1. The function f(x) = 49x³ - 35x² - 48x + 36 has a root of multiplicity 2 near x = 1. a) Can the (i) bisection method, (ii) regula falsi, (iii) standard Newton-Raphson method, (iv) secant method or (v) fixed-point iteration with g(x) = be used to determine = 35x²+48x-36 49x² this root? In each case, without doing any iterations, state why or why not. b) What is the order of iteration in finding this root using (i) the standard Newton-Raphson method, (ii) the Modified Newton-Raphson Method 1 [Xn+1 = xn - m and (iii) the u(xn) u'(xn) where u(x) = f(x)]. f'(x) From all the methods listed in parts (a) and (b), choose the method which you feel converges the fastest to determine the root to at least six significant figures. Prove that the multiplicity of the root found in part (c) is 2. d) c) Modified Newton-Raphson Method 2 [Xn+1 = Xn f(xn)] f'(xn) -
1. The function f(x) = 49x³ - 35x² - 48x + 36 has a root of multiplicity 2 near x = 1. a) Can the (i) bisection method, (ii) regula falsi, (iii) standard Newton-Raphson method, (iv) secant method or (v) fixed-point iteration with g(x) = be used to determine = 35x²+48x-36 49x² this root? In each case, without doing any iterations, state why or why not. b) What is the order of iteration in finding this root using (i) the standard Newton-Raphson method, (ii) the Modified Newton-Raphson Method 1 [Xn+1 = xn - m and (iii) the u(xn) u'(xn) where u(x) = f(x)]. f'(x) From all the methods listed in parts (a) and (b), choose the method which you feel converges the fastest to determine the root to at least six significant figures. Prove that the multiplicity of the root found in part (c) is 2. d) c) Modified Newton-Raphson Method 2 [Xn+1 = Xn f(xn)] f'(xn) -
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
Please only answer part(s) c and d. Do not answer parts a, b, or c
![1. The function f(x) = 49x³ - 35x² - 48x + 36 has a root of multiplicity 2 near x = 1.
a) Can the (i) bisection method, (ii) regula falsi, (iii) standard Newton-Raphson method, (iv)
secant method or (v) fixed-point iteration with g(x):
be used to determine
35x²+48x-36
49x²
this root? In each case, without doing any iterations, state why or why not.
=
b) What is the order of iteration in finding this root using (i) the standard Newton-Raphson
method, (ii) the Modified Newton-Raphson Method 1 [xn+1 = xn-m- f(x) and (iii) the
f'(xn)]
where u(x) = f(x)].
Modified Newton-Raphson Method 2 [Xn+1 = xn
u(xn)
u'(xn)
c)
From all the methods listed in parts (a) and (b), choose the method which you feel converges
the fastest to determine the root to at least six significant figures.
d) Prove that the multiplicity of the root found in part (c) is 2.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc98e05e-a67c-4d1e-9590-57e4be5283ff%2F6ddf6bf0-ddba-4a62-88ab-d6f8d9ec5a02%2Fqvlyij8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. The function f(x) = 49x³ - 35x² - 48x + 36 has a root of multiplicity 2 near x = 1.
a) Can the (i) bisection method, (ii) regula falsi, (iii) standard Newton-Raphson method, (iv)
secant method or (v) fixed-point iteration with g(x):
be used to determine
35x²+48x-36
49x²
this root? In each case, without doing any iterations, state why or why not.
=
b) What is the order of iteration in finding this root using (i) the standard Newton-Raphson
method, (ii) the Modified Newton-Raphson Method 1 [xn+1 = xn-m- f(x) and (iii) the
f'(xn)]
where u(x) = f(x)].
Modified Newton-Raphson Method 2 [Xn+1 = xn
u(xn)
u'(xn)
c)
From all the methods listed in parts (a) and (b), choose the method which you feel converges
the fastest to determine the root to at least six significant figures.
d) Prove that the multiplicity of the root found in part (c) is 2.
-
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