Most functions can be rearranged in several ways to give x = which to begin the fixed-point method. For f(x) = eª – 2x² = 0, one g(x) 9(x) with %3D is x =+. 2 (a) Show that this converges to the root near 1.5 if the positive value is used and to the root near -0.5 if the negative is used. (b) There is a third root near 2.6. Show that we do not converge to this root even though values near to the root such as Po = 2.5 or p, = 2.7 are used to begin the iterations. (c) Find another rearrangement that does converge correctly to the third root.
Most functions can be rearranged in several ways to give x = which to begin the fixed-point method. For f(x) = eª – 2x² = 0, one g(x) 9(x) with %3D is x =+. 2 (a) Show that this converges to the root near 1.5 if the positive value is used and to the root near -0.5 if the negative is used. (b) There is a third root near 2.6. Show that we do not converge to this root even though values near to the root such as Po = 2.5 or p, = 2.7 are used to begin the iterations. (c) Find another rearrangement that does converge correctly to the third root.
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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Transcribed Image Text:Most functions can be rearranged in several ways to give x =
g(x) with
which to begin the fixed-point method. For f(x) = e² – 2x² = 0, one g(x)
is
ex
x = ±.
2
(a) Show that this converges to the root near 1.5 if the positive value is
used and to the root near -0.5 if the negative is used.
(b) There is a third root near 2.6. Show that we do not converge to this
root even though values near to the root such as p.
o = 2.5 or p, = 2.7
are used to begin the iterations.
(c) Find another rearrangement that does converge correctly to the third
root.
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