31 Jul 2007 KKKQ 3013 PENGIRAAN BERANGKA Week 4 – Systems of Nonlinear Equations 31 July am – 9.00 am

31 Jul 2007 Week 4 Page 2 Topics Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations

31 Jul 2007 Week 4 Page 3 Open Method #1 : Newton-Raphson Therefore, x2x2 Repeat.. Derivative defines gradient

31 Jul 2007 Week 4 Page 4 Open Method #1 : Newton-Raphson In general, Iterate/repeat until | x i+1 | – |x i | <  (an appropriate error requirement) Need only 1 initial guess (i.e x 0 ) ! ………..(1)

31 Jul 2007 Week 4 Page 5 Open Method #2 : Secant Method Similar concept per Newton-Raphson, but gradient is estimated from its basic definition Therefore, substituting into equation (1) previously to Need 2 initial guess or points !!

31 Jul 2007 Week 4 Page 6 Tutorial Example 1 Determine the intersection of the following two equations: g(s) = s 6 h(s) = s + 1 using the secant method assumming initial values 1.0 & 1.05.

31 Jul 2007 Week 4 Page 7 Tutorial Example 1 Intersection of two equations means at a particular s, both g(s) and h(s) have the same value i.e. g(s) = h(s) s 6 = s + 1 ………..(1) Hence, need to find s which satisfies equation (1). Satisfying equation (1) is similar to solving : s 6 – s – 1 = 0 f(s) Therefore, need to find root of non-linear equation f(s).

31 Jul 2007 Week 4 Page 8 Tutorial Example 1 If we plot f(s) : f(s) s ? f(s) = s 6 -s-1

31 Jul 2007 Week 4 Page 9 Tutorial Example 1