Ch4_5 - 1 Section 4.5 Root Finding II More root finding...

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Next Previous Section 4.5: Root Finding II 11 ® More root finding methods ® Newton’s method ® Very fast way to find roots ® Requires taking the derivative of f(x) ® Can be unstable if ‘unattended’ ® Secant method ® Similar to Newton’s method, ® Approximates the derivative numerically, not analytically l does not require knowing the function ® Usually fast and very stable technique Section 4.5: Root Finding II
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Next Previous Section 4.5: Root Finding II 22 3. Newton's method ® Very fast way to find roots ® Must know the derivative of f(x) ® Can be unstable if 'unattended ' - 15 - 10 - 5 0 5 10 15 - 10 - 5 0 5 10 f(x ) x 1 x 2 x 5 0 x 18 0 x f 2 3 + - - = . . ) (
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Next Previous Section 4.5: Root Finding II 33 -6 -4 -2 0 2 4 6 8 10 12 3 4 5 6 7 f(x ) x 2 x x 54 0 dx x df 2 - - = . ) ( 1 x 2 x 5 0 x 18 0 x f 2 3 + - - = . . ) ( Xold = 5.318  f(xold) = 3.296 df/dx@ xold =7.955 Xnew = 5.318 Newton's method: the process Xnew = 4.839 ( 29 xold old old new dx x df x f x x - = ) ( ( 29 ( 29 xold new old old dx x df x x f x f = - - ) ( = 0 Xold = 4  f(xold) = -3.48 df/dx@ xold = 2.64
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Ch4_5 - 1 Section 4.5 Root Finding II More root finding...

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