Newton-Raphson method multivariable

Three methods that can be used to find the values of xj,..., that satisfy n simultaneous equations are extensions of methods given previously for single-variable problems. They are (a) successive substitution, (b) the Wegstein algorithm, and (c) the Newton-Raphson method (a multivariable extension of Newton s rule). The example that concludes this section illustrates all three algorithms. 

The material balance equation for component 3 can be solved separately once V is determined. The above three simultaneous equations are first solved for Yy Y2, and t /- They are nonlinear, involving products yy, and ti/Vj. Although they may be solved by elimination, a more general method is Newton-Raphson s multivariable iterative technique. First, rewrite the equations in the residual form ... 

The results shown for these examples are in good agreement with the fact that the Newton-Raphson method is said to exhibit quadratic convergence. For a single variable problem, quadratic convergence means that the error for the nth trial is proportional to the square of the error for the previous trial. The error for the nth trial is defined as the correct value of the variable minus the value predicted by the nth trial. For a multivariable problem, quadratic convergence means that the norm of the errors given by... 

Three well-known numerical methods for solving multivariable problems are presented as well as their convergence characteristics. The methods considered are direct iteration, the Newton-Raphson method, and Broyden s method. 

Simpson in 1740 was the first to formulate the Newton-Raphson method on the basis of calculus. He applied the iterative scheme for solving general systems of nonlinear equations. In addition to this important extension of the method to nonlinear systems, Simpson extended the iterative solver to multivariate minimization, noting that by setting the gradient to zero the same method can be applied. 

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