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Use fixed-point iteration and graphical method to determine a root of f(x) = -x² + 1.8x+2.5 using x = 5. Perform the computation until & is less than & = 0.05%

Use fixed-point iteration and graphical method to determine a root of f(x) = -x² + 1.8x+2.5 using x = 5. Perform the computation until & is less than & = 0.05%

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Formulas:

xi+1=g(xi)

   ℇa=|xi+1-xi / xi+1 | 100%

 

Use fixed-point iteration and graphical method to determine a root of f(x) = -x² + 1.8x+2.5 using x = 5. Perform the computation until & is less than & = 0.05%
Transcribed Image Text:Use fixed-point iteration and graphical method to determine a root of f(x) = -x² + 1.8x+2.5 using x = 5. Perform the computation until & is less than & = 0.05%
● Simple Fixed-point Method Example 1: Use simple fixed-point iteration to locate the root of f(x) = e¯x - x ● Solution: The function can be separated directly and expressed in the form of Eqn. I as Xj+1 = e-xi Starting with an initial guess of x = 0, this iterative equation can be applied to compute: &a (%) i O 1 2 3 4 Each iteration brings the estimate 8 closer to the true value of the root: 9 0.56714329. 10 O Xi 1.000000 0.367879 0.692201 0.500473 0.606244 0.545396 0.579612 0.560115 0.571143 0.564879 E.B.P. 100.0 171.8 46.9 38.3 17.4 11.2 5.90 3.48 1.93 1.11 &t (%) 100.0 76.3 35.1 22.1 11.8 6.89 3.83 2.20 1.24 0.705 0.399
Transcribed Image Text:● Simple Fixed-point Method Example 1: Use simple fixed-point iteration to locate the root of f(x) = e¯x - x ● Solution: The function can be separated directly and expressed in the form of Eqn. I as Xj+1 = e-xi Starting with an initial guess of x = 0, this iterative equation can be applied to compute: &a (%) i O 1 2 3 4 Each iteration brings the estimate 8 closer to the true value of the root: 9 0.56714329. 10 O Xi 1.000000 0.367879 0.692201 0.500473 0.606244 0.545396 0.579612 0.560115 0.571143 0.564879 E.B.P. 100.0 171.8 46.9 38.3 17.4 11.2 5.90 3.48 1.93 1.11 &t (%) 100.0 76.3 35.1 22.1 11.8 6.89 3.83 2.20 1.24 0.705 0.399
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