Lecture 17

# Lecture 17 - Newtons Method Need to compute f(x You may...

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Newton’s Method Need to compute f (x) You may have to do this numerically Open Method – Convergence depends on the initial guess (not guaranteed) However, Newton method can converge

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Newton’s Method Example : f (x) = 3x + sin(x) - exp(x) Solution : derivative f ' (x) = 3 + cos(x) – exp(x) start with x 0 = 0 f ( 0 ) = 0 + 0 -1 = -1 f ' ( 0 ) = 3 +1-1 = 3 x 1 = x 0 - f 0 / f 0 ' = 0 - (-1)/3 = 0.33333 f (0.33333) = -0.068417728 &
Newton’s Method Example : f(x) = 3x + sin(x) - exp(x) Iter. # x n f(x n ) e n =x n -r 1 0.333333333 -6.84E-02 2.71E-02 2 0.360170714 -6.28E-04 2.52E-04 0.346 3 0.360421681 -5.63E-08 2.25E-08 0.354 4 0.360421703 -6.66E-16 e n /e 2 n-1 Comments: * It may not converge if x 0 is too far from x = r. * If x 0 is close to r, Newton's method converges very fast. x 0 = 0

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Newton’s Method --Matlab code >> newtraph(@(x) 3*x+sin(x)-exp(x), @(x) 3+cos(x)-exp(x),1) ans = 0.360421702960200
Newton’s Method Quadratic convergence: = constant e n /e 2 n-1 Illustration of quadratic convergence: 1.E-11 1.E-09 1.E-07 1.E-05 1.E-03 1.E-01 1.E+01 1 10 |r-xn| n parabola linear

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Newton-Raphson Method Examples of poor convergence or failure
Secant Method Use secant line instead of tangent line at f ( x i )

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Secant Method The formula for the secant method is Notice that this is very similar to the false
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Lecture 17 - Newtons Method Need to compute f(x You may...

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