Nonlinear Algebraic Systems

1
Nonlinear Algebraic Systems

• Iterative solution methods
• Fixed-point iteration
• Newton-Raphson method
• Secant method
• Matlab tutorial
• Matlab exercise

2
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

3
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

4
Newton-Raphson Method

• Iterative equation derived from first-order
  Taylor series expansion
• Algorithm
• Input data f(x), df(x)/dx, x0, tolerance (d),
  maximum number of iterations (N)
• Given xn, compute xn1 as
• Continue until xn1-xn lt dxn or n N

5
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)

6
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)

7
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.

8
Matlab Tutorial cont.

• 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

gtgt xo 0 gtgt fzero('fun',xo) ans 0.5376
gtgt fzero('fun',4) ans 3.4015
gtgt fzero('fun',1) ans 1.2694
9
Nonisothermal Chemical Reactor

• Reaction A ? B
• Assumptions
• Pure A in feed
• Perfect mixing
• Negligible heat losses
• Constant properties (r, Cp, DH, U)
• Constant cooling jacket temperature (Tj)

• Constitutive relations
• Reaction rate/volume r kcA k0exp(-E/RT)cA
• Heat transfer rate Q UA(Tj-T)

10
Model Formulation

• Mass balance
• Component balance
• Energy balance

11
Matlab Exercise

• Steady-state model
• Parameter values
• k0 3.493x107 h-1, E 11843 kcal/kmol
• (-DH) 5960 kcal/kmol, rCp 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

12
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)

13
Matlab Exercise Solution with fsolve

• 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

gtgt xo 10 300' gtgt x fsolve('cstr',xo,optimse
t('Display','iter'))
Norm of First-order
Trust-region Iteration Func-count f(x)
step optimality radius 0
3 1.29531e007
1.76e006 1 1 6
8.99169e006 1 1.52e006
1 2 9 1.91379e006
2.5 7.71e005 2.5 3
12 574729 6.25
6.2e005 6.25 4 15
5605.19 2.90576 7.34e004
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) -
Vkoexp(-E/R/T)Ca f(2) rhoCpq(Tf - T)
-HVkoexp(-E/R/T)Ca UA(Tj-T) ff'