Non-linear equations, Lecture notes of Mathematics

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Applied Numerical Mathematics

Non-Linear Equations

Chemical Engineering Department / Transport Phenomena Luis M. Portela [email protected]

Week Overview

• Linear & Non-Linear Behaviour
  • Formulation
  • Approaches
• Single Non-Linear Equation
  • Bracketing Methods
  • Open Methods
• System of Non-Linear Equations
  • Newton-Raphson
  • Fixed-Point Iteration

System Behavior

Forcing (^) System Response

Parameters: characterization of the system

Response: dependent variables=f (independent variables; forcing)

Linear & Non-Linear Behavior

• f can be linear or non-linear
• Sources of non-linearity:
  • Non-linear parameter-function
  • Forcing <-> Parameters
• Linearized behavior:
  • Small differences
  • Near-equilibrium behav

Roots & Extrema

2 H x ( )  f ( )x f ( )^ x  0; f ( )x  0  f ( ) is a extremumx H x( ) is a extremum  f ( )x  0

Multiple Solutions

Bracketing & Open Methods

Bracketing Open

Single Non-Linear Equation

Bisection Method basic idea: f ( xa )  f ( xb )  0 half the interval (^)  xa ,xb

Error Estimate & Convergence

error:

k k ex   x  x 1 2 k (^) k b a   x  x 1

k k

u

u



uncertainty: 2 2 k k k ex k u u x   x  x  convergence criterion: k u  

Bracketing Methods Evaluation

• Robust
• Slow convergence: does not make good use of the function
  • Bisection: jump function interpolation
  • Regular Falsi: linear interpolation
• Other possibilities: quadratic interpolation, …
• Not practical for system of equations

17 Open Methods basic idea: use the function to approach the root bisection open method

Convergence Criterion 1 : root; ; ( ) i i i i X e x X e g x X     

i i

g x  g X  g  X  x  X

( ) ( ) ( ) i i g x  X  g  X  x  X 1 ( ) i i e g X e     convergence:^ g^ (^ X)^ ^1

Newton-Raphson Method basic idea: use the derivative to approach the root

f x

g x x f x

f x