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Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked ã, and a double root, marked x.. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration n+1 = g(n), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn - X*).

Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked ã, and a double root, marked x.. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration n+1 = g(n), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn - X*).

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 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked x, and a double root, marked x. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration Xn+1 = g(en), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn xx).
Transcribed Image Text:Question 2: (a) Give the definition of the double root of a function and sketch the graph of a function with two roots: a simple root, marked x, and a double root, marked x. (b) Find an unseen example of a function, f, with a double root, x. The function f should not be a polynomial but could be any rational or transcendental function. Perform enough iterations of Newton's algorithm, written as a fixed-point iteration Xn+1 = g(en), to infer the order and rate of convergence of the sequence of data produced. Prove this assertion by writing a Taylor series for g about x, in powers of (xn xx).
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