1. Nonlinear Algebraic Systems 1.Iterative solution methods 2.Fixed-point iteration 3.Newton-Raphson method 4.Secant method 5.Matlab tutorial 6.Matlab exercise

2. Single Nonlinear Equation l Nonlinear algebraic equation: f(x) = 0 »Analytical solution rarely possible »Need numerical techniques »Multiples solutions may exist l Iterative solution »Start with an initial guess x 0 »Algorithm generates x 1 from x 0 »Repeat to generate sequence x 0, x 1, x 2, … »Assume sequence convergences to solution »Terminate algorithm at iteration N when: l Many iterative algorithms available »Fixed-point iteration, Newton-Raphson method, secant method

3. Fixed-Point Iteration l Formulation of iterative equation »A solution of x = g(x) is called a fixed point l Convergence »The iterative process is convergent if the sequence x 0, x 1, x 2, … converges: »Let x = g(x) have a solution x = s and assume that g(x) has a continuous first-order derivative on some interval J containing s, then the fixed-point iteration converges for any x 0 in J & the limit of the sequence {x n } is s if: »A function satisfying the theorem is called a contraction mapping: »K determines the rate of convergence

4. Newton-Raphson Method l Iterative equation derived from first-order Taylor series expansion l Algorithm »Input data: f(x), df(x)/dx, x 0, tolerance (  ), maximum number of iterations (N) »Given x n, compute x n+1 as: »Continue until | x n+1 -x n | <  |x n | or n = N

5. Convergence of the Newton-Raphson Method l Order »Provides a measure of convergence rate »Newton-Raphson method is second-order l Assume f(x) is three times differentiable, its first- and second-order derivatives are non-zero at the solution x = s & x 0 is sufficiently close to s, then the Newton method is second-order & exhibit quadratic converge to s l Caveats »The method can converge slowly or even diverge for poorly chosen x 0 »The solution obtained can depend on x 0 »The method fails if the first-order derivative becomes zero (singularity)

6. Secant Method l Motivation »Evaluation of df/dx may be computationally expensive »Want efficient, derivative-free method l Derivative approximation l Secant algorithm l Convergence »Superlinear: »Similar to Newton-Raphson (m = 2)

7. Matlab Tutorial l Solution of nonlinear algebraic equations with Matlab l FZERO – scalar nonlinear zero finding »Matlab function for solving a single nonlinear algebraic equation »Finds the root of a continuous function of one variable »Syntax: x = fzero(‘fun’,xo) –‘fun’ is the name of the user provided Matlab m-file function (fun.m) that evaluates & returns the LHS of f(x) = 0. –xo is an initial guess for the solution of f(x) = 0. »Algorithm uses a combination of bisection, secant, and inverse quadratic interpolation methods.

8. l Solution of a single nonlinear algebraic equation: l Write Matlab m-file function, fun.m: l Call fzero from the Matlab command line to find the solution: l Different initial guesses, xo, can give different solutions: Matlab Tutorial cont. >> xo = 0; >> fzero('fun',xo) ans = 0.5376 >> fzero('fun',1) ans = 1.2694 >> fzero('fun',4) ans = 3.4015

9. Nonisothermal Chemical Reactor l Reaction: A  B l Assumptions »Pure A in feed »Perfect mixing » Negligible heat losses »Constant properties ( , C p,  H, U) »Constant cooling jacket temperature (T j ) l Constitutive relations »Reaction rate/volume: r = kc A = k 0 exp(-E/RT)c A »Heat transfer rate: Q = UA(T j -T)

10. Model Formulation l Mass balance l Component balance l Energy balance

11. Matlab Exercise l Steady-state model l Parameter values »k 0 = 3.493x10 7 h -1, E = 11843 kcal/kmol »(-  H) = 5960 kcal/kmol,  C p = 500 kcal/m 3 /K »UA = 150 kcal/h/K, R = 1.987 kcal/kmol/K »V = 1 m 3, q =1 m 3 /h, »C Af = 10 kmol/m 3, T f = 298 K, T j = 298 K. l Problem »Find the three steady-state points:

12. Matlab Tutorial cont. l FSOLVE – multivariable nonlinear zero finding »Matlab function for solving a system of nonlinear algebraic equations »Syntax: x = fsolve(‘fun’,xo) –Same syntax as fzero, but x is a vector of variables and the function, ‘fun’, returns a vector of equation values, f(x). »Part of the Matlab Optimization toolbox »Multiple algorithms available in options settings (e.g. trust- region dogleg, Gauss-Newton, Levenberg-Marquardt)

13. l Syntax for fsolve »x = fsolve('cstr',xo,options) »'cstr' – name of the Matlab m-file function (cstr.m) for the CSTR model »xo – initial guess for the steady state, xo = [C A T] '; »options – Matlab structure of optimization parameter values created with the optimset function l Solution for first steady state, Matlab command line input and output: Matlab Exercise: Solution with fsolve >> xo = [10 300]'; >> x = fsolve('cstr',xo,optimset('Display','iter')) Norm of First-order Trust-region Iteration Func-count f(x) step optimality radius 0 3 1.29531e+007 1.76e+006 1 1 6 8.99169e+006 1 1.52e+006 1 2 9 1.91379e+006 2.5 7.71e+005 2.5 3 12 574729 6.25 6.2e+005 6.25 4 15 5605.19 2.90576 7.34e+004 6.25 5 18 0.602702 0.317716 776 7.26 6 21 7.59906e-009 0.00336439 0.0871 7.26 7 24 2.98612e-022 3.77868e-007 1.73e-008 7.26 Optimization terminated: first-order optimality is less than options.TolFun. x = 8.5637 311.1702

14. Matlab Exercise: cstr.m function f = cstr(x) ko = 3.493e7; E = 11843; H = -5960; rhoCp = 500; UA = 150; R = 1.987; V = 1; q = 1; Caf = 10; Tf = 298; Tj = 298; Ca = x(1); T = x(2); f(1) = q*(Caf - Ca) - V*ko*exp(-E/R/T)*Ca; f(2) = rhoCp*q*(Tf - T) + -H*V*ko*exp(-E/R/T)*Ca + UA*(Tj-T); f=f';