Nonlinear Algebraic Systems

1. Iterative solution methods

2. Fixed-point iteration

3. Newton-Raphson method

4. Secant method

5. Matlab tutorial

6. Matlab exercise

Single Nonlinear Equation

Nonlinear algebraic equation: f(x) = 0» Analytical solution rarely possible» Need numerical techniques» Multiples solutions may exist

Iterative solution» Start with an initial guess x0

» Algorithm generates x1 from x0

» Repeat to generate sequence x0, x1, x2, …» Assume sequence convergences to solution» Terminate algorithm at iteration N when:

Many iterative algorithms available» Fixed-point iteration, Newton-Raphson method, secant

method

0)( Nxf

Fixed-Point Iteration

Formulation of iterative equation

» A solution of x = g(x) is called a fixed point

Convergence» The iterative process is convergent if the sequence x0, x1, x2, …

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 x0 in J & the limit of the sequence {xn} is s if:

» A function satisfying the theorem is called a contraction mapping:

» K determines the rate of convergence

)()()(0)( 1 nn xgxxgxxfxxf

0)(lim 1 nnn

xx

JxKx

g

1

vxKvgxg )()(

Newton-Raphson Method

Iterative equation derived from first-order Taylor series expansion

Algorithm» Input data: f(x), df(x)/dx, x0, tolerance (), maximum

number of iterations (N)» Given xn, compute xn+1 as:

» Continue until | xn+1-xn| < |xn| or n = N

dxxdf

xfxx

n

nnn )(

)(1

0)()(

)()()()()( 11)(

nn

nnn

x

xxdx

xdfxfxfxx

dx

dfxfxf

Convergence of the Newton-Raphson Method

Order» Provides a measure of convergence rate

» Newton-Raphson method is second-order

Assume f(x) is three times differentiable, its first- and second-order derivatives are non-zero at the solution x = s & x0 is sufficiently close to s, then the Newton method is second-order & exhibit quadratic converge to s

Caveats» The method can converge slowly or even diverge for poorly chosen x0

» The solution obtained can depend on x0

» The method fails if the first-order derivative becomes zero (singularity)

2

1 nnnn cxs

Secant Method

Motivation» Evaluation of df/dx may be computationally expensive» Want efficient, derivative-free method

Derivative approximation

Secant algorithm

Convergence» Superlinear: » Similar to Newton-Raphson (m = 2)

1

1)()()(

nn

nnn

xx

xfxf

dx

xdf

)()()(

1

11

nn

nnnnn xfxf

xxxfxx

211 mcm

nn

Matlab Tutorial

Solution of nonlinear algebraic equations with Matlab

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.

Solution of a single nonlinear algebraic equation:

Write Matlab m-file function, fun.m:

Call fzero from the Matlab command line to find the solution:

Different initial guesses, xo, can give different solutions:

Matlab Tutorial cont.

0604.0)9.0(

1

01.0)3(

1)(

22

xxxf

>> xo = 0;>> fzero('fun',xo)ans = 0.5376

>> fzero('fun',1)ans = 1.2694

>> fzero('fun',4)ans = 3.4015

Nonisothermal Chemical Reactor

Reaction: A B Assumptions

» Pure A in feed» Perfect mixing» Negligible heat losses» Constant properties (, Cp,

H, U)» Constant cooling jacket

temperature (Tj)

Constitutive relations» Reaction rate/volume: r = kcA = k0exp(-E/RT)cA

» Heat transfer rate: Q = UA(Tj-T)

Model Formulation

Mass balance

Component balance

Energy balance

qqqqwwdt

Vdiii

0)(

AAAiA

AAAAiiAAA

CRTEVkCCqdt

dCV

VrMqCMCqMdt

VCMd

)/exp()(

)(

0

)()/exp()()(

)()()()(

0 TTUACRTEVkHTTqCdt

dTVC

QrVHTTwCTTCwTTVCdt

d

jAipp

refprefipirefp

Matlab Exercise

Steady-state model

Parameter values» k0 = 3.493x107 h-1, E = 11843 kcal/kmol

» (-H) = 5960 kcal/kmol, Cp = 500 kcal/m3/K

» UA = 150 kcal/h/K, R = 1.987 kcal/kmol/K

» V = 1 m3, q =1 m3/h,

» CAf = 10 kmol/m3, Tf = 298 K, Tj = 298 K.

Problem» Find the three steady-state points:

0

0

0 ( ) exp( / )

0 ( ) ( ) exp( / ) ( )Af A A

p f A j

q C C Vk E RT C

qC T T H Vk E RT C UA T T

),( TCA

Matlab Tutorial cont.

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)

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 = [CA T] ';» options – Matlab structure of optimization parameter values created with the

optimset function

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 7.26Optimization terminated: first-order optimality is less than options.TolFun.

x =

8.5637 311.1702

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';