Solving Nonlinear Equation Chapter 6 Solving Nonlinear Equation System

Content Nonlinear equation system Successive substitution Newton-Raphson Example

Introduction (1) Nonlinear equation in n variables f1(x1,x2,…,xn) = 0 fn(x1,x2,…,xn) = 0 For example: x21+x1x2 = 10 x2+3x1x22 = 57 Solve by graphical method ?

Introduction (2) x21+x1x2 = 10 x2+3x1x22 = 57

Successive sub…(1) Each one of the nonlinear equations can be solved for one of the unknowns. Example x21+x1x2 = 10 x2+3x1x22 = 57 Initial guess with x1=1.5, x2 = 3.5 (1) (2) Substitute with x1=1.5, x2 = 3.5 into (1) substitute with new x1 and x2 = 3.5 into (2)

Successive sub…(2) Example (cont’d) Substitute so on…. not converge Now try with these two eqs This will converge to x1=2 and x2 =3, which are answers!!!! Problem is u should formulate eqns in appropriate forms.

Newton-Raphson (1) For nonlinear equation in n variables where If the derivatives of f exists, then we can estimate x from is the initial guess vector x

Newton-Raphson (2) is the vector x of ith estimation is the (updated) vector x of i+1th estimation is the function of x of ith estimation Named Jacobian matrix