NumMeth_Handout_5 - Numerical Methods in Chemical...

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Numerical Methods in Chemical Engineering 5 Solving systems of non-linear equations 5 Solving systems of non-linear equations. ...................................................... 1 5.1 Overview. ................................................................................................. 2 5.2 Passing functions. ..................................................................................... 3 5.3 1D Newtons Method (something you did at school). .............................. 4 5.4 Newton's method in more than one dimension. ....................................... 6 5.4.1 Solving the system of two coupled equations. .................................. 9 5.4.2 Solution trajectories . ....................................................................... 11 5.4.3 The reduced Newton step method. .................................................. 13 5.4.4 Problems with Newton's Method. ................................................... 14 5.5 A chemical engineering example - equilibrium. ................................... 16 5.5.1 The element balance equations . ...................................................... 17 5.5.2 The equilibrium equations. .............................................................. 17 5.5.3 Thermodynamic data. ...................................................................... 18 5.5.4 Solving the equations . .................................................................... 19 5.5.5 Full solution to the equilibrium problem in Matlab. ....................... 21 5.6 Convergence of the Newton-method . .................................................... 23 5.7 Calculating the Jacobian Matrix . ........................................................... 24 5.7.1 Finite differences. ............................................................................ 24 5.7.2 Broydens Method. ........................................................................... 25 5.7.3 Taking advantage of sparsity . ......................................................... 26 5.8 Trust region method (Used by fsolve) . .................................................. 27 5.9 Summary . ............................................................................................... 28 5.10 Amendments. ...................................................................................... 29 5.10.1 Updated version of Newton1D. ................................................... 29 5.10.2 Updated version of Newton. ........................................................ 30 - 1 -
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Numerical Methods in Chemical Engineering 5.1 Overview Now we will look at solving systems of non-linear equations. We will look at one method in particular, Newton's method. We are going to write our own routine for solving non-linear equations using Newton's method. Matlab has its own routines for solving systems of non-linear equations (e.g. fsolve, which is loosely based on Newton's method). You are encouraged to use these Matlab routines as they are likely to be more robust than anything we can write in a lecture! Additional reading for this lecture:
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NumMeth_Handout_5 - Numerical Methods in Chemical...

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