ENG5000 - Iteration

ENG5000 – Advanced Engineering Analysis:Numerical Methods - Iteration - : 

ENG5000 – Advanced Engineering Analysis:Numerical Methods - Iteration - “Mathematics for Graduate Engineers”

Objectives : 

Objectives Background to Numerical Methods Steps in Numerical Methods Iteration Bisection Fixed Point (Substitutive) Iteration Newton (Newton-Raphson) Method Secant Method Modified Newton-Raphson Method Method of False Position (Regula Falsi) Assignment

Background to Numerical Methods : 

Background to Numerical Methods The goal of engineering – exploit, control or manipulate the characteristics/behaviours of physical systems to suite our own objectives. Characteristics/Behaviours subject to fundamental mathematical and physical laws. Possibly subject to non-physical laws(social, spiritual) Engineers describe char/behav and theire interrelatedness as “Relationships”

Background to Numerical Methods(cont’d) : 

Background to Numerical Methods(cont’d) “Relationships” are codified as Functions Equations (Mathematical) Systems of equations Prediction of characteristics/behaviours called “Solving an equation/system of equations”

Solving Engineering Problems : 

Solving Engineering Problems Analytical Solution Methods:Manipulation of algebraic or other mathematical expressions into a form that expresses the desired answer. Numeric(al) Methods (NM):Preforming numeric calculations in a careful sequence on a computer or calculator until the desired answer is arrived at. All solution methods are neither strictly one or other; rather a combination of both approaches.

Steps of Solving Using NM : 

Steps of Solving Using NM 1. Interpretation:Review of the problem and identification of phenomena governing the system of concern. 2. Modelling:Expression of the fundamental laws governing system in the form of mathematical expressions.May be one or multiple expressions in various forms (not exclusive): Algebraic Differential Integration Summation/Series Vector/Matrix

Steps of Solving Using NM (cont’d) : 

Steps of Solving Using NM (cont’d) 3. Choose Numerical Method/Strategy:Review of the problem and identification of phenomena governing the system of concern. 4. Execute Numerical Sequence: Manually – e.g. with a calculator Automatically – using computer programmingC++/Fortran – pure programming languagesMatlab/Octave – psuedo programming for EngineersMathematica/Maple/Excel 5. Run Programme/SoftwareIncludes the process of debugging 6. InterpretationTranslate numeric output of programme into terminology relevant to the original problem

Iteration : 

Iteration

Problems with Iteration : 

Problems with Iteration Need good initial guess Criteria for reasonable initial guess might help Can diverge from answer Even if answer exists, sometimes iteration process drifts from answer Cannot confirm existence of an answer Sometimes there is no true answer. Iteration process look for what doesn’t exist Cannot confirm existence of many answers Sometimes answer found is not the best answer or the full answer.

Common Iteration Strategies : 

Common Iteration Strategies Bisection Fixed-Point (Substitutive Iteration) Newton (Newton-Raphson) Secant Modified Newton-Raphson False Position (Reguli Falsi)

Bisection : 

Bisection

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Execute bisection procedure Programming xn = 0; xp = 10; xm = 0.5*(xn+xp); fxn = xn^2 – 2; fxp = xp^2 – 2; xerr = 10; n=0; while xerr > 0.001 fxm = xm^2 – 2; if fxm > 0 xp = xm; else xn = xm; end xm = 0.5*(xn+xp); xerr = abs((xp – xn)/xm); n = n+1; end xm n

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Execute bisection procedure Excel

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Execute bisection procedure Programming: Incorporating “function” files

Properties of Bisection : 

Properties of Bisection Finding initial guesses (+ve and –ve) might be challenging. For continuous functions, once initial guess is determined, finding a solution is certain. Convergence can be slow Discontinuous functions can pose problems.

Fixed-point/Substitutive Iteration : 

Fixed-point/Substitutive Iteration

Fixed-point Iteration (FPI) Example : 

Fixed-point Iteration (FPI) Example

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Execute FPI procedure Excel

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Execute FPI procedure Programming

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Execute FPI procedure Programming. Do we get the same result when programming?

Theorem of Convergence for FPI : 

Theorem of Convergence for FPI

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