06-Open Methods

06-Open Methods - 6 Roots of Equations Open Methods Chapter...

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6: Roots of Equations: 6: Roots of Equations: Open Methods Open Methods
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives Recognize difference between bracketing and open methods. Know how to use Newton-Raphson method. Know how to use secant and modified secant method. Know how to use MATLAB's fzero() function. Know how to determine roots of polynomials with MATLAB.
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 3 Bracketing vs Open Methods bracketing needs 2 start pts on either side of a root bracketing methods are convergent each iteration improves estimate open methods start with 1 or 2 pts not necessarily a bracket E.G. Newton-Raphson, in (b) and (c) use 1 pt + slope to predict root open methods often converge faster methods may diverge estimates worsen, as in (b) Figure 6.1
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 4 6.2: Newton-Raphson Given one estimate: tangent extrapolated to x -axis: Figure 6.4 x i , f x i , f ' x i f ' x i  = f x i − 0 x i x i 1 x i x i 1 = f x i f ' x i x i 1 = x i f x i f ' x i (6.6)
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 5 Example 6.2 Find a root of: First derivative: Starting value: Iteration 1: f x  = e x x f ' x  =− e x 1 x 0 = 0 f x 0  = f 0  = e 0 0 = 1 f ' x 0  = f ' 0  =− e 0 1 =− 2 x 1 = x 0 f 0 f ' 0 = 0 1 2 = 0.5 a = x 1 x 0 x 1 × 100 % = 0.5 0 0.5 × 100 % = 100 %
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 6 Example 6.2 (contd.) Iteration 2: a = x 2 x 1 x 2 × 100 % = 0.566311 0.5 0.566311 × 100 % = 11.7 % f x 1  = f 0.5  = e 0.5 0.5 = 0.106531 f ' x 1  = f ' 0.5  =− e 0.5 1 =− 1.606531 x 2 = x 1 f x 1 f ' x 1 = 0.5 0.106531 1.606531 = 0.566311 i 0 0.000000 1.000000 -2.000000 1 0.500000 0.106531 -1.606531 100 2 0.566311 0.001305 -1.567616 11.7 3 0.567143 0.000000 -0.567143 0.15 4 0.567143 2.20E-005 x i f(x i ) f''(x i ) a |,% Summary:
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 7 Example 6.2 (contd.) Observations: convergence is rapid from Chapra and Canale: error is proportional to square of previous error when previous E is small, new E is very small this is quadratic convergence. E t ,i 1 = f ' ' x r 2 f ' x r E t ,i 2 (6.7)
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 8 Newton-Raphson Newton-Raphson can have slow convergence: Figure 6.3
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 9 Newton-Raphson Newton-Raphson can: diverge : oscillate: miss roots: Figure 6.6
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 10 Matlab (1) function root = newtraph(func,dfunc,xr,es,maxit) % root = newtraph(func,dfunc,xguess,es,maxit): % uses Newton-Raphson method to find the root of a function % input: % func = function % dfunc = derivative of function % xguess = initial guess % es = (optional) stopping criterion (%) % maxit = (optional) maximum allowable iterations % output: % root = real root % if necessary, assign default values if nargin<5, maxit=50; end %if maxit blank set to 50 if nargin<4, es=0.001; end %if es blank set to 0.001
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Oct 13, 2006 16:32 ECOR2606 -- Hassan & Holtz 11 Matlab (2) % Newton-Raphson iter = 0; while (1) xrold = xr; xr = xr - func(xr)/dfunc(xr);
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