGIRAAN BERANGKA Week 5 – Systems of Nonlinear Equations 07 August am – 9.00 am

07 Aug 2007 Week 5 Page 2 Topics Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations

07 Aug 2007 Week 5 Page 3 Newton-Raphson : Relative error See example 2.7 in Ch. 2 of lecture notes A measure of relative error between current approximation relative to previous approximation NOT similar to relative error (  ) defined from r = x k +  where r is the real root i.e. Here, it is relative error between approximation ( x k ) and true solution ( r ) estimated via truncation of Taylor series expansion Essence or basis of Newton- Raphson iteration i.e.

07 Aug 2007 Week 5 Page 4 Tutorial Example 1 Given f(x) = x 4 – x 3 + x 2 – x +1 Determine the solution for f(x) = 2 using the false position method. Consider termination criteria of 0.05% for the approximate error.

07 Aug 2007 Week 5 Page 5 Tutorial Example 1 x 4 – x 3 + x 2 – x + 1 = 2 Finding x when f(x) above is equals to 2 is similar to solving x 4 – x 3 + x 2 – x = 0 x 4 – x 3 + x 2 – x - 1 = 0 Now, we’ve got it in the form of g(x) = 0. The objective now is to find root of g(x) using the many possible method which the student has learned. g(x)

07 Aug 2007 Week 5 Page 6 Tutorial Example 1 False position method:

07 Aug 2007 Week 5 Page 7 Tutorial Example 2 The strain of an aluminium structure needs to be maintained below before it buckles catastrophically (where f is stress and E t is the tangential modulus). What is the max. allowed applied stress if the material parameters in the equation m = and f n = MPa. Given the aluminium modulus is MPa. The non-linear stress vs. strain relationship of metallic materials could be approximated quite well using the relationship shown in the next page.

07 Aug 2007 Week 5 Page 8 Tutorial Example 2 [1] British Aircraft Corporation, Structural Design Data SDD (1964) Generalized curves of normalized stress vs. normalized strain (courtesy ref. [1])

07 Aug 2007 Week 5 Page 9 Tutorial Example 2 Rearranging: Consider using fixed point iteration or succesive iteration Thus, 8000  f crit = MPa TRY OTHER METHODS !!