Numerical Solutions of Nonlinear Equations

We discuss here the use of two most prominent iterative methods of solutions of nonlinear equations through illustrations that are relevant to chemical engineers. 

The steady-state mass and energy balances in a CSTR are given by Equations (2.20) and (2.21), respectively, which are reproduced here for ready reference  

These can be combined by eliminating the reaction term to express the relationship between the dimensionless concentration and the dimensionless temperature as 

Given an initial guess Xq, g(Xo) is evaluated to give Xj. The process is repeated to obtain a sequence Xq,Xi. that may converge to the solution. The condition for the convergence of a sequence is given by 

The criterion for the convergence of such a sequence is that the absolute eigenvalues of the following Jacobian matrix evaluated at the root are less than 1  

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References General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equa-... 

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