6_Newton-Raphson_method.pdf

17 figure 5 estimate of the root for the first

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17 Figure 5 Estimate of the root for the first iteration.

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Example 1 Cont. % 90 . 19 100 06242 . 0 05 . 0 06242 . 0 100 1 0 1 x x x a 18 The absolute relative approximate error at the end of Iteration 1 is a The number of significant digits at least correct is 0, as you need an absolute relative approximate error of 5% or less for at least one significant digits to be correct in your result.
Example 1 Cont. 06238 . 0 10 4646 . 4 06242 . 0 10 90973 . 8 10 97781 . 3 06242 . 0 06242 . 0 33 . 0 06242 . 0 3 10 .993 3 06242 . 0 165 . 0 06242 . 0 06242 . 0 ' 5 3 7 2 4 2 3 1 1 1 2 x f x f x x 19 Iteration 2 The estimate of the root is

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Example 1 Cont. 20 Figure 6 Estimate of the root for the Iteration 2.
Example 1 Cont. % 0716 . 0 100 06238 . 0 06242 . 0 06238 . 0 100 2 1 2 x x x a 21 The absolute relative approximate error at the end of Iteration 2 is a The maximum value of m for which is 2.844. Hence, the number of significant digits at least correct in the answer is 2. m a 2 10 5 . 0

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Example 1 Cont. 06238 . 0 10 9822 . 4 06238 . 0 10 91171 . 8 10 44 . 4 06238 . 0 06238 . 0 33 . 0 06238 . 0 3 10 .993 3 06238 . 0 165 . 0 06238 . 0 06238 . 0 ' 9 3 11 2 4 2 3 2 2 2 3 x f x f x x 22 Iteration 3 The estimate of the root is
Example 1 Cont. 23 Figure 7 Estimate of the root for the Iteration 3.

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Example 1 Cont. % 0 100 06238 . 0 06238 . 0 06238 . 0 100 2 1 2 x x x a 24 The absolute relative approximate error at the end of Iteration 3 is a The number of significant digits at least correct is 4, as only 4 significant digits are carried through all the calculations.
Advantages and Drawbacks of Newton Raphson Method 25

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Advantages Converges fast (quadratic convergence), if it converges. Requires only one guess 26
Drawbacks 27 1. Divergence at inflection points Selection of the initial guess or an iteration value of the root that is close to the inflection point of the function may start diverging away from the root in ther Newton-Raphson method.

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