To find the unique solution p ∗ ∈ [0, 1] of the equation x^3 + 9x − 4 = 0, you can try to rewrite the equation in the fixed-point form x = g(x). You are to choose the function g such that the sequence {pn} from the fixed-point iteration pn = g(pn−1) will converge to p^∗ when p0 is sufficiently close to p^∗(but not equal to p^∗). Give three different choices of g, say g1, g2, and g3, such that g1 wouldn’t work, and g2 and g3 both work. Moreover, g3 will give faster convergence than g2.
To find the unique solution p ∗ ∈ [0, 1] of the equation x^3 + 9x − 4 = 0, you can try to rewrite the equation in the fixed-point form x = g(x). You are to choose the function g such that the sequence {pn} from the fixed-point iteration pn = g(pn−1) will converge to p^∗ when p0 is sufficiently close to p^∗(but not equal to p^∗). Give three different choices of g, say g1, g2, and g3, such that g1 wouldn’t work, and g2 and g3 both work. Moreover, g3 will give faster convergence than g2.
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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To find the unique solution p ∗ ∈ [0, 1] of the equation x^3 + 9x − 4 = 0, you can try to rewrite the equation in the fixed-point form x = g(x). You are to choose the function g such that the sequence {pn} from the fixed-point iteration pn = g(pn−1) will converge to p^∗ when p0 is sufficiently close to p^∗(but not equal to p^∗). Give three different choices of g, say g1, g2, and g3, such that g1 wouldn’t work, and g2 and g3 both work. Moreover, g3 will give faster convergence than g2.
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