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# lect_7 - Introduction to Numerical Analysis for Engineers...

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Numerical Methods for Engineers 13.002 Lecture 7 Introduction to Numerical Analysis for Engineers Roots of Non-linear Equations 2.1-2.4 – Heron’s formula – Stop criteria – General method 2.1-2.3 • Convergence • Examples – Newton-Raphson’s Method 2.4 • Convergence Speed • Examples – Secant Method 2.4 • Convergence and efficiency • Examples – Multiple roots 2.4 – Bisection 2.2 Mathews

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Numerical Methods for Engineers 13.002 Lecture 7 Roots of Nonlinear Equations Example – Square root Heron’s Principle a=2; n=6; g=2; % Number of Digits dig=5; sq(1)=g; for i=2:n sq(i)= 0.5*radd(sq(i-1),a/sq(i-1),dig); end ' i value ' [ [1:n]' sq'] hold off plot([0 n],[sqrt(a) sqrt(a)],'b') hold on plot(sq,'r') plot(a./sq,'r-.') plot((sq-sqrt(a))/sqrt(a),'g') grid on heron.m i value 1.0000 2.0000 2.0000 1.5000 3.0000 1.4167 4.0000 1.4143 5.0000 1.4143 6.0000 1.4143 Guess root Mean is better guess Iteration Formula ( )/2 ( )/2
Numerical Methods for Engineers 13.002 Lecture 7 Roots of Nonlinear Equations Stop-criteria Unrealistic stop-criteria Realistic stop-criteria Machine Accuracy x f(x) ‘flat’ f(x) G x f(x) ‘steep’ f(x) Cannot require Cannot require Use combination of the two criteria

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Numerical Methods for Engineers 13.002 Lecture 7 Roots of Nonlinear Equations General Method Non-linear Equation Goal: Converging series Rewrite Problem Example Iteration
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lect_7 - Introduction to Numerical Analysis for Engineers...

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