The iterative behavior is shown in the figure. The change in the solution from one iteration to another was less than 10 -12, and the function itself is smaller than 10 -14 for the first element and zero to machine accuracy for the second element. (If it takes hundreds of iterations you probably made a mistake. We already know that MATLAB solves the linear equations correctly, but watch that you have the minus sign! Now put back in the ' ' and run the problem for as many iterations as it needs. The values of f and the Jacobian are correct. (Alternatively, insert the command 'pause' in the code where you would like to stop and look at the results.) Look at the results on the screen for the first time through the calculation. % check if error1 nr and quickly press the apple-period. S=sprintf('****Did not converge within %3.0f iterations.****',itermax) % move the solution, xx(k+1) - xx(k), to xx(k+1) % Newton Raphson solution of two nonlinear algebraic equations Prepare the following script (but without the ' ' at the end of each line). Next let us apply the Newton-Raphson method to the system of two nonlinear equations solved above using optimization methods. The second one is such that upon convergence the right-hand side approaches a constant that doesn't change with further iterations. The first one of these is useful in theoretical developments (although an LU decomposition (link) is used rather than a matrix inverse (link)). There are two alternate ways to write the Newton-Raphson method. In this formulation of the method the right-hand side gradually (hopefully!) goes to zero. Sometimes the method converges even though the Jacobian is not reevaluated at each iteration. Since the Jacobian depends on the iterate, it must be evaluated at each iteration. Write a Taylor expansion in several variables. Multiple Nonlinear Equations using the Newton-Raphson Methodįor systems of equations the Newton-Raphson method is widely used, especially for the equations arising from solution of differential equations.
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